FPT Algorithms for Finding Near-Cliques in $c$-Closed Graphs

TL;DR

This paper introduces fixed-parameter tractable algorithms for enumerating near-cliques in c-closed graphs, providing theoretical foundations for analyzing noisy real-world social networks.

Contribution

It proves polynomial-time enumeration of various near-cliques in c-closed graphs, extending the fixed-parameter tractability results to noisy and approximate clique structures.

Findings

Polynomial-time enumeration of near-cliques in c-closed graphs.

Algorithms are simple backtracking procedures similar to practical heuristics.

Highlights the importance of c-closed graphs in social network analysis.

Abstract

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.