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.
Another classical problem in graph theory is finding a minimum vertex cover (MVC). In distributed computing, the time complexity of approximating MVC has been addressed in several cornerstone studies @cite_20 @cite_5 @cite_1 @cite_2 @cite_3 @cite_9 @cite_50 @cite_13 @cite_65 @cite_25 @cite_38 @cite_45 @cite_17 .
@cite_38 @cite_9 @cite_1 @cite_65 @cite_3 @cite_45 @cite_50 @cite_2 @cite_5 @cite_13 @cite_25 @cite_20 @cite_17
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Discusses time complexity analysis of the minimum vertex cover and minimum connected vertex cover problems for 3-connected graphs. A vertex cover of a graph G=(V, E) is a subset N of V such that each element of E is incident upon some element of N, where V and E are the sets of vertices and of edges of G, respectively. A connected vertex cover of a graph G is a vertex cover of G such that the subgraph G(N) induced by N of G is connected. The minimum vertex cover problem (VCP) is the problem of finding a vertex cover of minimum cardinality, and the minimum connected vertex cover problem is similarly defined. > Given a simple undirected graph @math and a positive integer @math , the maximum vertex coverage problem (MVC) is the problem of finding a set @math of @math vertices of @math such that the number of edges having at least one endpoint in @math is as large as possible. The problem is NP-hard even in bipartite graphs, as shown in two recent papers [N. Apollonio and B. Simeone, Discrete Appl. Math., 165 (2014), pp. 37--48; G. Joret and A. Vetta, Reducing the Rank of a Matroid, preprint, arXiv:1211.4853v1 [cs.DS], 2012]. By exploiting the structure of the fractional optimal solutions of a linear programming formulation for the maximum coverage problem, we provide a @math -approximation algorithm for the problem. The algorithm immediately extends to the weighted version of MVC. Abstract We consider the max-vertex-cover (MVC) problem, i.e., find k vertices from an undirected and edge-weighted graph G =( V , E ), where | V |= n ⩾ k , such that the total edge weight covered by the k vertices is maximized. There is a 3 4-approximation algorithm for MVC, based on a linear programming relaxation. We show that the guaranteed ratio can be improved by a simple greedy algorithm for k >(3 4) n , and a simple randomized algorithm for k >(1 2) n . Furthermore, we study a semidefinite programming (SDP) relaxation based approximation algorithms for MVC. We show that, for a range of k , our SDP-based algorithm achieves the best performance guarantee among the four types of algorithms mentioned in this paper. We present a distributed 2-approximation algorithm for the minimum vertex cover problem. The algorithm is deterministic, and it runs in (Δ + 1)2 synchronous communication rounds, where Δ is the maximum degree of the graph. For Δ = 3, we give a 2-approximation algorithm also for the weighted version of the problem.