104 research outputs found
FPT algorithms for finding near-cliques in c-closed graphs
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 -Plex Algorithm Parameterized by the Degeneracy Gap
Given a graph, the -plex is a vertex set in which each vertex is not
adjacent to at most other vertices in the set. The maximum -plex
problem, which asks for the largest -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,
, the gap between the degeneracy bound and the size of maximum -plex
in the given graph, and presenting an exact algorithm parameterized by
. In other words, we design an algorithm with running time polynomial
in the size of input graph and exponential in where is a constant.
Usually, is small and bounded by 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 values such as and
, our algorithm has superior performance over existing algorithms.Comment: IJCAI'202
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