We give deterministic distributed algorithms that given δ > 0 find in a planar graph G, (1 ± δ)-approximations of a maximum independent set, a maximum matching, and a minimum dominating set. The algorithms run in O(log * |G|) rounds. In addition, we prove that no faster deterministic approximation is possible and show that if randomization is allowed it is possible to beat the lower bound for deterministic algorithms.
Abstract. Hajnal and Szemerédi proved that every graph G with |G| = ks and δ(G) ≥ k(s − 1) contains k disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graphOur result implies the Hajnal-Szemerédi Theorem, and its degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.
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