Combinatorics Seminar 2010/2011

Archive of old talks. The current schedule can be found here.

Date Speaker Title
13.07.2011 Kevin Milans
University of South Carolina
 
06.07.2011 Uri Zwick
Tel Aviv University
All-Pairs Shortest Paths in O(n2) Expected Time
29.06.2011 Benny Sudakov
UCLA
Nonnegative k-sums, fractional covers, and probability of small deviations
22.06.2011 Hanno Lefmann
TU Chemnitz
Edge Colorings of Graphs and Fixed Forbidden Monochromatic Subgraphs
15.06.2011 Anita Liebenau
FU Berlin
A survey on the 'Cops-and-robbers problem'
09.06.2011 Endre Szemerédi
Rényi Institute of Mathematics
Long Arithmetic Progression in Sumsets
20.05.2011 Timothy Gowers
University of Cambridge
A combinatorial proof of the density Hales-Jewett theorem
13.05.2011 Dan Spielman
Yale University
On some problems that have been bugging me
12.05.2011 Dennis Clemens
FU Berlin
The uniqueness conjecture of Markoff numbers and equivalent problems: Part II
04.05.2011 Dennis Clemens
FU Berlin
The uniqueness conjecture of Markoff numbers and equivalent problems: Part I
24.02.2011 Fiona Skerman
Australian National University
Row and column sums of random 0-1 matrices
23.02.2011 Katherine Edwards
McGill University
Packing T-joins in Planar Graphs
16.02.2011 Oliver Friedman
LMU München
On Exponential Lower Bounds for Solving Infinitary Payoff Games and Linear Programs: Part II
28.01.2011 Dmitry Shabanov
Moscow State University
The Problem of Erdős and Hajnal Concerning Colorings of Hypergraphs and its Generalizations
21.01.2011 Tibor Szabó
FU Berlin
The Local Lemma is tight for SAT II
14.01.2011 Tibor Szabó
FU Berlin
Gábor Tardos
Rényi Institute of Mathematics
The Oberwolfach Problems
17.12.2010 Tibor Szabó
FU Berlin
The Local Lemma is tight for SAT
10.12.2010 Yury Person
FU Berlin
Extremal Hypergraphs for Hamilton Cycles
03.12.2010 Michael Krivelevich
Tel Aviv University
The Size Ramsey Number of a Directed Path
04.10.2010 - 26.11.2010 (Pre-)Doc-Course

Abstracts:

06.07.2011
Uri Zwick (Tel Aviv University)
All-Pairs Shortest Paths in O(n2) Expected Time

Abstract: We present an All-Pairs Shortest Paths (APSP) algorithm whose expected running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from [0, 1] is O(n2). This resolves a long standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano. The analysis relies on a proof that the expected number of locally shortest paths in such randomly weighted graphs is O(n2). We also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in O(log2 n) expected time.

Joint work with Yuval Peres, Dmitry Sotnikov and Benny Sudakov.
29.06.2011
Benny Sudakov (UCLA)
Nonnegative k-sums, fractional covers, and probability of small deviations

Abstract: More than twenty years ago, Manickam, Miklos, and Singhi conjectured that for any integers n ≥ 4k, every set of n real numbers with nonnegative sum has at least (n - 1 choose k - 1) k-element subsets whose sum is also nonnegative. In this talk we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for n ≥ 33k2. This substantially improves the best previously known exponential lower bound on n.

If time permits we will also discuss applications to finding the optimal data allocation for a distributed storage system and how our approach can be used to prove some theorems on minimum degree ensuring the existence of perfect matchings in hypergraphs.

Joint work with N. Alon and H. Huang, last part is also joint with P. Frankl, V. Rodl and A. Rucinski.
22.06.2011
Hanno Lefmann (TU Chemnitz)
Edge Colorings of Graphs and Fixed Forbidden Monochromatic Subgraphs

Abstract: Given a fixed positive integer r and a fixed graph F, we consider for host-graphs H on n vertices, n large, the number cr,F(H) of r-colorings of the set of edges of H such that no monochromatic copy of F arises. In particular, we are looking at the maximum cr,F(n) of cr,F(H) over all host-graphs H on n vertices. For forbidden fixed complete graphs F = K the quantity cr,F(n) has been investigated by Yuster, and Alon et al. and they proved for r = 2 or r = 3 colors that the maximum number of r-colorings is achieved by the corresponding Turán graph for F, while for r ≥ 4 colors this does not hold anymore. Based on similar results for some special forbidden hypergraphs, one might suspect that such a phaenomenon holds in general.

However, it seems this is not valid for (at least some) classes of forbidden bipartite graphs.
15.06.2011
Anita Liebenau (FU Berlin)
A survey on the 'Cops-and-robbers problem'

Abstract: Cops and robbers - in theory almost as thrilling as in an action film. The game was introduced by Nowakowski and Winkler, and by Quilliot more than thirty years ago: A number of cops moves along the edges of a graph and try to catch the robber. First, the cops move (at most) one step, then the robber. The cop number c(G), introduced by Aigner and Fromme in 1982, is the minimal number of cops needed in G in order to catch the robber. Numerous questions connected with this game have been posed and partly answered:

What kind of graphs can be covered by one cop alone (the so-called cop-win graphs)?

How many cops are enough to catch the robber on a planar graph?

What can we say about graphs embeddable in orientable surfaces of genus g?

How many cops do you need/are enough on general graphs?

Variations include a drunken robber (who moves randomly, and not in an intelligent way), a very fast robber (who may move along longer paths in one step), and non-perfect information.

In the talk, I will define all necessary concepts, and give an overview of what has been done in these last three decades. Further, I will sketch a proof by Scott and Sudakov (2011) of one of the most recent upper bounds on c(G).
09.06.2011
Endre Szemerédi (Rényi Institute of Mathematics)
Long Arithmetic Progression in Sumsets
Abstract: We are going to give exact bound for the size of longest arithmetic progression in sumset sums. In addition we describe the structure of the subset sums and give applications in number theory and probability theory. (this part is partially joint work with Van Vu)
04.05.2011, 12.05.2011
Dennis Clemens (FU Berlin)
The uniqueness conjecture of Markoff numbers and equivalent problems

Abstract: In 1879/80 Andrei A. Markoff studied the minima of indefinite quadratic forms and found an amazing relationship to the Diophantine equation x2 + y2 + z2 = 3xyz, which today is named the Markoff equation. Later Georg Frobenius conjectured that each solution of this equation is determined uniquely by its maximal component. Defining these components as the so-called Markoff numbers this conjecture means that for each Markoff number m there exists up to permutation exactly one triple (m, p, q) with maximum m solving the Markoff equation.

Till this day it is not known whether the conjecture is true, or not. However, we are able to prove uniqueness for certain Markoff numbers and we can find different statements, such as a statement in Markoff's theory on the minima of quaratic forms, being equivalent to the uniqueness conjecture of the Markoff numbers.

At first I will give a short overview on the properties of Markoff numbers as well as some easier results concerning the uniqueness conjecture. We will see how the solutions of Markoff's equation, called Markoff triples, can be computed recursively such that we will get a tree of infinitely many Markoff triples. Then, by using correspondences to other trees we will get different statements from Number Theory, Graph Theory and Linear Algebra being equivalent to the uniqueness conjecture.

The second part of my presentation will be concerned with the best known result on this conjecture proven by J. O. Button in 2001. With a bijection between Markoff triples and elements in certain number fields we will see how the uniqueness problem can be reformulated as a question in Ideal Theory. Taking results on ideals and continued fractions we finally can conclude a criterion proving uniqueness for a big subset of Markoff numbers.
24.02.2011
Fiona Skerman (Australian National University)
Row and column sums of random 0-1 matrices

Abstract: Construct a random m × n matrix by independently setting each entry to 1 with probability p and to 0 otherwise. We study the joint distribution of the row sums s = (s1, ..., sm) and column sums t = (t1, ..., tn). Clearly s and t have the same sum, but otherwise their dependencies are complicated. We prove that under certain conditions the distribution of (st) is accurately modelled by (S1, ..., SmT1, ..., Tn), where each Sj has the binomial distribution Binom(np'), each Tk has the binomial distribution Binom(mp'), p' is drawn from a truncated normal distribution, and S1, ..., Sm, T1, ..., Tn are independent apart from satisfying Σj=1,...,m Sj = Σk=1,...,n Tk. We also consider the case of random 0-1 matrices where only the number of 1s is specified, and also the distribution of s when t is specified. In the seminar I will include details of one of the bounding arguments used in this last case. This bounding argument is an application of the generalised Doob's martingale process.

These results can also be expressed in the language of random bipartite graphs.

Joint work with Brendan McKay.
23.02.2011
Katherine Edwards (McGill University)
Packing T-joins in Planar Graphs
Abstract: Let G be a graph and T an even sized subset of its vertices. A T-join is a subgraph of G whose odd-degree vertices are precisely those in T, and a T-cut is a cut δ(S) where S contains an odd number of vertices of T. It has been conjectured by Guenin that if all T-cuts of G have the same parity and the size of every T-cut is at least k, then G contains k edge-disjoint T-joins. We discuss some recent progress on this conjecture and related results.
16.02.2011
Oliver Friedman (LMU München)
On Exponential Lower Bounds for Solving Infinitary Payoff Games and Linear Programs: Part II

Abstract: Policy iteration is one of the most important algorithmic schemes for solving problems in the domain of determined game theory such as parity games, stochastic games and Markov decision processes, and many more. It is parameterized by an improvement rule that determines how to proceed in the iteration from one policy to the next. It is a major open problem whether there is an improvement rule that results in a polynomial time algorithm for solving one of the considered game classes.

Simplex algorithms for solving linear programs are closely related to policy iteration algorithms. Like policy iteration, the simplex algorithm is parameterized by a so-called pivoting rule that describes how to proceed from one basic feasible solution in the linear program to the next. Also, it is a major open problem whether there is a pivoting rule that results in a (strongly) polynomial time algorithm for solving linear programs. We describe our recent constructions for parity games that give rise to superpolynomial and exponential lower bounds for all major improvement rules, and how to extend these lower bounds to more expressive game classes like stochastic games. We show that our constructions for parity games can be translated to Markov decision processes, transferring our lower bounds to their domain, and finally show how the lower bounds for the MDPs can be transferred to the linear programming domain, solving problems that have been open for several decades.
28.01.2011
Dmitry Shabanov (Moscow State University)
The Problem of Erdős and Hajnal Concerning Colorings of Hypergraphs and its Generalizations
Abstract: The talk is devoted to the classical exremal combinatorial problem that was stated by P.Erdős and A.Hajnal in the 60-s. The task is to find the value m(n, r) equal to the minimum number of edges in an n-uniform non-r-colorable hypergraph. This problem has a lot of generalizations and is closely related to the classical problems of Ramsey theory. We shall discuss the known bounds in the Erdős-Hajnal problem, present some new results and pay special attention to the probabilistic methods, by which these results have been obtained.
14.01.2011
Tibor Szabó (FU Berlin) & Gabór Tardos (Rényi Institute of Mathematics)
The Oberwolfach Problems
Abstract: We plan to state and initiate an informal discussion on (an admittedly subjective selection of) the open problems raised in last week's Combinatorics workshop.
21.01.2011
Tibor Szabó (FU Berlin)
The Local Lemma is tight for SAT II
Abstract: I will give a construction of unsatisfiable k-SAT formulas, where each variable is contained in only a few clauses. The numerical value of "few" is asymptotically best possible. Joint work with Heidi Gebauer and Gabór Tardos.
17.12.2010
Tibor Szabó (FU Berlin)
The Local Lemma is tight for SAT
Abstract: We construct unsatisfiable k-CNF formulas where every clause has k distinct literals and every variable appears in at most (2/e + o(1))2k/k clauses. The lopsided Loca Lemma shows that our result is asymptotically best possible. The determination of this extremal function is particularly important as it represents the value where the k-SAT problem exhibits its complexity hardness jump: from having every instance being a YES-instance it becomes NP-hard just by allowing each variable to occur in one more clause. We also consider the related extremal function l(k) which denotes the maximum number, such that every k-CNF formula with each clause containing k distinct literals and each clause having a common variable with at most l(k) other clauses, is satisfiable. We establish that l(k) = (1/e + o(1))2k The SAT-formulas are constructed via special binary trees. In order to construct the trees a continuous setting of the problem is defined, giving rise to a differential equation. The solution at 0 diverges, which in turn implies that the binary tree obtained from the discretization of this solution has the required properties. Joint work with Heidi Gebauer and Gabór Tardos.
10.12.2010
Yury Person (FU Berlin)
Extremal Hypergraphs for Hamilton Cycles
Abstract: We study sufficient conditions for various Hamilton cycles in k-uniform hypergraphs and obtain both Turán- and Dirac-type results. In particular, we show that the only extremal 3-uniform hypergraph (for n moderately large) not containing a loose Hamilton cycle on n vertices consists of the complete hypergraph on n - 1 vertices and an isolated vertex (thus answering a question of Woitas). More generally, we determine extremal hypergraphs for so-called l-tight Hamilton cycles and we give first sufficient conditions on the minimum degree δ of type c(n choose k - 1), with fixed c < 1 and n sufficiently large, that ensure the existence of Hamilton cycles. Joint work with Roman Glebov and Wilma Weps.
03.12.2010
Michael Krivelevich (Tel Aviv University)
The Size Ramsey Number of a Directed Path

Abstract: Given a (di)graph H and an integer q ≥ 2, the size Ramsey number re(Hq) is the minimal number m for which there is a (di)graph G with m edges such that every q-coloring of G contains a monochromatic copy of H. We study the size Ramsey number of the directed path of length n in oriented graphs, where no antiparallel edges are allowed. We give nearly tight bounds for every fixed number of colors, showing that for every q ≥ 2, the corresponding number re(Hq) has asymptotic order of magnitude n2q - 2 + o(1).

A joint work with Ido Ben-Eliezer (Tel Aviv U.) and Benny Sudakov (UCLA).