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.

 

IPAM remote blogging: The Many Facets of Linear Programming

The many facets of Linear Programming

Here is an extremely nice paper by Michael Todd from 2001. It gives useful background for many lectures and it can serve as a good base point to examine last decade’s progress.

Background post for today’s morning session talks: Is Backgammon in P?

And here is a link to Ye’s paper giving a strongly polynomial algorithm for Markov decision processes with constant discounting. Update: The slides for Ye’s lecture are now posted and they are very detailed and clear.

Günter Ziegler: 1000$ from Beverly Hills for a Math Problem. (IPAM remote blogging.)

Scanned letter by Zadeh. (c) Günter M. Ziegler

left-to-right: David Avis, Norman Zadeh,  Oliver Friedmann, and Russ Caflish (IPAM director). Photo courtesy Eddie Kim.

Update: The slides for Friedmann’s talk are now available. The conference schedule page contains now the slides for most presentations.

This post is authoured by Günter Ziegler with some help by David Avis. A German (slightly expanded) version can be found on Günter’s blog.

Oliver Friedmann, a Computer Science graduate student at Munich University, will defend his thesis next month. It contains a proof, that “Zadeh’s rule” for linear optimization is “exponential”, that is, it may take an awfully long time on relatively small problems.

 

Why is this remarkable?

“Linear Optimization” problems are extremely important computational tasks that arise in all kinds of larger planning processes. Such tasks are usually solved on a computer using a method known as the “simplex algorithm”, invented by George Dantzig in 1947. The simplex method needs a prescription for making local choices in its search, known as the “pivot rule”. And it is known about mostsuch pivot rules that they can be extremely slow and inefficient on particular optimization problems. One would like (and need) a rule that is “provably fast”.

One candidate for this was the “Zadeh rule” — until today. When Zadeh was a postdoc in the Department of Operations Research at Stanford he published a technical report on the worst case examples of the simplex method. Continue reading

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.

Continue reading

The Polynomial Hirsch Conjecture: A proposal for Polymath3

pball

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 <t 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!

Continue reading

A Diameter Problem (6): Abstract Objective Functions

Dantzig and Khachyan

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 G(P) of a d-polytope P with n facets is at most n-d. Hirsch, in fact, made the conjecture also for graphs of unbounded polyhedra – namely the intersection of n closed halfspaces in R^d. But in the unbounded case, Klee and Walkup found a counterexample with diameter n-d+ [d/5]. The abstract problem we considered extends also to the unbounded case and  n-d+ [d/5] is the best known lower bound for the abstract case as well. It is not known if there is a polynomial (in terms of d and n) 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 \phi(x)=b_1x_1+ b_2 x_2 \dots +b_dx_d subject to a system of n linear inequalities in the variables x_1,x_2,\dots,x_d.

 a_{11}x_1 + x_{12}x_2 + \dots + x_{1d}x_d \le c_1,

a_{21}x_1 + x_{22}x_2 + \dots + x_{2d}x_d \le c_2,

a_{n1}x_1 + x_{n2}x_2 + \dots + x_{nd}x_d \le c_n,

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 P. (This polyhedron is called the feasible polyhedron.) The maximum of \phi is attained at a face F of P. We start with an initial vertex (extreme point) v of the polyhedron and look at its neighbors in G(P). Unless v \in F there is a neighbor u of v that satisfies \phi(u) > \phi (v). When you find such a vertex move from v to u and repeat!

8. Abstract objective functions and unique sink orientation.

Let P be a simple d-polytope and let \phi be a linear objective function which is not constant on any edge of the polytope. Remember, the graph of P, G(P) is a d-regular graph. We can now direct every edge u,v from u to v if \phi (v) > \phi (u). 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 F and its own graph G(F) then again our linear objective function induces an orientation of the edges of G(F) 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 F 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 d-polytope is an ordering < of the vertices of the polytope such that the directed graph obtained by directing an edge from u to v if  u<v is a unique sink acyclic orientation. (Of course, coming from an ordering the orientation is automatically acyclic.) 

Continue reading

Diameter Problem (3)

3. What we will do in this post and and in future posts

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 \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.

Continue reading