104 research outputs found

    FPT algorithms for finding near-cliques in c-closed graphs

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    Finding large cliques or cliques missing a few edges is a fundamental algorithmic task in the study of real-world graphs, with applications in community detection, pattern recognition, and clustering. A number of effective backtracking-based heuristics for these problems have emerged from recent empirical work in social network analysis. Given the NP-hardness of variants of clique counting, these results raise a challenge for beyond worst-case analysis of these problems. Inspired by the triadic closure of real-world graphs, Fox et al. (SICOMP 2020) introduced the notion of c-closed graphs and proved that maximal clique enumeration is fixed-parameter tractable with respect to c. In practice, due to noise in data, one wishes to actually discover “near-cliques”, which can be characterized as cliques with a sparse subgraph removed. In this work, we prove that many different kinds of maximal near-cliques can be enumerated in polynomial time (and FPT in c) for c-closed graphs. We study various established notions of such substructures, including k-plexes, complements of bounded-degeneracy and bounded-treewidth graphs. Interestingly, our algorithms follow relatively simple backtracking procedures, analogous to what is done in practice. Our results underscore the significance of the c-closed graph class for theoretical understanding of social network analysis

    A Fast Maximum kk-Plex Algorithm Parameterized by the Degeneracy Gap

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    Given a graph, the kk-plex is a vertex set in which each vertex is not adjacent to at most k1k-1 other vertices in the set. The maximum kk-plex problem, which asks for the largest kk-plex from a given graph, is an important but computationally challenging problem in applications like graph search and community detection. So far, there is a number of empirical algorithms without sufficient theoretical explanations on the efficiency. We try to bridge this gap by defining a novel parameter of the input instance, gk(G)g_k(G), the gap between the degeneracy bound and the size of maximum kk-plex in the given graph, and presenting an exact algorithm parameterized by gk(G)g_k(G). In other words, we design an algorithm with running time polynomial in the size of input graph and exponential in gk(G)g_k(G) where kk is a constant. Usually, gk(G)g_k(G) is small and bounded by O(log(V))O(\log{(|V|)}) in real-world graphs, indicating that the algorithm runs in polynomial time. We also carry out massive experiments and show that the algorithm is competitive with the state-of-the-art solvers. Additionally, for large kk values such as 1515 and 2020, our algorithm has superior performance over existing algorithms.Comment: IJCAI'202
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