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 .
Recently, @cite_36 , showed that there is an algorithm for the LOCAL model that finds a @math -approximation to @math in @math rounds, for a constant @math . The results in @cite_13 @cite_17 give a lower bound of @math rounds for any deterministic algorithm returning an independent set of size at least @math on a cycle. Furthermore, @cite_47 provide a deterministic @math algorithm, and a randomized @math rounds algorithm, for @math -approximations in planar graphs.
@cite_36 @cite_47 @cite_13 @cite_17
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We give deterministic distributed algorithms that given i¾?> 0 find in a planar graph G, (1±i¾?)-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. This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and (Δ+1)-vertex coloring), the randomized complexity is at most polylogarithmic in the size n of the network, while the best deterministic complexity is typically 2O(√logn). Understanding and potentially narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in logn rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and (Δ+1)-coloring) in logn rounds in the LOCAL model. Perhaps most surprisingly, we show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values. In addition, our formal framework also allows us to develop polylogarithmic-time randomized distributed algorithms in a simpler way. As a result, we provide a polylog-time distributed approximation scheme for arbitrary distributed covering and packing integer linear programs. Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish connections between the deterministic and randomized complexities in the LOCAL model. Each result has a very compelling take-away message: 1. Fast @math -coloring of trees requires random bits: Building on the recent lower bounds of , we prove that the randomized complexity of @math -coloring a tree with maximum degree @math is @math , whereas its deterministic complexity is @math for any @math . This also establishes a large separation between the deterministic complexity of @math -coloring and @math -coloring trees. 2. Randomized lower bounds imply deterministic lower bounds: We prove that any deterministic algorithm for a natural class of problems that runs in @math rounds can be transformed to run in @math rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires @math time deterministically. 3. Deterministic lower bounds imply randomized lower bounds: We prove that the randomized complexity of any natural problem on instances of size @math is at least its deterministic complexity on instances of size @math . This shows that a deterministic @math lower bound for any problem implies a randomized @math lower bound. It also illustrates that the graph shattering technique is absolutely essential to the LOCAL model. We present a simple distributed @math -approximation algorithm for maximum weight independent set (MaxIS) in the @math model which completes in @math rounds, where @math is the maximum degree, @math is the number of rounds needed to compute a maximal independent set (MIS) on @math , and @math is the maximum weight of a node. Whether our algorithm is randomized or deterministic depends on the MIS algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in @math rounds, where @math is the number of nodes. We also present a deterministic @math -round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a @math -approximation for maximum weight matching without incurring any additional round penalty in the @math model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of for which we describe a mechanism that allows the simulation to run in the @math model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ( @math ) allows us to devise a distributed algorithm requiring @math rounds for any constant @math . For the unweighted case, we can even obtain a @math -approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on @math .