We present improved results for approximating Maximum Independent Set ( @math ) in the standard LOCAL and CONGEST models of distributed computing. Let @math and @math be the number of nodes and maximum degree in the input graph, respectively. Bar- [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a @math -approximation to @math in @math rounds, where @math is the running time for finding a independent set, and @math is the maximum weight of a node in the network. Whether their algorithm is randomized or deterministic depends on the @math algorithm that they use as a black-box. Our results: (1) A deterministic @math rounds algorithm for @math -approximation to @math in the CONGEST model. (2) A randomized @math rounds algorithm that finds, with high probability, an @math -approximation to @math in the CONGEST model. (3) An @math lower bound for any randomized algorithm that finds an independent set of size @math that succeeds with probability at least @math , even for the LOCAL model. This hardness result applies for graphs of maximum degree @math . One might wonder whether the same hardness result applies for low degree graphs. We rule out this possibility with our next result. (4) An @math rounds algorithm that finds an independent set of size @math in graphs with maximum degree @math , with high probability. Due to a lower bound of @math that was given by Kuhn, Moscibroda and Wattenhofer [JACM, 2016] on the number of rounds for finding a maximal independent set ( @math ) in the LOCAL model, even for randomized algorithms, our second result implies that finding an @math -approximation to @math is strictly easier than @math .
In the sequential setting, an excellent summary of the known results is given by @cite_11 , which we overview in what follows. For general graphs, the best known algorithm achieves an @math -approximation factor @cite_18 . Assuming @math , @cite_29 shows that there is no efficient @math -approximation algorithm for every constant @math .
@cite_18 @cite_29 @cite_11
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We show an algorithm that finds cliques of size (log n log log n)2 whenever a graph has a clique of size at least n (log n)b for an arbitrary constant b. This leads to an algorithm that approximates max clique within a factor of O(n(log log n)2 (log n)3), which matches the best approximation ratio known for the chromatic number. The previously best approximation ratio known for max clique was O(n (log n)2). We consider the maximum independent set problem on sparse graphs with maximum degree d. The best known result for the problem is an SDP based O(d log log d log d) approximation due to Halperin. It is also known that no o(d log2 d) approximation exists assuming the Unique Games Conjecture. We show the following two results: (i) The natural LP formulation for the problem strengthened by O(log4(d)) levels of the mixed-hierarchy has an integrality gap of O(d log2 d), where O(·) ignores some log log d factors. However, our proof is non-constructive, in particular it uses an entropy based approach due to Shearer, and does not give a O(d log2 d) approximation algorithm with sub-exponential running time. (ii) We give an O(d log d) approximation based on polylog(d)-levels of the mixed hierarchy that runs in nO(1) exp(logO(1) d) time, improving upon Halperin's bound by a modest log log d factor. Our algorithm is based on combining Halperin's approach together with an idea used by Ajtai, Erdos, Komlos and Szemeredi to show that Kr-free, degree-d graphs have independent sets of size Ωr(n log log d d).