Category Archives: Convex polytopes

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

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.

Thinking about multisets (monomials).

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

More examples, please…

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<j<k, 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.

polymath3

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.

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.

Continue reading

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.