# Plans for polymath3

Polymath3 is planned to study the polynomial Hirsch conjecture. In order not to conflict with Tim Gowers’s next polymath project which I suppose will start around January, I propose that we will start polymath3 in mid April 2010. I plan to write a few posts on the problem until then. We had a long and interesting discussion regarding the Hirsch conjecture followed by another interesting discussion. We can continue the discussion here.

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?)

# The Polynomial Hirsch Conjecture: Discussion Thread, Continued

Here is a  link for the just-posted paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle,  Sasha Razborov, and Thomas Rothvoss.

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.

# The Polynomial Hirsch Conjecture: Discussion Thread

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.)

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 $d$ and number of facets $n$.

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? Continue reading

# The Polynomial Hirsch Conjecture – How to Improve the Upper Bounds.

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 $\Delta (d,n)$ is the maximum diameter of the graph of a $d$-dimensional polyhedron with $n$ 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 $v$ in all possible directions $\Delta(d,k)$ steps we can reach vertices on at least $k+1$ facets.

But it stands to reason that we can do better.

Suppose that $n$ is not too small (say $n=d^2$.). Suppose that we start from a vertex $v$ and walk in all possible directions $t$ steps for

$t=\Delta (d,10d)+\Delta (d-1,10d)+\Delta(d-2,10d)+\dots +\Delta(2,10d)$.  (We can simply take the larget quantity $t = d \Delta (d,11d)$.)

The main observation we just mentioned implies that with paths of this length starting with the vertex $v$ we can reach vertices on $10d$ facets  and on every facet we reach we can reach vertices on $10d$ facets and in every facet of a facet we can reach vertices on $10d$ facets and so on. It seems that following all these paths we will be able to reach vertices on many many more than $10d$ facets. (Maybe a power greater than one of $d$ or more.)  Unless, unless something very peculiar happens that perhaps we can analyze as well.

# The Polynomial Hirsch Conjecture, a Proposal for Polymath3 (Cont.)

## The Abstract Polynomial Hirsch Conjecture

A convex polytope $P$ 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 $P$ is the intersection of  $P$ with a supporting hyperplane. (A hyperplane $H$ is a supporting hyperplane of $P$ if $P$ is contained in a closed halfspace bounded by $H$, and the intersection of $H$ and $P$ is not empty.) We regard the empty face and the entire polytope as trivial faces. The extreme points of a polytope $P$ are called its vertices. The one-dimensional faces of $P$ are called edges. The edges are line intervals connecting a pair of vertices. The graph $G(P)$ of a polytope $P$  is a graph whose vertices are the vertices of $P$ and two vertices are adjacent in $G(P)$ if there is an edge of $P$ connecting them. The $(d-1)$-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 ${\cal G}(d,n)$ of graphs $G$ whose vertices are labeled by $d$-subsets of an $n$ element set.

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

Abstract Polynomial Hirsch Conjecture (APHC): Let $G \in {\cal G}(d,n)$  then the diameter of $G$ is bounded above by a polynomial in $d$ and $n$.

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 $n^2$ 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, Continue reading

# The Polynomial Hirsch Conjecture: A proposal for Polymath3

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 $n^{3/2}$.

Update (July 20) An improved lower bound of $\Omega(n^2/\log n)$ 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: Continue reading

# A Diameter problem (7): The Best Known Bound

### Our Diameter problem for families of sets

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

Associate to $\cal F$ a graph $G({\cal F})$ as follows: The vertices of  $G({\cal F})$ are simply the sets in $\cal F$. Two vertices $S$ and $T$ are adjacent if $|S \cap T|=d-1$.

For a subset $A \subset N$ let ${\cal F}[A]$ denote the subfamily of all subsets of $\cal F$ which contain $A$

MAIN ASSUMPTION: Suppose that for every $A$ for which ${\cal F}[A]$ is not empty $G({\cal F}[A])$ is connected.

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

MAIN QUESTION:   How large can the diameter of $G({\cal F})$ be in terms of $d$ and $n$?

We denote the answer by $F(d,n)$.

For $v \in \{1,2,\dots,n\}$ let ${\cal F'}[v]$ be the family obtained from ${\cal F}[\{v\}]$ by removing $v$  from every set. Since $G({\cal F}[v]) = G({\cal F}' [v])$, the diameter of  $G({\cal F}[\{v\}])$ is at most $F(d-1,n-1)$.

### 8. A slight generalization

Let ${\cal F}$ be an hereditarily connected family of $d$-subsets of a set $X$. Let $Y$ be a subset of $X$. The length of a path of sets $S_1,S_2,\dots, S_t$ modulo Y  (where $|S_i \cap S_{i+1}|=d-1$ for every $i$) is the number of $j, 1 \le j such that both $S_j$ and $S_{j+1}$ are subsets of $Y$. (In other words, in $G({\cal F})$ we consider edges between subsets of Y as having length 1 and other edges as having length 0.)

Let $T(d,n)$ be the largest diameter of an hereditarily connected family of $d$-subsets of an arbitrary set $X$ modulo a set Y , $Y \subset X$ with $|Y|=n$.

Since we can always take $X=Y$ we have $F(d,n) \le T(d,n)$.

### 9.  A quasi-polynomial upper bound

We will now describe an argument giving a quasi-polynomial upper bound for $T(d,n)$. This is an abstract version of a geometric argument of Kleitmen and me.

Let ${\cal F}$ be a hereditarily connected family of $d$-subsets of some set $X$, let $Y \subset X$, $|Y|=n$, and let $S$ and $T$ be two sets in the family.

Claim: We can always either

1) find paths of length at most $T(d,k)$  modulo from $S$ to $d$-subsets of $Y$ whose union has more than $k$ elements.

or

2) we can find a path of this length $T(d,k)$  modulo Y   from $S$ to $T$.

Proof of the claimLet $Z$ be the set of elements from $Y$ that we can reach in $T(d,k)$ steps modulo Y   from $S$. (Let me explain it better: $Z$ is the elements of $Y$ in the union of all sets that can be reached in $T(d,k)$ steps modulo Y from $S$. Or even better: $Z$ is the intersection of $Y$ with the union of all sets in $\cal F$ which can be reached from $S$ in $T(d,k)$ steps modulo Y. )

The distance of $S$ from $T$ modulo Z  is at most $T(d,|Z|)$.

Now, if $|Z|>k$ we are in case 1).

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

Corollary: $T(d,n) \le 2T(d,n/2)+T(d-1,n-1)$.

By a path of length $T(d,n/2)$ modulo Y  we reach from $S$ at least $n/2$ elements in $Y$, (or $T$).  By a path of length $T(d,n/2)$ modulo Y  we reach from $T$ at least $n/2$ elements in $Y$, (or $S$). So unless we can go from $S$ to $T$ in $T(d,n/2)$ steps modulo Y  we can reach more than $n/2$ elements from both $S$ and $T$ by paths of length $T(d,n/2)$ modulo Y ,hence there is some element we can reach from both.

In other words in $T(d,n/2)$ steps modulo Y  we go from $S$ to $S'$ and from $T$ to $T'$ so that $S'$ and $T'$ share an element $u$.

But the distance from $S'$ to $T'$ modulo Y  (which is the same as the distance modulo Y  from $S' \backslash u$ to $T' \backslash u$ in ${\cal F}'[\{u\}]$ is at most $T(d-1,n-1)$.  (We use here the fact that $u \in Y$) Ahla!

To solve the recurrence, first for convenience replace $T(d-1,n-1)$ by $T(d-1,n)$. (You get a weaker inequality.) Then write $G(d,n)=T(d,2n)$ to get $G(d,n) \le G(d,n/2)+G(d-1,n)$ and $H(d,x)=G(d,2^x)$ to get $H(d,x) \le H(d-1,x) + H(d,x-1)$ which gives $H(d,x) \le {{d+x} \choose {d}}$ which in turn gives $G(d,n) \le {{log n+d} \choose {d}}$ and $T(d,n) \le n {{log n +d} \choose {d}}$. Sababa!