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 .
On the other hand, @cite_8 showed a lower bound of @math , even for the LOCAL model. All the algorithms mentioned earlier for finding an @math are randomized that succeed with high probability. For deterministic algorithms, in @cite_28 a @math -round algorithm is given using network decomposition, in the LOCAL model. In @cite_38 a coloring-based @math -round CONGEST algorithm is given.
@cite_28 @cite_38 @cite_8
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In this paper, we improve the bounds for computing a network decomposition distributively and deterministically. Our algorithm computes an (n?(n),n?(n))-decomposition innO(?(n))time, whereformula. As a corollary we obtain improved deterministic bounds for distributively computing several graph structures such as maximal independent sets and ?-vertex colorings. We also show that the class of graphs G whose maximum degree isnO(?(n))where ?(n)=1 log lognis complete for the task of computing a near-optimal decomposition, i.e., a (logn, logn)-decomposition, in polylog(n) time. This is a corollary of a more general characterization, which pinpoints the weak points of existing network decomposition algorithms. Completeness is to be intended in the following sense: if we have an algorithmAthat computes a near-optimal decomposition in polylog(n) time for graphs inG, then we can compute a near-optimal decomposition in polylog(n) time for all graphs. The distributed @math -coloring problem is one of the most fundamental and well-studied problems in distributed algorithms. Starting with the work of Cole and Vishkin in 1986, a long line of gradually improving algorithms has been published. The state-of-the-art running time, prior to our work, is @math , due to Kuhn and Wattenhofer [Proceedings of the @math th Annual ACM Symposium on Principles of Distributed Computing, Denver, CO, 2006, pp. 7--15]. Linial [Proceedings of the @math th Annual IEEE Symposium on Foundation of Computer Science, Los Angeles, CA, 1987, pp. 331--335] proved a lower bound of @math for the problem, and Szegedy and Vishwanathan [Proceedings of the 25th Annual ACM Symposium on Theory of Computing, San Diego, CA, 1993, pp. 201--207] provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are unlikely to achieve a running time smaller than @math . We present a de... The question of what can be computed, and how efficiently, is at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a distributed fashion. More precisely, if nodes of a network must base their decision on information in their local neighborhood only, how well can they compute or approximate a global (optimization) problem? In this paper we give the first polylogarithmic lower bound on such local computation for (optimization) problems including minimum vertex cover, minimum (connected) dominating set, maximum matching, maximal independent set, and maximal matching. In addition, we present a new distributed algorithm for solving general covering and packing linear programs. For some problems this algorithm is tight with the lower bounds, whereas for others it is a distributed approximation scheme. Together, our lower and upper bounds establish the local computability and approximability of a large class of problems, characterizing how much local information is required to solve these tasks.