ABSTRACT:We study biased Maker/Breaker games on the edges of the complete graph, as introduced by Chvátal and Erdős. We show that Maker, occupying one edge in each of his turns, can build a spanning tree, even if Breaker occupies bedges in each turn. This improves a result of Beck, and is asymptotically best possible as witnessed by the Breaker-strategy of Chvátal and Erdős. We also give a strategy for Maker to occupy a graph with minimum degree c (where c = c(n) is a slowly growing function of n) while playing against a Breaker who takes b ≤ (1 − o(1)) · n ln n edges in each turn. This result improves earlier bounds by Krivelevich and Szabó. Both of our results support the surprising random graph intuition: the threshold bias is asymptotically the same for the game played by two "clever" players and the game played by two "random" players.
We study a Maker/Breaker game described by Beck. As a result we disprove a conjecture of Beck on positional games, establish a connection between this game and SAT 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.The Maker/Breaker game we study is as follows. Maker and Breaker take turns in choosing vertices from a given n-uniform hypergraph F , with Maker going first. Maker's goal is to completely occupy a hyperedge and Breaker tries to avoid this. Beck conjectures that if the maximum neighborhood size of F is at most 2 n−1 then Breaker has a winning strategy. We disprove this conjecture by establishing an n-uniform hypergraph with maximum neighborhood size 3 · 2 n−3 where Maker has a winning strategy. Moreover, we show how to construct an n-uniform hypergraph with maximum degree 2 n−1 n where Maker has a winning strategy. In addition we show that each n-uniform hypergraph with maximum degree at most 2 n−2 en has a proper halving 2-coloring, which solves another open problem posed by Beck related to the Neighborhood Conjecture.Finally, we establish a connection between SAT and the Maker/Breaker game we study. We can use this connection to derive new results in SAT. A (k, s)
The Local Lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science, yet there are hardly any natural problems known where it provides an asymptotically tight answer. The main theme of our paper is to identify several of these problems, among them a couple of widely studied extremal functions related to certain restricted versions of the k-SAT problem, where the Local Lemma does give essentially optimal answers.As our main contribution, we construct unsatisfiable k-CNF formulas where every clause has k distinct literals and every variable appears in at most * A preliminary version of this paper has appeared as an extended abstract in the Proceedings of SODA 2011 known bounds on the maximum degree and maximum edge-degree of a kuniform Maker's win hypergraph in the Neighborhood Conjecture of Beck.
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. Advice complexity is a recently introduced method for a fine-grained analysis of the hardness of online problems. We consider the advice complexity of disjoint path allocation, measured in the length L of the path. We show that asking for a bit of advice for every edge is necessary to be optimal and give online algorithms with advice achieving a constant competitive ratio using much less advice. Furthermore, we consider the case of using less than log log L advice bits, where we prove almost matching lower and upper bounds on the competitive ratio. In the latter case, we moreover show that randomness is as powerful as advice by designing a barely random online algorithm achieving almost the same competitive ratio.
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