Abstract
Given a vertex-weighted undirected graph \(G=(V,E,w)\) and a positive integer \(k\), we consider the \(k\)-separator problem: it consists in finding a minimum-weight subset of vertices whose removal leads to a graph where the size of each connected component is less than or equal to \(k\). We show that this problem can be solved in polynomial time for some graph classes including bounded treewidth, \(m K_2\)-free, \((G_1, G_2, G_3, P_6)\)-free, interval-filament, asteroidal triple-free, weakly chordal, interval and circular-arc graphs. Polyhedral results with respect to the convex hull of the incidence vectors of \(k\)-separators are reported. Approximation algorithms are also presented.
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References
Arnborg S, Lagergren J, Seese D (1991) Easy problems for tree-decomposable graphs. J Algorithms 12:308–340
Balas E (1998) Disjunctive programming: properties of the convex hull of feasible points. Discret Appl Math 89:1–44
Balas E, de Souza C (2005) The vertex separator problem: a polyhedral investigation. Math Program 103:583–608
Balas E, Yu C (1989) On graphs with polynomially solvable maximum-weight clique problem. Networks 19:247–253
Ben-Ameur W, Mohamed-Sidi M-A., Neto J (2013) The k-separator problem. In: Proceedings of COCOON 2013, Springer LNCS 7936, pp 337–348
Bodlaender HL (1993) A linear time algorithm for finding tree-decompositions of small treewidth. In: Proceedings of STOC’93, pp 226–234
Boliac R, Cameron K, Lozin V (2004) On computing the dissociation number and the induced matching number of bipartite graphs. Ars Combin 72:241253
Borie RB (1995) Generation of polynomial-time algorithms for some optimization problems on tree-decomposable graphs. Algorithmica 14:123–137
Borndörfer R, Ferreira CE, Martin A (1998) Decomposing matrices into blocks. SIAM J Optim 9:236–269
Broersma H, Kloks T, Kratsch D, Müller H (1999) Independent sets in asteroidal triple-free graphs. SIAM J Discret Math 12:276–287
Cameron K, Hell P (2006) Independent packings in structured graphs. Math Program Ser B 105:201–213
Dinur I, Safra S (2005) On the hardness of approximating minimum vertex cover. Ann Math 162:439–485
Gavril F (2000) Maximum weight independent sets and cliques in intersection graphs of filaments. Inf Process Lett 73:181–188
Goldschmidt O, Hochbaum D (1997) K-edge subgraphs problems. Discree Appl Math 74:159–169
Habib M, McConnell R, Paul C, Viennot L (2000) Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition, and consecutive ones testing. Theor Comput Sci 234:59–84
Hastad J (1999) Clique is hard to approximate within \(n^{1-\epsilon }\). Acta Math 182:105–142
Hayard RB (1985) Weakly triangulated graphs. J Combin Theory Ser B 39:200–209
Khot S, Regev O (2008) Vertex cover might be hard to approximate to within \(2- \epsilon \). J Comput Syst Sci 74:335–349
Kloks T (1994) Treewidth: computations and approximations, vol 842. Lecture Notes in Computer Science, Springer, Berlin
Korte B, Vygen J (2005) Combinatorial optimization: theory and algorithms. Springer, New York
Lozin V, Milanic M (2008) A polynomial algorithm to find an independent set of maximum weight in a fork-free graph. J Discret Algorithms 6:595–604
Lozin V, Rautenbach D (2003) Some results on graphs without long induced paths. Inform Process Lett 88:167–171
Minty G (1980) On maximal independent sets of vertices in claw-free graphs. J Combin Theory Ser B 28:284–304
Oosten M, Rutten J, Spiksma F (2007) Disconnecting graphs by removing vertices: a polyhedral approach. Stat Neerl 61:35–60
Orlovich Y, Dolgui A, Finke G, Gordon V, Werner F (2011) The complexity of dissociation set problems in graphs. Discret Appl Math 159:1352–1366
Papadimitriou CH, Yannakakis M (1982) The complexity of restricted spanning tree problems. J Assoc Comput Mach 29:285309
Sbihi N (1980) Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discret Math 29:53–76
Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. Springer, Berlin
Shmoys D (1997) Cut problems and their application to divide-and-conquer. In: Dorit SH (ed) Approximation algorithms for NP-hard problems. PWS, Boston, pp 192–235
Spinrad JP, Sritharan R (1995) Algorithms for weakly triangulated graphs. Discret Appl Math 59:181–191
Telle JA, Proskurowski A (1997) Algorithms for vertex partitioning problems on partial k-trees. SIAM J Discret Math 10:529–550
Williamson D (2002) The primal-dual method for approximation algorithms. Math Program Ser B 91:447–478
Yannakakis M (1981) Node-deletion problems on bipartite graphs. SIAM J Comput 10:310–327
Zuckerman D (2007) Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput 3:103–138
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Ben-Ameur, W., Mohamed-Sidi, MA. & Neto, J. The \(k\)-separator problem: polyhedra, complexity and approximation results. J Comb Optim 29, 276–307 (2015). https://doi.org/10.1007/s10878-014-9753-x
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DOI: https://doi.org/10.1007/s10878-014-9753-x