Further algorithmic aspects of the local lemma
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Improved bounds and algorithms for hypergraph 2-coloring
Random Structures & Algorithms
Efficient proper 2-coloring of almost disjoint hypergraphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Improved algorithmic versions of the Lovász Local Lemma
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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Using the symmetric form of the Lovász Local Lemma, one can conclude that a k-uniform hypergraph $\mathcal{H}$ admits a proper 2-colouring if the maximum degree (denoted by Δ) of $\mathcal{H}$ is at most $\frac{2^k}{8k}$ independently of the size of the hypergraph. However, this argument does not give us an algorithm to find a proper 2-colouring of such hypergraphs. We call a hypergraph linearif no two hyperedges have more than one vertex in common.In this paper, we present a deterministic polynomial time algorithm for 2-colouring every k-uniform linear hypergraph with $\Delta \le 2^{k-k^{\epsilon}}$, where 1/2 茂戮驴kis larger than a certain constant that depends on 茂戮驴. The previous best algorithm for 2-colouring linear hypergraphs is due to Beck and Lodha [4]. They showed that for every 茂戮驴 0 there exists a c 0 such that every linear hypergraph with Δ≤ 2k茂戮驴 茂戮驴kand$k c\log\log(|E(\mathcal{H})|)$, can be properly 2-coloured deterministically in polynomial time.