Heidi Gebauer

Disproof of the Neighborhood Conjecture with Implications to SAT

We study a special class of binary trees. Our results have implications on Maker/Breaker games and SAT: We disprove a conjecture of Beck on positional games and construct an unsatisfiable k-CNF formula with few occurrences per variable, thereby improving a previous result by Hoory and Szeider and showing that the bound obtained from the Lovasz Local Lemma is tight up to a constant factor.

On the Power of Advice and Randomization for the Disjoint Path Allocation Problem

In the disjoint path allocation problem, we consider a path of L + 1 vertices, representing the nodes in a communication network. Requests for an unbounded-time communication between pairs of vertices arrive in an online fashion and a central authority has to decide which of these calls to admit. The constraint is that each edge in the path can serve only one call and the goal is to admit as many calls as possible.

The Lovász Local Lemma and Satisfiability

We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovasz Local Lemma, a probabilistic lemma with many applications in combinatorics. In positive terms, we are interested in simple combinatorial conditions which guarantee for a CNF formula to be satisfiable. The criteria obtained are nontrivial in the sense that even though they are easy to check, it is by far not obvious how to compute a satisfying assignment efficiently in case the conditions are fulfilled; until recently, it was not known how to do so. It is also remarkable that while deciding satisfiability is trivial for formulas that satisfy the conditions, a slightest relaxation of the conditions leads us into the territory of NP-completeness. Several open problems remain, some of which we mention in the concluding section.

On the construction of 3-chromatic hypergraphs with few edges

Abstract The minimum number m ( n ) of edges in a 3-chromatic n-uniform hypergraph has been widely studied in the literature. The best known upper bound is due to Erdős, who showed, using the probabilistic method, that m ( n ) ⩽ O ( n 2 2 n ) . Abbott and Moser gave an explicit construction of a 3-chromatic n-uniform hypergraph with at most ( 7 ) n ≈ 2.65 n hyperedges, which is the best known constructive upper bound. In this paper we improve this bound to 2 ( 1 + o ( 1 ) ) n . Our technique can also be used to describe n-uniform hypergraphs with chromatic number at least r + 1 and at most r ( 1 + o ( 1 ) ) n hyperedges, for every r ⩾ 3 .

Disproof of the neighborhood conjecture with implications to SAT

AbstractWe study a special class of binary trees. Our results have implications on Maker/Breaker games and SAT: We disprove a conjecture of Beck on positional games and construct an unsatisfiable k-CNF formula with few occurrences per variable, thereby improving a previous result by Hoory and Szeider and showing that the bound obtained from the Lovász Local Lemma is tight up to a constant factor.A (k, s)-CNF formula is a boolean formula in conjunctive normal form where every clause contains exactly k distinct literals and every variable occurs in at most s clauses. The (k, s)-SAT problem is the satisfiability problem restricted to (k, s)-CNF formulas. Kratochvíl, Savický and Tuza showed that for every k≥3 there is an integer f(k) such that every (k, f(k))-CNF formula is satisfiable, but (k, f(k) + 1)-SAT is already NP-complete (it is not known whether f(k) is computable). Kratochvíl, Savický and Tuza also gave the best known lower bound

The Random Graph Intuition for the Tournament Game

f(k) = \Omega \left( {\tfrac{{2^k }} {k}} \right)

FAST EXPONENTIAL-TIME ALGORITHMS FOR THE FOREST COUNTING AND THE TUTTE POLYNOMIAL COMPUTATION IN GRAPH CLASSES

, which is a consequence of the Lovász Local Lemma. We prove that, in fact,