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The past months have seen some exciting progress on diameter bounds for polytopes and polytopal complexes, both in the negative and in the positive direction.  Jesus de Loera and Steve Klee described simplicial polytopes which are not  weakly vertex decomposable and the existence of non weakly k-vertex decomposable polytopes for k up to about  was proved  by Hähnle, Klee, and Pilaud in the paper  Obstructions to weak decomposability for simplicial polytopes. In this post I want to outline a generalization of a beautiful result of Billera and Provan in support of the Hirsch conjecture.

I will consider the simplicial version of the Hirsch conjecture, dual to the classic formulation of Hirsch conjecture. Furthermore, I will consider the Hirsch conjecture, and the non-revisiting path conjecture, for general simplicial complexes, as opposed to the classical formulation for polytopes.

Theorem [Billera & Provan `79] The barycentric subdivision of a shellable simplicial complex satisfies the Hirsch Conjecture.

The barycentric subdivision of a shellable complex is vertex decomposable. The Hirsch diameter bound for vertex decomposable complexes, in turn, can be proven easily by induction.

This is particularly interesting since polytopes, the objects for which the Hirsch conjecture was originally formulated, are shellable. So while in general polytopes do not satisfy the Hirsch conjecture, their barycentric subdivisions always do! That was a great news!

Shellability is a strong combinatorial property that enables us to decompose a complex nicely, so it does not come as a surprise that it can be used to give some diameter bounds on complexes. Suprisingly, however, shellability is not needed at all!  And neither is the barycentric subdivision!

A simplicial complex Σ is called flag if it is the clique complex of its 1-skeleton. It is called normal if it is pure and for every face F of Σ of codimension two or more, Lk(F,Σ) is connected.

Theorem (Adiprasito and Benedetti):  Any flag and normal simplicial complex Σ satisfies the non-revisiting path conjecture and, in particular, it satisfies the Hirsch conjecture.

This generalizes the Billera–Provan result in three ways:

– The barycentric subdivision of a simplicial complex is flag, but not all flag complexes are obtained by barycentric subdivisions.

– Shellability imposes strong topological and combinatorial restrictions on a complex; A shellable complex is always homotopy equivalent to a wedge of spheres of the same dimension, and even if a pure complex is topologically nice (if, for example, it is a PL ball) it may not be shellable, as classic examples of Goodrick, Lickorish and Rudin show. Being normal still poses a restriction, but include a far wider class of complexes. For example, every triangulation of a (connected) manifold is normal, and so are all homology manifolds.

– Instead of proving the Hirsch conjecture, we can actually obtain the stronger conclusion that the complex satisfies the non-revisiting path conjecture, which for a given complex is stronger than the Hirsch conjecture.

A geometric proof of our theorem appeared in a recent paper “Metric geometry and collapsibility”  with Bruno Benedetti. . I will give here a short combinatorial proof.

Construction of a combinatorial segment

Preliminaries

Lk(F,Σ) shall denote the link of a face F of Σ, and St(F,Σ) shall denote the star of F in Σ. Let denote the distance between two vertices in the 1-skeleton of Σ. Let denote the distance between two vertex sets S, T in Σ. Let denote the pairs of points in S, T that realize the distance , and let resp. denote the set of vertices of T realizing the distance resp. the set of points y in Lk(x,Σ) with the property that . A vertex path shall mean a path in the 1-skeleton of Σ, and facet path is short for facet-ridge path.

Part 1: From a facet X to a vertex set S.

We construct a facet path from a facet X of Σ to a subset S of the vertex set of Σ, i.e. a facet path from X to a facet intersecting S, with the property that S is intersected by the path Γ only in the last facet of the path.

If Σ is 1-dimensional, choose a shortest vertex path realizing the distance . The edges in that path, including X, give the desired facet path.

If Σ is of a dimension d larger than 1, set , and proceed as follows:

1. If intersects S, stop the algorithm. If not, proceed to step 2.

2. Let be any vertex of that minimizes the distance to S. Set . Using the construction for dimension d-1, we can construct a facet path in  from the facet to the vertex set . By considering the join of the elements of that path with , we obtain a facet path from to the vertex set . Call the last facet of the path , and the vertex of it intersects .

Repeat the procedure with instead of , instead of , and instead of . The process stops once the facet path constructed intersects S.

Part 2: From a facet X to another facet Y.

Using Part 1., construct a facet path from X to a vertex z of the vertex set of Y, and let Z denote the last facet of the path.

If Σ is of dimension 1, complete the path to a facet path from X to Y by adding the facet Y to the path.

If Σ is of dimension d greater than 1, apply the (d-1)-dimensional construction to construct a facet path in Lk(z,Σ) from Lk(z,Z) to Lk(z,Y), and lift this to a facet path in Σ by joining the elements of the path with .

This finishes the construction. We call the facet paths constructed combinatorial segments.

The combinatorial segment is non-revisiting.

We start off with some simple observations and notions for combinatorial segments:

1. A combinatorial segment Γ comes with a path (see Part 1. of the construction). This is a shortest path in the 1-skeleton, realizing the distance , called the thread t of the combinatorial segment Γ.

2. Every facet of the combinatorial segment Γ is associated to a vertex of the thread like this: F intersects the thread t, and there is a unique i such that F contains in t, but not . Call  the pearl of F in t.

We consider a combinatorial segment and its thread with the natural order from X to S resp. from X to Y.

Lemma S: If F is a facet of Γ, where F has pearl $latex x_i$ in t, and v is a vertex of Γ s.t. F and lie in St(v,Σ), then the first facet G of Γ whose pearl is is a facet of St(v,Σ) as well, and the part of the combinatorial segment from F to G lies in St(v,Σ).

Proof: The lemma is clear if v is in t (i.e. v coincides with or ). To see the case v not in t, we can use induction on the dimension of Σ:

For 1-dimensional complexes, this is again clear.

If Σ is of dimension d larger than 1, consider the (d-1)-complex . is a facet of the combinatorial segment . Since the complex contains and since Σ is flag, we obtain that St(v,Σ’) contains . Furthermore, F’ is clearly contained in St(v,Σ’). Thus, by induction assumption, the portion of from F’ to the first facet G’ of Γ’ containing is contained in St(v,Σ’). Since the combinatorial segment in the relevant part from to is obtained from  by join with (i.e. ), we have the desired statement. This finishes the proof of the Lemma.

This suffices to prove that a combinatorial segment Γ must be non-revisiting:

Proof of the Theorem: Consider a combinatorial segment Γ that connects a facet X with a facet Y of Σ. Let A, B be any two facets of Γ, with pearls in t respectively that both lie in the star of a vertex of v in Σ. Then the part of Γ from A to B (B coming, w.l.o.g., after A in Γ) lies in the star St(v,Σ) of v entirely.To see this, there are two cases to consider:

If i=j This case follows directly from Lemma S, since B is somewhere between A and the first facet G of Γ to be associated with , so , as desired.

A system of linear inequalities defines a polytope as a feasible region. The simplex algorithm begins at a starting vertex and moves along the edges of the polytope until it reaches the vertex of the optimum solution. (Photo credit: Wikipedia)

Inspired by an article in the New Scientist I am returning to a favourite subject – whether P = NP and what the implications would be in the (unlikely) case that this were so.

Here’s a crude but quick explanation of P and NP: P problems are those that can be solve in a known time based on a polynomial (hence P) of the problem’s complexity – ie., we know in advance how to solve the problem. NP (N standing for non-deterministic) problems are those for which we can quickly (ie in P) verify that a solution is correct but for which we don’t have an algorithmic solution to hand – in other words we have to try all the possible algorithmic solutions in the hope of hitting the right one. Reversing one-way functions (used to encrypt internet commerce) is an NP problem – hence, it is thought/hoped that internet commerce is secure. On the other hand drawing up a school timetable is also an NP problem so solving that would be a bonus. There are a set of problems, known as NP-complete, which if any one was shown to be, in reality a P problem would mean that P = NP – in other words there would be no NP problems as such (we are ignoring NP-hard problems).

If it was shown we lived in a world where P=NP then we would inhabit ‘algorithmica’ – a land where computers could solve complex problems with, it is said, relative ease.

But what if, actually, we have polynomial solutions to P class problems but there were too complex to be of much use? The New Scientist article – which examines the theoretical problems faced by users of the ‘simplex algorithm’ points to just such a case.

The simplex algorithm aims to optimise a multiple variable problem using linear programming – as in an example they suggest, how do you get bananas from 5 distribution centres with varying numbers of supplies to 200 shops with varying levels of demand – a 1000 dimensional problem.

The simplex algorithm involves seeking the optimal vertex in the geometrical representation of this problem. This was thought to be rendered as a problem in P via the ‘Hirsch conjecture‘ – that the maximum number of edges we must traverse to get between any two corners on a polyhedron is never greater than the number of faces of the polyhedron minus the number of dimensions in the problem.

While this is true in the three dimensional world a paper presented in 2010 and published last month in the Annals of Mathematics – A counterexample to the Hirsch Conjecture by Francisco Santos has knocked down its universal applicability. Santos found a 43 dimensional shape with 86 faces. If the Hirsch conjecture was valid then the maximum distance between two corners would be 43 steps, but he found a pair at least 44 steps apart.

That leaves another limit – devised by Gil Kalai of the Hebrew University of Jerusalem and Daniel Kleitman of MIT, but this, says the New Scientist is “too big, in fact, to guarantee a reasonable running time for the simplex method“. Their two page paper can be read here. They suggest the diameter (maximal number of steps) is where is the number of faces and the dimensions. (The Hirsch conjecture is instead .)

So for Santos’s shape we would have a maximal diameter of (this is the upper limit, rather than the actual diameter). A much bigger figure even for a small dimensional problem, the paper also refers to a linear programming method that would require, in this case, a maximum of steps. Not a practical proposition if the dimension count starts to rise. (NB I am not suggesting these are the real limits for Santos’s shape, I am merely using the figures as an illustration of the many orders of magnitude difference they suggest might apply).

I think these figures suggest that proving P = NP might not be enough even if it were possible. We might have algorithms in P, but the time required would be such that quicker, if somewhat less accurate, approximations (as often used today) would still be preferred.

Caveat: Some/much of the above is outside my maths comfort zone, so if you spot an error shout it out.

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there is a polynomial upper bound for the diameter of graphs of -dimensional polytopes with facets.

Here, I would like to mention a topological approach which follows a result that was discovered independently by Tamon Stephen and Hugh Thomas in their paper An Euler characteristic proof that 4-prismatoids have width at most 4,
and by Paco Santos in his paper Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids. This post is based on a discussion with Paco Santos at Oberwolfach.

Two maps on a two dimensional Sphere

Theorem: Given a red map and a blue map drawn in general position on there is an intersection point of two edges of different colors which is adjacent (in the refined map) to a red vertex and to a blue vertex.

Blue and black maps

There are two proofs for the theorem. The proof by Stephen and Thomas uses an Euler characteristic argument. The proof by Santos applies a connectivity argument. Both papers are short and elegant. Both papers point out that the result does not hold for maps on a torus.

Santos’ counterexample to the Hirsch conjecture is based on showing that a direct extension of this result to maps in three dimensions fails. (Even for two maps coming from fans based on polytopes.) Of course, first Paco found his counterexample and then the two-map theorem was found in response to his question  of whether one can find in dimension four counterexamples of the kind he presented in dimension five.

The theorem by Santos, Stephen, and Thomas is very elegant. The proofs are simple but far from obvious and it seems to me that the result will find interesting applications. Its elegance and depth reminded me of Anton Klyachko’s car crash theorem.

A topological question in high dimensions

Now we are ready to present a higher-dimensional analog:

Tentative Conjecture: Let be a red map and let  be a blue map drawn in general position on , and let $M$ be their common refinement.  There is a vertex of a blue vertex , a red vertex and two faces such that 1) , 2) , and 3) .

A simple (but perhaps not the most general) setting in which to ask this question is with regard to the red and blue maps  coming from red and blue polyhedral fans associated to red and blue convex polytopes. The common refinement will be the fan obtained by taking all intersections of cones, one from the first fan and one from the second.

(Perhaps when we can even guarantee that .)

Why the tentative conjecture implies that the diameter is polynomial

An affirmative answer to this conjecture will lead to a bound of the form for the graph of -polytopes with facets.

Here is why:

- It is known that the diameter of every polytope with facets and dimension is bounded above by the “length” of a Dantzig figure with facets and vertices.

Here a Dantzig figure is a simple polytope of dimension with facets and two complementary vertices. (i.e., two vertices such that each vertex lies in half of the facets, and they do not belong to any common facet).

The length of the Dantzig figure is the graph distance between these two vertices. This is the classical “d-step theorem” of Klee and Walkup, 1967.

- The length of a Dantzig figure of dimension is the same as the minimum distance between blue and red vertices in a pair of two maps in the -sphere, with  cells each.

- Our tentative conjecture implies, by dimension on , that the minimum distance between blue and red vertices in a pair of maps in the -sphere and with cells is bounded above by (essentially) . ( cells means “cells of the blue map plus cells of the red map”, not “cells of the common refinement”).

The abstract setting and other approaches

More comments, ideas, and updates on the abstract setting are of course very welcome.  Also very welcome are other approaches to the polynomial Hirsch conjecture, and discussion of related problems.

An example showing that the theorem fail for blue and red maps on a torus.

The 4th research thread was, in my opinion, fruitful. An interesting further abstraction was offered and for this abstraction a counterexample was found. I will review these developments below.

There are several reasons why the positive direction is more tempting than the negative one. (And as usual, it does not make much of a difference which direction you study. The practices for trying to prove a statement and trying to disprove it are quite similar.) But perhaps we should try to make also some more pointed attempts towards counterexamples?

Over the years, I devoted much effort including a few desperate attempts to try to come up with counterexamples. (For a slightly less abstract version than that of EHRR.) I tried to base one on the Towers of Hanoi game. One can translate the positions of the game into a graph labelled by subsets. But the diameter is exponential! So maybe there is a way to change the “ground set”? I did not find any. I even tried to look at games (in game stores!) where the player is required to move from one position to another to see if this leads to an interesting abstract example. These were, while romantic, very long shots.

Two more things: First, I enjoyed meeting in Lausanne for the first time Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss. (EHR of EHRR.) Second, Oliver Friedmann, Thomas Dueholm Hansen, and Uri Zwick proved (mildly) subexponential lower bounds for certain randomized pivot steps for the simplex algorithm. We discussed it in this post.  The underlying polytopes in their examples are combinatorial cubes. So this has no direct bearing on our problem. (But it is interesting to see if geometric or abstract examples coming from more general games of the type they consider may be relevant.)

So let me summarize PHC4 excitements and, as usual, if I missed something please add it.

The original abstract problem

Consider t disjoint families of subsets of {1,2,…,n}, .

Suppose that

(*) For every , and every and , there is which contains .

The basic question is: How large can t  be???

We denote the answer by . If we consider only sets of size then we denote the answer by . We add superscript * when we refer to monomials (multisets) rather than sets.

Yuri’s abstraction

Here we want to abstract properties of sequences of sets which are the “support” of such families.

Let’s look at “legal sequences” of subsets of defined inductively as follows:

0. The only legal sequence on elements is .

1. Any legal sequence on elements is also a legal sequence on elements.

2. A legal sequence must be convex, namely for .

3. A sequence on elements is legal if and only if
3a) every proper subsequence is legal (there are two possible versions of this rule: the less restrictive one only requires that intervals are legal, the more restrictive – that all subsequences are. The difference can be demonstrated by the sequence which is legal in the former sense but not the latter)

and

3b) if an element belongs to every then there are subsets such that is a legal sequence on elements.

We denote by the length of the largest legal sequence of subsets of  .

Paco’s example

(Quoting from his comment.)

Hello everyone,

I am afraid I can show that , which implies a super-polynomial lower bound. The exact inequalities I prove, which eventually give the one above, are:

,
,
,
,
, …

… and so on.

For the first one, we simply observe that the sequence with copies of [n+1] is valid on elements, and use two blocks of it to show . Since this “blocks” idea is crucial to the whole proof, let me formalize it a bit. I consider my set of symbols as consisting of two parts and of size , and my sequence is , with a first block of ‘s of length and a second block of ‘s of the same length.

Now, I increase my set of symbols by two, putting one in and one in . Then I can construct a valid sequence with *three* blocks of length each: a first block of ‘s, a second block of ‘s and a third block of ‘s.

But if I put one more symbol to and to , so that I now have in total, I can build a valid sequence with *four* blocks of length : a first block of ‘s, a second and third blocks of ‘s and a fourth block of ‘s.

And so on…

This gives at least – quite close to the upper bound.

Commutativity(?)

Concluding this part of the discussion Paco said: ”One thing we learned is that we can model by Yury’s axioms together with commutativity of the restrictions.”

Actually, I don’t understand this commutativity so well, so I will be happy if some participants will clarify it further.

“Another thing is that keeping track only of the intervals when individual elements are active will not be enough to prove polynomiality of .” So lets look at pairs of elements etc.

Adding another parameter and abstraction for the m-shadow

Suppose we would like to abstract not the support of our families but rather the shadow on sets of size . Let’s try to adapt Yuri’s axioms for the -shadows.

We would like to define legal sequences of families of sets of size . All these sets are from a ground set of elements.

0. The only legal sequence for n=0 is of length 1 and has the empty set as the only member of the family.

1. Any legal sequence on a ground set of   elements is also a legal sequence on a larger ground set of  elements.

2. A legal sequence  must be convex, namely for for every and there is such that .

3. A sequence of families of -sets  on elements is legal if and only if

3a) every proper subsequence is legal

and

3b) if an element belongs to the union of sets in every  then there are legal sequences of families   on the ground set   of  elements, such that is a family of -subsets of whose -shadow is included in the set of all subsets so that adding to them gives us a set in . Update: Also should be a subset of . 

Lets denote by the maximum length of a legal sequence of families of sets of size The case is especially simple. We can start with this case.

In this case we have a convex sequence of  families of pairs. Let me repeat what 4b) says in this case. Suppose that is supported by all families in the sequence. then you can have  new legal sequences of families of pairs from  such that if is a pair in then both and are pairs in the original family .

What I propose to discuss in this research thread.

1) Any ideas how to find good upper bounds for which will exploit the extra structure that we cannot use for ?

2) What about counterexamples? Can we find superpolynomial examples for for ? For fixed? Perhaps we can use Paco’s example as a base?

3) Other ideas, no matter how desparate, for counterexamples?

This post has three parts. 1) Around Nicolai’s conjecture; 1) Improving the upper bounds based on the original method; 3) How to find super-polynomial constructions?

1) Around Nicolai’s conjecture

Proving Nicolai’s conjecture

Nicolai conjectured that and this bound, if correct, is sharp as seen by several examples. Trying to prove this conjecture is still, I feel, the most tempting direction in our project. The conjecture is as elegant as Hirsch ‘s conjecture itself.

Some role models: I remember hard conjectures that were proved by amazingly simple arguments, like in Adam Marcus’s and Gabor Tardos’s proof of the Stanley-Wilf conjecture, or by an ingenious unexpected algebraic proof, like Reimer’s proof of the Butterfly lemma en route to the Van den Berg Kesten Conjecture. I don’t have the slightest idea how such proofs are found.

More general settings.

In some comments, participants offered even more general conjectures with the same bound which may allow some induction process to apply. (If somebody is willing to summarize these extensions, that would be useful.)

Do you think that there is some promising avenue to attack Nicolai’s conjecture?

Deciding the case d=3.

Not much has happened on the front.

What about f(d,n)?

ERSS do not give a quadratic lower bound for f(d,n) but only such a bound up to a logarithmic factor. Can the gap between sets and multisets be bridged?

And what about f(2,n); do we know the answer there?

Disproving Nicolai’s conjecture

This is a modest challenge in the negative direction. The conjecture is appealing but the evidence for it is minimal. This should be easier than disproving PHC.

2) Improving the upper bounds based on the original method.

Remember that the recurrence relation was based on reaching the same element in sets from the first families and from the last families.  The basic observation is that in the first families, we must have multisets covering at least k+1 elements altogether.

There should be some “tradeoff”: Either we can reach many elements much more quickly, or else we can say something about the structure of our families which will help us.

What will this buy us? If we replace f(n/2) by f(n/10) the effect is small, but replacing it by will lead to a substantial improvement (not yet PHC).

Maybe there is hope that inside the “do loop” we can cut back. We arrived at a common ‘m’ by going from both ends. We can even reach many ‘m’s by taking steps from both ends. But then when we restrict ourselves to sets containing ‘m’, do we really start from scratch? This is the part of the proof that looks most wasteful.

Maybe looking at the shadows of the families will help. There were a few suggestions along these lines.

What do you regard as a promising avenue for improving the arguments used in current upper bound proofs?

3) How to find super-polynomial constructions?

Well, I would take sets of small size compared to n. And we want the families to be larger as we go along, and perhaps also the sets in the families to be larger. What about taking, say, at random,  in a few small sets, and in ${\cal F}_2$ much larger sets and so on? Achieving convexity (condition (*)) is difficult.

Jeff Kahn has (privately) a general sanity test against such careless suggestions, even if you force this convexity somehow: See if the upper bound proof gives you a much better recurrence. In any case, perhaps we should carefully check such simple ideas before we try to move to more complicated ideas for constructions? Maybe we should try to base a construction on the upper bound ideas. In some sense, ERSS constructions and even Nicolai’s simple one resemble the proof a little. But it goes only “one level”. It takes a long time to reach from both ends sets containing the same element, but then multisets containing the common ‘m’  use very few elements. What about Terry’s examples of families according to the sum of indices? (By the way, does this example extend to d>3?) Can you base families on more complicated equations of a similar nature?

Anyway, it is perhaps time to talk seriously about strategies for counterexamples.

What do you think a counterexample will look like?

In the first post we described the combinatorial problem: Finding the largest possible number f(n) of disjoint families of subsets from an n-element set which satisfy a certain simple property (*).We denote by f(d,n) the largest possible number of families satisfying (*) of d-subsets from {1,2,…,n}.

The two principle questions we ask are:

Can the upper bounds be improved?

and

Can the lower bounds be improved?

What are the places that the upper bound argument is wasteful and how can we improve it? Can randomness help for constructions? How does a family for which the upper bound argument is rather sharp will look like?

We are also interested in the situation for small values of n and for small values of d. In particular, what is f(3,n)? Extending the problem to multisets (or monomials) instead of sets may be fruitful since there is a proposed suggestion for an answer.

The original aim of this project is to prove the polynomial Hirsch conjecture, which is a conjecture in the combinatorial geometry of polytopes.  However, there is a reduction due to Eisenbrand, Hahnle, Razborov, and Rothvoss that would deduce this conjecture from a purely combinatorial conjecture, which can be stated as follows.

Combinatorial polynomial Hirsch conjecture. Let be non-empty collections of subsets of with the following properties:

1. (Disjointness) for every .
2. (Connectedness)  If and , there exists such that .

Then t is of polynomial size in n (i.e. ).

For instance, when n=3, one can obtain such a family with t=6, e.g.

one can show that this is the best possible value of t for this choice of n.  The best possible value of t for n=4 is still not worked out; it is between 8 and 11.

One appealing thing about this problem is that there is a simple elementary argument that gives the bound for all ; and so in some sense one is “only a logarithm away” from proving the conjecture.  Anyway, the project is just starting, and does not require any particularly specialised background, so anyone who may be interested in this problem one may want to take a look at the research thread.

I propose to try to solve the following purely combinatorial problem.

Consider t disjoint families of subsets of {1,2,…,n}, .

Suppose that

(*) For every , and every and , there is which contains .

The basic question is: How large can t  be???

(When we say that the families are disjoint we mean that there is no set that belongs to two families. The sets in a single family need not be disjoint.)

In a recent post I showed the very simple argument for an upper bound . The major question is if there is a polynomial upper bound. I will repeat the argument below the dividing line and explain the connections between a few versions.

A polynomial upper bound for will imply a polynomial (in ) upper bound for the diameter of graphs of polytopes with facets. So the task we face is either to prove such a polynomial upper bound or give an example where is superpolynomial.

The abstract setting is taken from the paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle,  Sasha Razborov, and Thomas Rothvoss. They gave an example that can be quadratic.

We had many posts related to the Hirsch conjecture.

Remark: The comments for this post will serve both the research thread and for discussions. I suggested to concentrate on a rather focused problem but other directions/suggestions are welcome as well.

Let’s call the maximum t,  f(n).
 
Remark: If you restrict your attention  to sets in these families containing an element m and delete m from all of them, you get another example of such families of sets, possibly with smaller value of t. (Those families which do not include any set containing m will vanish.)
 
Theorem: .
 
Proof: Consider the largest s so that the union of all sets in is at most n/2.   Clearly, .
Consider the largest r so that the union of all sets in is at most n/2.   Clearly, .
 
Now, by the definition of s and r, there is an element m shared by a set in the first s+1 families and a set in the last r+1 families. Therefore (by (*)), when we restrict our attention to the sets containing ‘m’ the families all survive. We get that . Q.E.D.

Remarks:

1) The abstract setting is taken from the paper  by Eisenbrand,  Hahnle,  Razborov, and  Rothvoss (EHRR). We can consider families of d-subsets of {1,2,…, n}, and denote the maximum cardinality t by . The argument above gives the relation , which implies .

2) (and thus also ) are upper bounds for the diameter of graphs of d-polytopes with n facets. Let me explain this and also the relation with another abstract formulation. Start with a -polytope with facets. To every vertex v of the polytope associate the set of facets containing . Starting with a vertex we can consider as the family of sets which correspond to vertices of distance from $w$. So the number of such families (for an appropriate is as large as the diameter of the graph of the polytope. I will explain in a minute why condition (*) is satisfied.

3) For the diameter of graphs of polytopes we can restrict our attention to simple polytopes namely for the case that all sets have size .

4)  Why the families of graphs of simple polytopes satisfy (*)? Because if you have a vertex of distance from , and a vertex at distance . Then consider the shortest path from to in the smallest face containing both and . The sets for every vertex  in  (and hence on this path) satisfies . The distances from of adjacent vertices in the shortest path from to  differs by at most 1. So one vertex on the path must be at distance from .

5) EHRR considered also the following setting: consider a graph whose vertices are labeled by subsets of {1,2,…,n}. Assume that for every vertex v labelled by S(v) and every vertex u labelled by S(u)  there is a path so that all vertices are labelled by sets containing . Note that having such a labelling  is the only properties of graphs of simple -polytopes that we have used in remark 4.

Francisco (Paco) Santos’s paper “A Counterexample to the Hirsch Conjecture” is now out: 

For some further information and links to the media see also this page. Here is a link to a TV interview.

Abstract: The Hirsch Conjecture (1957) stated that the graph of a -dimensional polytope with facets cannot have (combinatorial) diameter greater than . That is, that any two vertices of the polytope can be connected to each other by a path of at most edges. This paper presents the first counter-example to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope with 48 facets which violates a certain generalization of the -step conjecture of Klee and Walkup.

This is certainly a major event. The Hirsch conjecture with the very appealing “non-revisiting” formulation, with the neat formula , and with the many equivalent forms, captured the imagination of many people.  We saw it here: When we tried to discuss related questions, most people and most comments targeted the original conjecture itself.

The proof demonstrates not only Paco’s own ingenuity and endurance but also the steady progress in constructing polytopes and related combinatorial objects. The computer package POLYMAKE created by  Ewgenij Gawrilow and Michael Joswig  played an important role in giving a different verification of the properties of the five-dimensional construction in the paper.

Paco quotes  Klee and Kleinschmidt who wrote that “finding a counterexample will be merely a small first step in the line of investigation related to the conjecture.” But it is left to be seen if related questions such as the “polynomial Hirsch conjecture” will also draw as much interest as the original conjecture did. Certainly, Paco’s achievement may lead to renewed interest in these related problems. 

So it is time to read, learn and discuss the paper. (It is good that our term just ended.)

One direction which I see as promising is to try to examine the known upper and lower bounds for the abstract problem.  Here is again a  link for the paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle,  Sasha Razborov, and Thomas Rothvoss.

I would also be happy to hear your thoughts about strong polynomial algorithms for linear programming either via randomised pivot rules for the simplex algorithm or by any other method.  It occured to me that I am not aware of any work trying to examine the possibility that there is no strongly polynomial algorithm for linear programming. Also I am not aware of any work which shows that strongly polynomial algorithm for LP is a consequence of some (however unlikely) computational assumption. (Is it a consequence of NP=P? of P=P-space?)

And here is a link to the paper  by Sandeep Koranne and Anand Kulkarni “The d-step Conjecture is Almost true”  – most of the discussion so far was in this direction.

We had a long and interesting discussion regarding the Hirsch conjecture and I would like to continue the discussion here.  

The way I regard the open collaborative efforts is as an open collective attempt to discuss and make progress on the problem (and to raise more problems), and also as a way to assist people who think or work (or will think or will work) on these problems on their own.

Most of the discussion in the previous thread was not about the various problems suggested there but rather was about trying to prove the Hirsch Conjecture precisely! In particular, the approach of Sandeep Koranne and Anand Kulkarni which attempts to prove the conjecture using “flips” (closely related to Pachner moves, or bistaller operations) was extensively discussed.  Here is the link to another paper by Koranne and Kulkarni ”Combinatorial Polytope Enumeration“. There is certainly more to be understood regarding flips, Pachner moves, the diameter, and related notions. For example, I was curious about for which Pachner moves  ”vertex decomposibility” (a strong form of shellability known to imply the Hirsch bound) is preserved. We also briefly discussed metric aspects of the Hirsch conjecture and random polytopes.

For general background: Here is a  chapter that I wrote about graphs, skeleta and paths of polytopes. Some papers on polytopes on Gunter Ziegler’s homepage  describe very interesting and possibly relevant current research in this area. Here is a  link to Eddie Kim and Francisco Santos’s survey article on the Hirsch Conjecture. 

Here is a link from the open problem garden to the continuous analog of the Hirsch conjecture proposed by Antoine Deza, Tamas Terlaky, and  Yuriy Zinchenko.

Earlier posts are: The polynomial Hirsch conjecture, a proposal for Polymath 3 , The polynomial Hirsch conjecture, a proposal for Polymath 3 cont. , The polynomial Hirsch conjecture – how to improve the upper bounds .

Here are again some basic problems around the Hirsch Conjecture. When we talk about polytopes we usually mean simple polytopes (although looking at general polytopes may be of interest).

Problem 0: Study various possible approaches for proving the Hirsch conjecture.

We mainly discussed this avenue, which is certainly the most tempting.

Problem 1: Improve the known upper bounds for the diameter of graphs of polytopes, perhaps even finding a polynomial upper bound in terms of the dimension and the number of facets .

Strategy 1: Study the problem in the purely combinatorial settings studied in the EHRR paper.

Strategy 2: Explore other avenues.

(Nicolai Hahnle remarked that the proof extends to families of monomials.)

Problem 2: Improve the known lower bounds for the problem in the abstract setting.

Strategy 3: Use the argument for upper bounds as some sort of a role model for an example. 

Strategy 4: Try to use recursively mesh constructions like those used by EHRR.

Problem 3: What is the diameter of a polytopal digraph for a polytope with n facets in dimension d?

A polytopal digraph is obtained by orienting edges according to some generic linear objective function. This problem can be studied also in the abstract setting of shellability (and even in the context of unique sink orientations).

Problem 4: Find a (possibly randomized) pivot rule for the simplex algorithm which requires, in the worse case, small number of pivot steps.

A “pivot rule” refers to a rule to walk on the polytopal digraph where each step can be performed efficiently.

Problem 5: Study the diameter of graphs (digraphs) of specific classes of polytopes. 

Problem 6: Study these problems in low dimensions.

Problem 7: What can be said about expansion properties of graphs of polytopes?

Problem 8: What is the maximum length of a directed path in a graph of a d-polytope with n facets?

Problem 9: Study (and find further) continuous analogs of the Hirsch conjecture.

Problem 10: Find “high dimensional” analogs for the diameter problem and for shellability.

Problem 11: Find conditions for rapid convergence of a random walk (or of other stochastic processes) on directed acyclic graphs.

Problem 12: Study these problems for random polytopes.

A polynomial upper bound for graphs of polytopes is not known also for random polytops.

Problem 13: How many dual graphs of simplicial d-spheres with n facets are there?

This post is devoted to the polymath-proposal about the polynomial Hirsch conjecture. My intention is to start here a discussion thread on the problem and related problems. (Perhaps identifying further interesting related problems and research directions.)

Earlier posts are: The polynomial Hirsch conjecture, a proposal for Polymath 3 , The polynomial Hirsch conjecture, a proposal for Polymath 3 cont. , The polynomial Hirsch conjecture – how to improve the upper bounds .

First, for general background: Here is a  chapter that I wrote about graphs, skeleta and paths of polytopes. Some papers on polytopes on Gunter Ziegler’s homepage  describe very interesting and possibly relevant current research in this area,  and also a few of the papers under “discrete geometry” (which follow the papers on polytopes) are relevant. Here are again links for the recent very short paper by  Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss, the 3-pages paper by Sasha Razborov,  and to Eddie Kim and Francisco Santos’s survey article on the Hirsch Conjecture.

Here are the basic problems and some related problems. When we talk about polytopes we usually mean simple polytopes. (Although looking at general polytopes may be of interest.)

Problem 1: Improve the known upper bounds for the diameter of graphs of polytopes, perhaps finding a polynomial upper bound in term of the dimension and number of facets .

Strategy 1: Study the problem in the purely combinatorial settings proposed in the EHR paper.

Strategy 2: Explore other avenues.

Problem 2: Improve  the known lower bounds for the problem in the abstract setting.

Strategy 3: Use the argument for upper bounds as some sort of a role model for an example. 

Strategy 4:Try to use recursively mesh constructions as those used by EHR.

Problem 3: What is the diameter of a polytopal digraph for a polytope with n facets in dimension d.

A polytopal digraph is obtained by orienting edges according to some generic linear objective function. This problem can be studied also in the abstract setting of shellability. (And even in the context of unique sink orientations.)

Problem 4: Find a (possibly randomized) pivot rule for the simplex algorithm which requires, in the worse case, small number of pivot steps.

A “pivot rule” refers to a rule to walk on the polytopal digraph where each step can be performed efficiently.

Problem 5: Study the diameter of graphs (digraphs) of specific classes of polytopes. 

Problem 6: Study these problems in low dimensions.

Here are seven additional relevant problems.

Problem 7: What can be said about expansion properties of graphs of polytopes?

Problem 8: What is the maximum length of a directed path in a graph of a d-polytope with n facets?

Problem 9: Study (and find further) continuous analogs of the Hirsch conjecture.

Problem 10: Find “high dimensional” analogs: for the diameter problem and for shellability.

(The diameter of a graph is a 1-dimensional notion; are there interesting high dimension analogs? Shellability is an abstraction of a 1-dimensional projection, are there interesting abstractions for projections to higher dimensions?)

Problem 11: Find conditions for rapid convergence of a random walk (or of other stochastic processes) on directed acyclic graphs.

Problem 12: Study these problems for random polytopes.

Problem 13: How many dual graphs of simplicial d-spheres with n facets are there?

The way I regard these open collaborative efforts is as an open collective attempt to discuss and have progress on these problems (and to raise more problems), also by helping people who think or work (or will think and will work) on these problems on their own.

I can see three main avenues toward making progress on the Polynomial Hirsch conjecture.

One direction is trying to improve the upper bounds, for example,  by looking at the current proof and trying to see if it is wasteful and if so where it can be pushed further.

Another direction is trying to improve the lower-bound constructions for the abstract setting, perhaps by trying to model an abstract construction on the ideas of the upper bound proof.

The third direction is to talk about entirely different avenues to the problem: new approaches for upper bounds, related problems, special classes of polytopes, expansion properties of graphs of polytopes, the relevance of shellability, can metric properties come to play, is the connection with toric varieties relevant, continuous analogs, and other things I cannot even imagine.

Reading the short  recent paper by  Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss will get you started both for the upper bounds and for the lower bounds.

I want to discuss here very briefly how the upper bounds could be improved. (Several people had ideas in this direction and it would be nice to discuss them as well as new ideas.) First, as an appetizer, the very basic argument described for polytopes. Here is the maximum diameter of the graph of a -dimensional polyhedron with facets.

(Click on the picture to get it readable.)

The main observation here (and also in the abstract versions of the proof) is that

if we walk from a vertex in all possible directions steps we can reach vertices on at least facets.

But it stands to reason that we can do better.

Suppose that  is not too small (say .). Suppose that we start from a vertex and walk in all possible directions steps for

.  (We can simply take the larget quantity .)

The main observation we just mentioned implies that with paths of this length starting with the vertex we can reach vertices on facets  and on every facet we reach we can reach vertices on facets and in every facet of a facet we can reach vertices on facets and so on. It seems that following all these paths we will be able to reach vertices on many many more than facets. (Maybe a power greater than one of  or more.)  Unless, unless something very peculiar happens that perhaps we can analyze as well.

A convex polytope is the convex hull of a finite set of points in a real vector space. A polytope can be described as the intersection of a finite number of closed halfspaces. Polytopes have a facial structure: A (proper) face of a polytope is the intersection of  with a supporting hyperplane. (A hyperplane is a supporting hyperplane of if is contained in a closed halfspace bounded by , and the intersection of and is not empty.) We regard the empty face and the entire polytope as trivial faces. The extreme points of a polytope are called its vertices. The one-dimensional faces of are called edges. The edges are line intervals connecting a pair of vertices. The graph of a polytope  is a graph whose vertices are the vertices of and two vertices are adjacent in if there is an edge of connecting them. The -dimensional faces of a polytop are called facets.  

The Hirsch conjecture: The graph of a d-polytope with n  facets has diameter at most n-d.

A weaker conjecture which is also open is:

Polynomial Hirsch Conjecture: Let G be the graph of a d-polytope with n facets. Then the diameter of G is bounded above by a polynomial in d and n.

The avenue which I consider most promising (but I may be wrong) is to replace “graphs of polytopes” by a larger class of graphs. Most known upper bound on the diameter of graphs of polytopes apply in much larger generality. Recently, interesting lower bounds were discovered and we can wonder what they mean for the geometric problem.    

Here is the (most recent) abstract setting:

Consider the collection of graphs whose vertices are labeled by -subsets of an element set.

The only condition we will require is that if   is a vertex labeled by and is a vertex labeled by the set , then there is a path between and so that all labels of its vertices are sets containing .

Abstract Polynomial Hirsch Conjecture (APHC): Let   then the diameter of is bounded above by a polynomial in and .

Everything that is known about the APHC can be described in a few pages. It requires only rather elementary combinatorics; No knowledge about convex polytopes is needed.

A positive answer to APHC (and some friends of mine believe that is the right upper bound) will apply automatically to convex polytopes.

A negative answer to APHC will be (in my opinion) extremely interesting as well,  but will leave the case of polytopes open.  (One of the most active areas of convex polytope theory is methods for constructing polytopes, and there may be several ways to move from an abstract combinatorial example to a geometric example.)

If indeed we will decide to go for a polymath3, the concrete problem which I propose attacking is the APHC. However, we can discuss possible arguments regarding diameter of polytopes which use geometry, and we can be open to even more general abstract forms of the problem. (Or other things that people suggest.)

Reading the recent very short paper by  Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss and the 3-pages paper by Sasha Razborov (the merged journal paper of these two contributions  will become available soon, ) will get you right to the front lines. (There is an argument from the first paper that uses the Hall-marriage theorem, and an argument from the second paper that uses the “Lovasz local lemma”.)

I will try to repeat in later posts the simple arguments from these  papers -  I plan to devote one post to the upper bounds, another post to the lower bounds, and yet another post to general background, motivation and cheerleading  for the problem. I will try to make the different posts self-contained.

Questions and remarks about polytopes, the problem, or these papers are welcome.

This post is continued here. 

Eddie Kim and Francisco Santos have just uploaded a survey article on the Hirsch Conjecture.

The Hirsch conjecture: The graph of a d-polytope with n vertices  facets has diameter at most n-d.

We devoted several posts (the two most recent ones were part 6 and part  7) to the Hirsch conjecture and related combinatorial problems.

A weaker conjecture which is also open is:

Polynomial Diameter Conjecture: Let G be the graph of a d-polytope with n facets. Then the diameter of G is bounded above by a polynomial of d and n.

One remarkable result that I learned from the survey paper is in a recent paper by  Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss who proved that: 

Eisenbrand, Hahnle, and Rothvoss’s theorem: There is an abstract example of graphs for which the known upper bounds on the diameter of polytopes apply, where the actual diameter is .

Update (July 20) An improved lower bound of can be found in this 3-page note by Rasborov. A merged paper by Eisenbrand, Hahnle, Razborov, and Rothvoss is coming soon. The short paper of Eisenbrand,  Hahnle, and Rothvoss contains also short proofs in the most abstract setting of the known upper bounds for the diameter.

This is something I tried to prove (with no success) for a long time and it looks impressive. I will describe the abstract setting of Eisenbrand,  Hahnle, and Rothvoss (which is also new) below the dividing line.

I was playing with the idea of attempting a “polymath”-style  open collaboration (see here, here and here) aiming to have some progress for these conjectures. (The Hirsch conjecture and the polynomial diameter conjecture for graphs of polytopes as well as for more abstract settings.) Would you be interested in such an endeavor? If yes, add a comment here or email me privately. (Also let me know if you think this is a bad idea.) If there will be some interest, I propose to get matters started around mid-August. 

 Here is the abstract setting of Eisenbrand, Hahnle, and Rothvoss:

Consider the collection of graphs whose vertices are labeled by -subsets of an element set. The only condition is that if   is a vertex labeled by and is a vertex labelled by the set , then there is a path between and so that all labelling of its vertices are sets containing .

The main difference between this abstraction and the one we considered in the series of posts (and my old papers) is that it is not assumed that if two vertices are labeled by sets which share elements then these two vertices are adjacent.

Our Diameter problem for families of sets

Consider a family of subsets of size d of the set N={1,2,…,n}.

Associate to a graph as follows: The vertices of  are simply the sets in . Two vertices and are adjacent if .

For a subset let denote the subfamily of all subsets of which contain . 

MAIN ASSUMPTION: Suppose that for every for which is not empty is connected.

We will call a family satisfying this assumption “hereditarily connected”.

MAIN QUESTION:   How large can the diameter of be in terms of and ?

We denote the answer by .

For let be the family obtained from by removing   from every set. Since , the diameter of   is at most .

8. A slight generalization

Let be an hereditarily connected family of -subsets of a set . Let be a subset of . The length of a path of sets modulo Y  (where for every ) is the number of such that both and are subsets of . (In other words, in we consider edges between subsets of Y as having length 1 and other edges as having length 0.)

Let be the largest diameter of an hereditarily connected family of -subsets of an arbitrary set  modulo a set Y , with .

Since we can always take we have .  

9.  A quasi-polynomial upper bound

We will now describe an argument giving a quasi-polynomial upper bound for . This is an abstract version of a geometric argument of Kleitmen and me. 

Let be a hereditarily connected family of -subsets of some set , let , , and let and be two sets in the family.

Claim: We can always either

1) find paths of length at most  modulo Y  from to -subsets of whose union has more than elements.

or

2) we can find a path of this length   modulo Y   from to .  

Proof of the claim: Let be the set of elements from that we can reach in steps modulo Y   from . (Let me explain it better: is the elements of in the union of all sets that can be reached in steps modulo Y from . Or even better: is the intersection of with the union of all sets in which can be reached from in steps modulo Y. ) 

The distance of from modulo Z  is at most .

Now, if we are in case 1).

If then there is a path from to modulo Z of length . If this path reaches no set containing a point in we are in case 1).  (Because this path is actually a path of length from to modulo Y).  Otherwise, we reached via a path of length modulo Y from a set containing a point in , in contradiction to the definition of .  Walla.

Corollary: .

By a path of length modulo Y  we reach from at least elements in , (or ).  By a path of length modulo Y  we reach from at least elements in , (or ). So unless we can go from to in steps modulo Y  we can reach more than elements from both and by paths of length modulo Y ,hence there is some element we can reach from both.

In other words in steps modulo Y  we go from to and from to so that and share an element .

But the distance from to modulo Y  (which is the same as the distance modulo Y  from to in is at most .  (We use here the fact that ) Ahla!

To solve the recurrence, first for convenience replace by . (You get a weaker inequality.) Then write to get and to get which gives which in turn gives and . Sababa!

This is the last post in the series. The proof presented here is an abstract version of a geometric proof for graphs of polytopes by Kalai and Kleitman. Different paths to weaker quasi-polynomial upper bounds can be found here. These bounds are linear when is fixed.  A similar (even a bit simpler) argument under an even more general context was found by Razborov. (But I don’t remember it at present.) The argument above extends to the directed case. But finding an actual pivot rule for the simplex algorithm which comes close to this bound is out of reach.

I conjecture that is polynomial (and that this holds for and even in the greater generality considered by Razborov). I also conjectured before that it is not a polynomial, but changed my mind. So frankly, I do not have a clue. Remember that it is even possible that .

Summary of earlier posts: Part 1 describes the problem. (It is repeated here.) Part 2 describe the connection to the Hirsch Conjecture.  Part 3 describes linear bound when is fixed. It also raises the question if past (or future) developements on the problem can be quasi-automatize.  Part 5 follows a question from part 4 and describes a subexponential upper bound. Part 6 describes further the connection with linear programming and with shellability, and poses a directed version of the problem.

George Dantzig and Leonid Khachyan

In this part we will not progress on the diameter problem that we discussed in the earlier posts but will rather describe a closely related problem for directed graphs associated with ordered families of sets. The role models for these directed graphs are the directed graphs of polytopes where the direction of the edges is described by a linear objective function.  

7. Linear programming and the simplex algorithm.

Our diameter problem for families of sets was based on a mathematical abstraction (and a generalization) of the Hirsch Conjecture which asserts that the diameter of the graph of a -polytope with facets is at most . Hirsch, in fact, made the conjecture also for graphs of unbounded polyhedra – namely the intersection of closed halfspaces in . But in the unbounded case, Klee and Walkup found a counterexample with diameter []. The abstract problem we considered extends also to the unbounded case and   [] is the best known lower bound for the abstract case as well. It is not known if there is a polynomial (in terms of and ) upper bound for the diameter of graphs of d-polytopes with n facets.

Hirsch’s conjecture was motivated by the simplex algorithm for linear programming.  Let us talk a little more about it: Linear programming is the problem of maximizing a linear objective function subject to a system of n linear inequalities in the variables .

 ,

,

…

,

The set of solutions to the system of inequalities is a convex polyhedron. (If it is bounded it is a polytope.)  A linear objective function makes a graph of a polytope (or a polyhedron) into a digraph (directed graph). If you like graphs you would love digraphs, and if you like graphs of polytopes, you would like the digraphs associated with them.  

The geometric description of Dantzig’s simplex algorithm is as follows: the system of inequalities describes a convex d-dimensional polyhedron . (This polyhedron is called the feasible polyhedron.) The maximum of is attained at a face of . We start with an initial vertex (extreme point) of the polyhedron and look at its neighbors in . Unless there is a neighbor of that satisfies . When you find such a vertex move from to and repeat!

8. Abstract objective functions and unique sink orientation.

Let be a simple d-polytope and let be a linear objective function which is not constant on any edge of the polytope. Remember, the graph of , is a -regular graph. We can now direct every edge from to if . Here are two important properties of this digraph.

(AC) It is acyclic! (no cycles)

(US’) It has a unique SINK, namely a unique vertex such that all edges containing it are directed towards it.

The unique sink property is in fact the property that enables the simplex algorithm to work!

When we consider a face of the polytope and its own graph then again our linear objective function induces an orientation of the edges of which is acyclic and also has the unique sink property. Every subgraph of an acyclic graph is acyclic. But having the unique sink property for a graph does not imply it for a subgraph. We can now describe the general unique sink properties of digraphs of polytopes:

(US) For every face F of the polytope, the directed graph induced on the vertices of has a unique sink.

A unique sink acyclic orientation of the graph of a polytope is an orientation of the edges of the graph which satisfies properties (AC) and (US). 

An abstract objective function of a -polytope is an ordering of the vertices of the polytope such that the directed graph obtained by directing an edge from to  if   is a unique sink acyclic orientation. (Of course, coming from an ordering the orientation is automatically acyclic.) 

9. Questions and answers regarding the simplex algorithm.

Q: What is a polyhedron, is it just a fancy name for a polytope?

A: A polyhedron is the intersection of closed half spaces in . A bounded polyhedron is a polytope. 

Q: How do you find the initial feasible vertex v?

A: Ohh, good point. Usually you need a first stage of the algorithm to reach a feasible vertex. This is sometimes referred to as Phase 1 of the algorithm, and moving from a feasible vertex to the optimal one is called Phase 2. But you can transform every LP problem to another one in which the origin is a feasible polyhedron so for the purpose of studying the worst-case behavior of the simplex algorithm it is enough to study phase 2.  

Q: How do you choose to which neighbor to move?

A: Ahh, this is a good question. Often, there are many ways to do it and a rule for making the choice is called a pivot rule. Which pivot rule to take, for theoretical purposes as well as practical purposes is important.

Q: Is the simplex algorithm a polynomial time algorithm?

A: We do not know any pivot rule that leads to a polynomial algorithm in the sense that the number of pivot steps is bounded above by a polynomial function of and .

Q: Is there a polynomial algorithm for LP?

A: Yes, Katchian proved in 1979 that the Nemirovski-Shor ellipsoid algorithm is a polynomial time algorithm for LP.

Q: Didn’t you neglect to mention some important things?

A: Quite a few. In particular, I ignored issues of degeneracy, for example if the feasible polyhedron is not simple.

Before we go on to describe an even more abstract objective functions, let me recall section 2 about the connection between the abstract combinatorial graphs based on families of sets and the graphs of polytopes. If this is already fresh in your memory you can safely skip it.

2. The connection of our abstract setting with the Hirsch’s Conjecture reminded

The Hirsch Conjecture asserts that the diameter of the graph G(P) of a d-polytope P with n facets is at most n-d. Not even a polynomial upper bound for the diameter in terms of d and n is known. Finding good upper bounds for the diameter of graphs of d-polytopes is one of the central open problems in the study of convex polytopes. If d is fixed then a linear bound in n is known, and the best bound in terms of d and n is . We will come back to these results later.

One basic fact to remember is that for every d-polytope P, G(P) is a connected graph. As a matter of fact, a theorem of Balinski asserts that G(P)$ is d-connected.

The combinatorial diameter problem I mentioned in an earlier post (and which is repeated below) is closely related. Let me now explain the connection. 

Let P be a simple d-polytope. Suppose that P is determined by n inequalities, and that each inequality describes a facet of P. Now we can define a family of subsets of {1,2,…,n} as follows. Let be the n inequalities defining the polytope P, and let be the n corresponding facets. Every vertex v of P belongs to precisely d facets (this is equivalent to P being a simple polytope). Let be the indices of the facets containing v, or, equivalently, the indices of the inequalities which are satisfied as equalities at v. Now, let be the family of all sets for all vertices of the polytope P.

The following observations are easy.  

(1) Two vertices v and w of P are adjacent in the graph of P if and only if . Therefore, .

(2)  If A is a set of indices, then the vertices v of P such that are precisely the set of vertices of a lower dimensional face of P. This face is described by all the vertices of P which satisfy all the inequalities indexed by , or equivalently all vertices in P which belong to the intersection of the facets for .

Therefore, for every , if is not empty the graph is connected – this graph is just the graph of some lower dimensional polytope. This was the main assumption in our abstract problem.

10. Even more abstract objective functions.

 We will not discuss actual pivot rules for linear programming in this thread of posts.  This is an interesting topic that we may discuss separately. Linear objective functions transform the graph of the polytope into a directed graph. We replaced graphs of polytopes by very abstract and general graphs associated to families of sets. What about digraphs of polytopes? 

Let be an ordered family of -subsets of {1,2,…,n}. Define a digraph or a directed graph  as a digraph whose vertex set is latex \cal F$ and which has a directed edge from to if .   

Let .

Make the following assumption:

(*) For every , is connected. Moreover, for every and every subset the graph is connected. (In words, the graph which corresponds to all sets in the family that come after and contain is connected.)

Now we can define a directed graph by orienting an edge   from to if .

Starting from a simple d-polytope with facets, we associated to a family of sets that correspond to the vertices of . When we have an objective function , we can order the vertices of and thus obtain an ordered family that satisfies the assumption (*).  If we start with a simple polytope with facets and order the sets which correspond to its vertices according to an abstract objective function we get a more general class of ordered families satisfying (*).

11. Shellability again

Remember the notion of shellability in Kimmo Errikson’s poem? Let be the ideal or simplicial complex spanned by the family . So  for some . To say that the ordering of is an abstract objective function is equivalent to the statement that is shellable and the ordering of is a shelling order on .

One important consequence of this observation is that not every family of sets satisfying our connectivity conditions can be ordered as to satisfy our new connectivity relation (*).

12. Short directed path?

Here is the directed version of our diameter problem. Given an ordered family of sets satisfying our condition (*), we can always have a directed path from every to . Can we always guarantee a path of length ? A path of length bounded by some for some ?

Proposition 1:

How to prove it: This is easy to prove: Given two sets and in our family , we first find a path of the form where, and . We let with and consider the family . This is a family of -subsets of an set () . It follows that we can have a path from to in   of length at most . Putting all these paths together gives us the required result. (We remind the notations at the end of this post.)

How to use it: It is not obvious how to use Proposition 1. Barnette’s argument from part 3 was about , and it used something a bit more sophisticated. Applying Proposition 1 directly for does not give anything non trivial. However, when is small compared to and is a small fraction of we can say something.

Let us start with an example: . let us choose . Consider a path in from two sets and . Suppose also that in this path 

(*) , for every . 

Let be the last set in the path whose intersection with has at least elements.  Let be the last set in the path whose intersection with has at least elements. I claim that is a path in .

To see this note that must contain new elements not already in . Next must contain at least d/2 elements not already in and . Together the three sets must therefore contain at least elements. This means that their union has at least  elements, hence their union contains at least elements from and by (*)  and share at least  elements. Sababa.

This argument extends to the following proposition: 

Proposition 2: .

So what? How to use these propositions: Remember that the bound to beat was (actually, Larman improved it to , but in any case, it is exponential in .) Applying the two propositions and the trivial bound we can get

Proposition 3: . 

Can we automatize it? Let me return to the question of whether such arguments can be automatized.  The above argument for Proposition 2 was simple but somewhat ad hoc. You can get a slightly worse upper bound by noting that if is a family of -subsets of an n-set and , then does not contain an independent set of size . Just use the fact that . This implies that its diameter is not larger than . The bound on the independence set based on a standard estimetes for union of sets and the connection between the diameter and the independence number both look rather automatable. So is “thinking about” and proving Proposition 1 and deducing Proposition 3. (But I must admit that overall I am less optimistic about the ability to make these very elementary attacks on the problem automatic.)

What else? There is a little more to be said. The problem we face using Propositions 1 and 2 is that the ratio between and may deteriorate. Once is large compared with  the situation is hopeless. But if we force the ratio between and to be bounded also for families we can get better (polynomial!!) bounds. I will state these bounds for polytopes keeping in mind the simple connection between the abstract problem and the diameter problem for graphs of polytopes.

Proposition: Let  be a simple -polytope and suppose that for every face of the number of facets of is at most . Then the diameter of the graph of is at most . Here .

For it is not hard to see that has a diameter at most 2 and to then deduce that the graph of the polytope has diameter at most .

Reminder: Our Diameter problem for families of sets and some notations and basic observations.

Consider a family of subsets of size d of the set N={1,2,…,n}.

Associate to a graph as follows: The vertices of  are simply the sets in . Two vertices and are adjacent if .

For a subset let denote the subfamily of all subsets of which contain . 

MAIN ASSUMPTION: Suppose that for every for which is not empty is connected.

MAIN QUESTION:   How large can the diameter of be in terms of and .

Let us denote the answer by .

More notations and a basic observation from previous parts:

Let be the family obtained from by removing the elements of A from every set. Note that . Therefore, the diameter of   is at most , where and is the number of elements in the union of all the sets in .

We associated more general graphs to as follows: For an integer define  as follows: The vertices of  are simply the sets in . Two vertices and are adjacent if . Our original problem dealt with the case .  Thus, .   

Let be the maximum diameter of   in terms of and , for all families of -subsets  of satisfying our connectivity relations.

Recall that we consider a family of subsets of size  of the set .

Let us now associate  more general graphs to as follows: For an integer define  as follows: The vertices of  are simply the sets in . Two vertices and are adjacent if . Our original problem dealt with the case .  Thus, . Barnette proof presented in the previous part refers to and to paths in this graph.  

As before for a subset let denote the subfamily of all subsets of which contain . Of course, the smaller is the more edges you have in . It is easy to see that assuming that is connected for every for which is not empty already implies our condition that is connected for every for which is not empty.

Let be the maximum diameter of   in terms of and , for all families of -subsets  of satisfying our connectivity relations.

Here is a simple claim:

Can you prove it? Can you use it? 

We will now try all sorts of ideas to give good upper bounds for the abstract diameter problem that we described. As we explained, such bounds apply to the diameter of graphs of simple d-polytopes.

All the methods I am aware of for providing upper bounds are fairly simple.

(1) You think about a strategy from moving from one set to another, 

(2) You use this strategy to get a recursive bound,

(3) You solve the recursion and hope for the best.

What I would like you to think about, along with reading these posts, is the following questions:

(a) Can I come up with a different/better strategy for moving from one set to the other?

(b) Can I think about a mathematically more sophisticated way to get an upper bound for the diameter?

(c) Can this process of finding a strategy/writing the associated recurrence/solving the recurrence be automatized? The type of proofs we will describe are very simple and this looks like a nice example for a “quasi-automatic” proof process.

Let me repeat the problem and prove to you a nice upper bound:

Reminder: Our Diameter problem for families of sets

Consider a family of subsets of size d of the set N={1,2,…,n}.

Associate to a graph as follows: The vertices of  are simply the sets in . Two vertices and are adjacent if .

For a subset let denote the subfamily of all subsets of which contain . 

MAIN ASSUMPTION: Suppose that for every for which is not empty is connected.

MAIN QUESTION:   How large can the diameter of be in terms of and .

Let us denote the answer by .

4. A one line observation.

What the upper bound tells us about the diameter of  ? Let be the family obtained from by removing the elements of A from every set. Note that . Therefore, the diameter of   is at most , where and is the number of elements in the union of all the sets in .

5. Linear bounds for a fixed dimension

Let’s use the following strategy to move from one set to the other.

Given two sets S and T in we first try to move from S to T using a different type of path. ,  where this time .  We will choose such a path with t being as small as possible.

Let . We will consider the families .

The one line observation tells us that the diameter of is bounded from above by where the number of elements in the union of all the sets in .

We want to prove an upper bound on of the form . For this purpose, let us have a closer look at these sets .

Claim: if .

Proof: Suppose that   and . So there is a set which contains and , and there is a set   which contains both and . Now we can shortcut! We replace the segment by . This will give us a shorter path of the peculiar type we consider here.

The claim implies that every element of N is included in at most three s. We are done! If then we get that the distance between and in is at most . This gives us .  

This argument is due to Barnette.

Reminder: The connection with Hirsch’s Conjecture

The Hirsch Conjecture asserts that the diameter of the graph G(P) of a d-polytope P with n facets is at most n-d. Not even a polynomial upper bound for the diameter in terms of d and n is known. Finding good upper bounds for the diameter of graphs of d-polytopes is one of the central open problems in the study of convex polytopes. If d is fixed then a linear bound in n is known, and the best bound in terms of d and n is . We will come back to these results later.

One basic fact to remember is that for every d-polytope P, G(P) is a connected graph. As a matter of fact, a theorem of Balinski asserts that G(P)$ is d-connected.

The combinatorial diameter problem I mentioned in an earlier post (and which is repeated below) is closely related. Let me now explain the connection. 

Let P be a simple d-polytope. Suppose that P is determined by n inequalities, and that each inequality describes a facet of P. Now we can define a family of subsets of {1,2,…,n} as follows. Let be the n inequalities defining the polytopeP, and let be the n corresponding facets. Every vertex v of P belongs to precisely d facets (this is equivalent to P being a simple polytope). Let be the indices of the facets containing v, or, equivalently, the indices of the inequalities which are satisfied as equalities at v. Now, let be the family of all sets for all vertices of the polytope P.

The following observations are easy.  

(1) Two vertices v and w of P are adjacent in the graph of P if and only if . Therefore, .

(2)  If A is a set of indices. The vertices v of P such that are precisely the set of vertices of a lower dimensional face of P. This face is described by all the vertices of P which satisfies all the inequalities indexed by , or equivalently all vertices in P which belong to the intersection of the facets for .

Therefore, for every if is not empty the graph is connected – this graph is just the graph of some lower dimensional polytope. This was the main assumption in our abstract problem.

Remark: It is known that the assertion of the Hirsch Conjecture fails for the abstract setting. There are examples of families where the diameter is as large as n-(4/5)d.