On the tractability of finding disjoint clubs in a network

Abstract We study a variant of the problem of finding a collection of disjoint s-clubs in a given network. Given a graph, the problem asks whether there exists a collection of at most r disjoint s-clubs that covers at least k vertices of the network. An s-club is a connected graph that has diameter bounded by s, for a positive integer s. We demand that each club is non-trivial, that is it has order at least t ≥ 2 , for some positive integer t. We prove that the problem is APX-hard even when the input graph has bounded degree, s = 2 , t = 3 and r = | V | . Moreover, we show that the problem is polynomial-time solvable when s ≥ 4 , t = 3 and r = | V | , and when s ≥ 3 , t = 2 and r = | V | . Finally, for s ≥ 2 , we present a fixed-parameter algorithm for the problem, when parameterized by the number of covered vertices.

[1]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[2]  Christian Komusiewicz,et al.  Parameterized computational complexity of finding small-diameter subgraphs , 2012, Optim. Lett..

[3]  Piotr Sankowski,et al.  Maximum matchings via Gaussian elimination , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[4]  Srinivas Pasupuleti,et al.  Detection of Protein Complexes in Protein Interaction Networks Using n-Clubs , 2008, EvoBIO.

[5]  Viggo Kann,et al.  Some APX-completeness results for cubic graphs , 2000, Theor. Comput. Sci..

[6]  Petr A. Golovach,et al.  Finding clubs in graph classes , 2014, Discret. Appl. Math..

[7]  Riccardo Dondi,et al.  Optimizing Social Interaction - A Computational Approach to Support Patient Engagement , 2018, HEALTHINF.

[8]  Wencheng Wang,et al.  Finding Disjoint Dense Clubs in an Undirected Graph , 2016, FAW.

[9]  Gilbert Laporte,et al.  An exact algorithm for the maximum k-club problem in an undirected graph , 1999, Eur. J. Oper. Res..

[10]  Charalampos E. Tsourakakis,et al.  Denser than the densest subgraph: extracting optimal quasi-cliques with quality guarantees , 2013, KDD.

[11]  Yuichi Asahiro,et al.  Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems , 2017, Algorithmica.

[12]  R. Alba.  A graph‐theoretic definition of a sociometric clique† , 1973 .

[13]  Noga Alon,et al.  Color-coding , 1995, JACM.

[14]  Riccardo Dondi,et al.  Covering with Clubs: Complexity and Approximability , 2018, IWOCA.

[15]  R. J. Mokken,et al.  Close communication and 2-clubs in corporate networks: Europe 2010 , 2015, Social Network Analysis and Mining.

[16]  Christian Komusiewicz,et al.  Multivariate Algorithmics for Finding Cohesive Subnetworks , 2016, Algorithms.

[17]  R. J. Mokken,et al.  Cliques, clubs and clans , 1979 .

[18]  Maw-Shang Chang,et al.  Finding large $$k$$-clubs in undirected graphs , 2013, Computing.

[19]  Christian Komusiewicz,et al.  Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs , 2015, J. Graph Algorithms Appl..

[20]  Maarten Marx,et al.  Close Communities in Social Networks: Boroughs and 2-Clubs , 2015 .

[21]  Bruce A. Reed,et al.  Partition into cliques for cubic graphs: Planar case, complexity and approximation , 2008, Discret. Appl. Math..

[22]  David P. Williamson,et al.  The Design of Approximation Algorithms , 2011 .

[23]  Albert-Lszl Barabsi,et al.  Network Science , 2016, Encyclopedia of Big Data.

[24]  Sergiy Butenko,et al.  Novel Approaches for Analyzing Biological Networks , 2005, J. Comb. Optim..

[25]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[26]  Christian Komusiewicz,et al.  An algorithmic framework for fixed-cardinality optimization in sparse graphs applied to dense subgraph problems , 2015, Discret. Appl. Math..

[27]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .