Projections to the TSP Polytope

Michael Ben Or told me about the following great paper Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds by Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary and Ronald de Wolf. The paper solves an old conjecture of Yannakakis about projections of polytopes.

From the abstract: “We solve a 20-year old problem posed by M. Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the maximum cut problem and the stable set problem. These results follow from a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.”

There are many interesting aspects to this story. The starting point was a series of papers in the 80s trying to prove that P=NP by solving TSP using linear programming. The idea was to present the TSP polytope as a projection of a larger dimensional polytope described by  polynomially many linear inequalities, and solve the LP problem on that larger polytope.  Yannakakis proved that such attempts are doomed to fail, when the larger LP problem keep the symmetry of the original TSP polytope.

Yannakakis asked if the symmetry condition can be removed and this is what the new paper shows. This is a very interesting result also from the point of view of convex polytope theory.

Another exciting aspect of the paper is the use of methods from quantum communication complexity.

Update: See this post over GLL for discussion and a description of a follow up paper.

Polymath3 (PHC6): The Polynomial Hirsch Conjecture – A Topological Approach

This is a new polymath3 research thread. Our aim is to tackle the polynomial Hirsch conjecture which asserts that there is a polynomial upper bound for the diameter of graphs of $d$-dimensional polytopes with $n$ facets. Our research so far was devoted to an abstract combinatorial setting. We studied an appealing conjecture by Nicolai Hahnle and considered an even more general abstraction proposed by Yury Volvovskiy. Comments towards this abstract conjecture are most welcome!

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 $S^2$ 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 $M_1$ be a red map and let  $M_2$ be a blue map drawn in general position on $S^{n}$, and let $M$ be their common refinement.  There is a vertex $w$ of $M$ a blue vertex $v \in M_1$, a red vertex $u \in M_2$ and two faces $F,~G \in M$ such that 1) $v,w \in F$, 2) $w,u \in G$, and 3) $\dim F + \dim G =n$.

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 $n=2k$ we can even guarantee that $\dim F=\dim G=k$.)

Why the tentative conjecture implies that the diameter is polynomial

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

Here is why:

– It is known that the diameter of every polytope with $n$ facets and dimension $d$ is bounded above by the “length” of a Dantzig figure with $2n-2d$ facets and $n-d$ vertices.

Here a Dantzig figure is a simple polytope of dimension $D$ with $2D$ 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 $d$ is the same as the minimum distance between blue and red vertices in a pair of two maps in the $(d-2)$-sphere, with $d$ cells each.

– Our tentative conjecture implies, by dimension on $d$, that the minimum distance between blue and red vertices in a pair of maps in the $d$-sphere and with $n$ cells is bounded above by (essentially) $nd$. ($n$ 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.

IPAM Remote Blogging: Santos-Weibel 25-Vertices Prismatoid and Prismatoids with large Width

Here is a web page by Christope Weibel on the improved counterexample.

The IPAM webpage contains now slides of some of the lectures. Here are Santos’s slides. The last section contains some recent results on the “width of 5-prismatoids”  A prismatoid is a polytope with two facets containing all the vertices. The width of a prismatoid is the number of steps needed to go between these two facets where in each step we move from a facet to an adjacent one. Santos’s counterexample is based on findng 5-dimensional prismatoid with width larger than 5. It is observed that the width of a 5-prismatoid with n vertices cannot exceed 3n/2 and it is shown (by rather involved constructions) that there are examples where the width is as large as $\sqrt n$.

Remote Blogging: Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?

Here are some links and posts related to some of the talks in IPAM’s workshop “Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?” I will be happy to add links to pdf’s of the presentations and to relevant papers. Descriptions and remarks on individual lectures are very welcome. In particular you are most welcome to post here the posters/abstract/papers from the poster session. (Because of some technical matters I will have to miss the workshop. I hope to be able somehow to follow it from far away.)

Following is a description of some of the lectures and some useful links for a few lectures:

A counterexample to Hirsch’s conjecture: Posts related to Paco Santo’s lecture (Tuesday morning) describing a counter example to Hirsch’s conjecture are here and here The second post contains a link to Paco’s paper. Fred Holt’s second lecture will describe some new consequence to Paco’s construction.

Acyclic USO: Acyclic unique sink orientations (Acyclic USO) of cubes and more general polytopes are mentioned in this post about abstract linear objective functions as well as this one about telling simple polytopes from their graphs. Acyclic USO are mentioned discussed in David Avis’s Wednesday morning talk and a few others.

Stochastic games: Stochastic games relevant to Yinyu Ye’s and Oliver Friedmann’s Thursday morning talks are discussed in this post. Some background (and links to the papers) for the new subexponential lower bounds for randomized pivot rules that will be described in Friedmann’s lecture can be found in this post.

Polynomial Hirsch conjecture: Ed Kim’s (Thursday afternoon) and Nicolai Hähnle’s  (Friday’s morning) talks are related to polymath3. David Bremner will discuss (Tuesday afternoon) the combinatorics and geometry of path complexes. Jonathan Kelner will propose (Friday morning) a geometric/probabilistic method based on smoothed analysis to attack th epolynomial Hirsch conjecture.

Bad behavior of the simplex algorithms. Examples for the bad behavior of simplex type algorithms (mainly in three dimensions) will be described in Günter Ziegler’s talk (Tuesday afternoon). Here is the link to Günter’s slides which are rather detailed. Bernd Gärtner (Wednesday morning) will demonstrate how Goldfarb’s cubes can be used to refute a conjecture regarding an algorithm for machine learning.

Interior point methods: Since the mid 80s interior point methods for linear programming are as important theoretically and practically as simplex type algorithms. (I will add a link for a good wide-audience description of interior point methods HERE.) Jim Renegar’s Thursday afternoon lecture will describe some new advances on  Central Swaths (a generalization of the central path a central notion for interior point methods. ) Santosh Vampala closing lecture will propose a hybrid vertex-following interior-type algorithm.

Continuous analogs: There are several interesting continuous analogs for combinatorial notions and questions related to the simplex algorithms. Yuriy Zinchenko will discuss (Wednesday afternoon) continuous analogs of the Hirsch conjecture

Walking on the arragements: Consider the entire arrangement of hyperplanes described by the inequalities of an LP problem. The simplex algorithm can be described as a walk on vertices of this arrangements. Those are very special vertices – the vertices of the feasible polyhedron. The dual simplex algorithm can also be described as a walk on vertices of the arrangement. This time the relevant vertices (I think) are dual-feasible, namely those are vertices of the arrangement which optimize the objective function w.r.t. a subset of the inequalities.  What about LP algorithms based on more general type of walks?  Kumei Fukuda Thursday’s afternoon talk will discuss this issue.

A new polynomial LP algorithm:  Sergei Chubanov (Thursday afternoon) will propose a strongly polynomial relaxation-type algorithm which either finds a solution of a linear system or decides that the system has no 0,1-solutions. If the system is feasible and the bounds on variables are tight, the algorithm always finds a solution. Sergei will continue to show that the algorithm can be used as the basis for the construction of a polynomial algorithm for linear programming. Here is a link to the paper. Tamás Terlaky (Wednesday afternoon) will review several algorithms for linear programming including elimination and pivot algorithms, interior point methods and the perceptron algorithm.

Complexity of Delaunay Triangulation:  Nina Amenta’s Friday talk will describe what we recently learned about the the complexity of Delaunay triangulations as function of the distribution of their vertices, and will raise the question “how much of this can be applied to polytopes in general?”

Among the additional presentations: Christophe Weibel presented an improvement to Paco Santos’ counterexample. Jésus de Loera presented a paper with Bernd Sturmfels, and Cynthia Vinzant on the centeral curve in linear programming.

Subexponential Lower Bound for Randomized Pivot Rules!

Oliver Friedmann, Thomas Dueholm Hansen, and Uri Zwick have managed to prove subexponential lower bounds of the form $2^{n^{\alpha}}$ for the following two basic randomized pivot rules for the simplex algorithm! This is the first result of its kind and deciding if this is possible was an open problem for several decades. Here is a link to the paper.

Update: Oliver Friedmann have managed to use similar methods to find similar lower bounds also for Zadeh’s deterministic pivot rule. See this paper.

We can regard the simplex algorithm as starting from an arbitrary vertex of the feasible polytope and repeatedly moving to a neighboring vertex with a higher value of the objective function according to some pivot rule.

The pivot rules considered in the paper are

RANDOM EDGE– Choose an improving pivoting step uniformly at random.

RANDOM FACET– Choose at random a facet containing your vertex and apply the algorithm in that facet.

Polymath3: Polynomial Hirsch Conjecture 4

So where are we? I guess we are trying all sorts of things, and perhaps we should try even more things. I find it very difficult to choose the more promising ideas, directions and comments as Tim Gowers and Terry Tao did so effectively in Polymath 1,4 and 5.  Maybe this part of the moderator duty can also be outsourced. If you want to point out an idea that you find promising, even if it is your own idea, please, please do.

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? Continue reading

Polymath3 : Polynomial Hirsch Conjecture 3

Here is the third research thread for the polynomial Hirsch conjecture.  I hope that people will feel as comfortable as possible to offer ideas about the problem we discuss. Even more important, to think about the problem either in the directions suggested by others or on their own. Participants who follow the project and think about the issues without adding remarks are valuable.

The combinatorial problem is simple to state and also everything that we know about it is rather simple. At this stage joining the project should be easy.

Let me try to describe (without attemting to be complete) one main direction that we discuss. This direction started with the very first comment we had by Nicolai.

Please do not hesitate to repeat an idea raised by yourself or by other if you think it can be useful.

Let $f^*(d,n)$ be the largest number of disjoint families $F_1, F_2, ..., F_t$ of degree d monomials in the variables $x_1,\dots,x_n$ such that

(*) for i < j < k, whenever $m_1 \in F_i$ and $m_2 \in F_k$, then there exists a monomial $u \in F_j$ such that $gcd(m_1, m_2) | u$.

Nicolai’s conjecture:

$f^*(d,n)=d(n-1)+1$.

The example that supports this conjecture consists of families with a single monomial in every family.

The monomials are

$x_1^d,$

$x_1^{d-1}x_2$,

$\dots$,

$x_2^d$,

$x_2^{d-1}x_3$,

$\dots$,

$x_n^d$.

There are other examples that achieve the same bound. The bound can be achieved by families whose union include all monomials, and for such families the conjecture is correct.

The case d=3.

An upper bound by EHRR (that can be extended to monomials) following works of Barnette and Larman on polytopes is $f^*(d,n) \le 2^{d-1}n$. For degree 3 monomials we have a gap

$3n-2\le f^*(3,n) \le 4n-1$.

It may be the case that understanding the situation for $d=3$ is the key for the whole problem.

There is another example achieving the lower bound that Terry found

$F_i := \{ \{a,b,c\}: a+b+c = i+2 \}$ $i=1,2,\dots 3n-2$

Various approaches to the conjecture

Several approaches to the cojecture were proposed. Using clever reccurence relations, finding useful ordering, applying the method of compression, and algebraic methods. In a series of remarks Tim is trying to prove Nicolai’s conjecture. An encouraging sign is that both examples of Nicolai, Klas, and Terry come up naturally. One way to help the project at this stage would be to try to enter Tim’s mind and find ways to help him “push the car”. In any case, if Nicolai’s conjecture is correct I see no reason why it shouldn’t have a simple proof (of course we will be happy with long proofs as well).

Constructions

Something that is also on the back of our minds is the idea to find examples that are inspired from the upper bound proofs. We do not know yet what direction is going to prevail so it is useful to remember that every proof of a weaker result and every difficulty in attempts to proof the hoped-for result can give some ideas for disproving what we are trying to prove.

Some preliminary attempts were made to examine what are the properties of examples for d=3 which will come close to the 4n bound. It may also be the case that counterexamples to Nicolai’s conjecture can be found for rather small values of n and d.

Two polls:

Polymath 3: The Polynomial Hirsch Conjecture 2

Here we start the second research thread about the polynomial Hirsch conjecture.  I hope that people will feel as comfortable as possible to offer ideas about the problem. The combinatorial problem looks simple and also everything that we know about it is rather simple: At this stage joining the project should be very easy. If you have an idea (and certainly a question or a request,) please don’t feel necessary to read all earlier comments to see if it is already there.

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.

Polymath 3: Polynomial Hirsch Conjecture

I would like to start here a research thread of the long-promised Polymath3 on the polynomial Hirsch conjecture.

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

Consider t disjoint families of subsets of {1,2,…,n}, $F_1, F_2, ..., F_t$.

Suppose that

(*) For every $i, and every $S \in F_i$ and $T \in F_k$, there is $R\in F_j$ which contains $S\cap T$.

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 $n^{\log n+1}$. 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 $f(n)$ will imply a polynomial (in $n$) upper bound for the diameter of graphs of polytopes with $n$ facets. So the task we face is either to prove such a polynomial upper bound or give an example where $t$ 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 $f(n)$ can be quadratic.

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.

Faces of Simple 4 Polytopes

In the conference celebrating Klee and Grünbaum’s mathematics at Seattle Günter Ziegler proposed the following bold conjecture about 4 polytopes.

Conjecture: A simple 4-polytope with $n$ facets has at most a linear number (in $n$)  two dimensional faces which are not 4-gons!

If the polytope is dual-to-neighborly then the number of 2-faces is quadratic in $n$. For the dual-to-cyclic polytope the assertion of the conjecture is true.