We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires @math rounds in the worst case in the CONGEST model, as well as any algorithm for @math -coloring a graph, where @math is the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of rounds. Finally, we address the problem of computing an exact solution to weighted all-pairs-shortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple @math lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a complexity of @math . We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question.
Distance computation problems have also been considered in the CONGESTED-CLIQUE model @cite_58 @cite_54 @cite_4 , in which the underlying communication network forms a clique. In this model @cite_54 showed that unweighted APSP, and a @math -approximation for weighted APSP, can be computed in @math rounds.
@cite_54 @cite_58 @cite_4
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In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an @math round matrix multiplication algorithm, where @math is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in @math rounds, improving upon the @math triangle detection algorithm of [DISC 2012], -- a @math -approximation of all-pairs shortest paths in @math rounds, improving upon the @math -round @math -approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in @math rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles. We study the broadcast version of the CONGEST CLIQUE model of distributed computing. In this model, in each round, any node in a network of size @math can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC'14] a randomized @math -approximation algorithm to compute all pairs shortest paths (APSP) in time @math on weighted graphs, where we use the convention that @math is essentially @math polylog @math and @math is essentially @math polylog @math . We complement this result by proving that any randomized @math -approximation of APSP and @math -approximation of the diameter of a graph takes @math time in the worst case. This demonstrates that getting a negligible improvement in the approximation factor requires significantly more time. Furthermore this bound implies that already computing a @math -approximation of all pairs shortest paths is among the hardest graph-problems in the broadcast-version of the CONGEST CLIQUE model and contrasts a recent @math -approximation for APSP that runs in time @math in the unicast version of the CONGEST CLIQUE model. On the positive side we provide a deterministic version of Nanongkai's @math -approximation algorithm for APSP. To do so we present a fast deterministic construction of small hitting sets. We also show how to replace another randomized part within Nanongkai's algorithm with a deterministic source-detection algorithm designed for the CONGEST model presented by Lenzen and Peleg at PODC'13.