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

Authors Balaram Behera, Edin Husić , Shweta Jain, Tim Roughgarden, C. Seshadhri

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Balaram Behera
  • Georgia Institute of Technology, Atlanta, GA, USA
Edin Husić
  • London School of Economics and Political Science, UK
Shweta Jain
  • University of Illinois, Urbana-Champaign, IL, USA
Tim Roughgarden
  • Columbia University, New York, NY, USA
C. Seshadhri
  • University of California, Santa Cruz, CA, USA


We would like to thank anonymous referees for their comments and suggestions.

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Balaram Behera, Edin Husić, Shweta Jain, Tim Roughgarden, and C. Seshadhri. FPT Algorithms for Finding Near-Cliques in c-Closed Graphs. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 17:1-17:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Social networks
  • c-closed graph
  • dense subgraphs
  • FPT algorithm
  • enumeration algorithm
  • k-plex
  • Moon-Moser theorem


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