Upper bounds and heuristics for the 2-club problem

Given an undirected graph G = ( V, E), a k-club is a subset of V that induces a subgraph of diameter at most k. The k-club problem is that of finding the maximum cardinality k-club in G. In this paper we present valid inequalities for the 2-club polytope and derive conditions for them to define face...

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Published in:	European journal of operational research Vol. 210; no. 3; pp. 489 - 494
Main Authors:	Carvalho, Filipa D., Almeida, M. Teresa
Format:	Journal Article
Language:	English
Published:	Amsterdam Elsevier B.V 01.05.2011 Elsevier Elsevier Sequoia S.A
Series:	European Journal of Operational Research
Subjects:	Applied sciences Combinatorial optimization Combinatorial optimization Integer programming k-club problem Heuristics Computer science; control theory; systems Cutting Density Exact sciences and technology Flows in networks. Combinatorial problems Graph theory Graphs Heuristic Heuristics Inequalities Information retrieval. Graph Integer programming k-club problem Linear programming Mathematical models Mathematical programming Operational research Operational research and scientific management Operational research. Management science Optimization Optimization algorithms Studies Theoretical computing Upper bounds Integer programming k-club problem Combinatorial optimization Heuristics Polytope Linear programming Cutting plane method Upper bound Cardinal number Optimal solution Heuristic method Subgraph Relaxation method Diameter Non directed graph
ISSN:	0377-2217, 1872-6860
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Abstract	Given an undirected graph G = ( V, E), a k-club is a subset of V that induces a subgraph of diameter at most k. The k-club problem is that of finding the maximum cardinality k-club in G. In this paper we present valid inequalities for the 2-club polytope and derive conditions for them to define facets. These inequalities are the basis of a strengthened formulation for the 2-club problem and a cutting plane algorithm. The LP relaxation of the strengthened formulation is used to compute upper bounds on the problem’s optimum and to guide the generation of near-optimal solutions. Numerical experiments indicate that this approach is quite effective in terms of solution quality and speed, especially for low density graphs.
AbstractList	Given an undirected graph G = (V, E), a k-club is a subset of V that induces a subgraph of diameter at most k. The k-club problem is that of finding the maximum cardinality k-club in G. In this paper we present valid inequalities for the 2-club polytope and derive conditions for them to define facets. These inequalities are the basis of a strengthened formulation for the 2-club problem and a cutting plane algorithm. The LP relaxation of the strengthened formulation is used to compute upper bounds on the problem's optimum and to guide the generation of near-optimal solutions. Numerical experiments indicate that this approach is quite effective in terms of solution quality and speed, especially for low density graphs. Given an undirected graph G = ( V, E), a k-club is a subset of V that induces a subgraph of diameter at most k. The k-club problem is that of finding the maximum cardinality k-club in G. In this paper we present valid inequalities for the 2-club polytope and derive conditions for them to define facets. These inequalities are the basis of a strengthened formulation for the 2-club problem and a cutting plane algorithm. The LP relaxation of the strengthened formulation is used to compute upper bounds on the problem’s optimum and to guide the generation of near-optimal solutions. Numerical experiments indicate that this approach is quite effective in terms of solution quality and speed, especially for low density graphs. Given an undirected graph G = (V, E), a k-club is a subset of V that induces a subgraph of diameter at most k. The k-club problem is that of finding the maximum cardinality k-club in G. In this paper we present valid inequalities for the 2-club polytope and derive conditions for them to define facets. These inequalities are the basis of a strengthened formulation for the 2-club problem and a cutting plane algorithm. The LP relaxation of the strengthened formulation is used to compute upper bounds on the problem's optimum and to guide the generation of near-optimal solutions. Numerical experiments indicate that this approach is quite effective in terms of solution quality and speed, especially for low density graphs. [PUBLICATION ABSTRACT]
Author	Carvalho, Filipa D. Almeida, M. Teresa
Author_xml	– sequence: 1 givenname: Filipa D. surname: Carvalho fullname: Carvalho, Filipa D. email: filipadc@iseg.utl.pt organization: ISEG, Universidade Técnica de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal – sequence: 2 givenname: M. Teresa surname: Almeida fullname: Almeida, M. Teresa email: talmeida@iseg.utl.pt organization: ISEG, Universidade Técnica de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal
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References	Nemhauser, Wolsey (b0075) 1988 Balasundaram, B., Butenko, S., Hicks, I.V., 2009. Clique Relaxations in Social Network Analysis: The Maximum k-plex Problem. Available at Martins (b0065) 2010; 37 Wasserman, Faust (b0090) 1994 Mokken (b0070) 1979; 13 Bomze, Budinich, Pardalos, Pelillo (b0035) 1999 Sorensen (b0085) 2004; 154 Alidaee, Glover, Kochenberger, Wang (b0010) 2007; 181 Park, Lee, Park (b0080) 1996; 95 Macambira, Souza (b0060) 2000; 123 (accessed December 2009). Bourjolly, Laporte, Pesant (b0040) 2000; 27 Carvalho, F.D., Almeida, M.T., 2008. Strong valid inequalities for the 2-club problem. Centro de Investigação Operacional, Working Paper 2/2008. Available at Balasundaram, Butenko, Trukhanov (b0015) 2005; 10 Bourjolly, Laporte, Pesant (b0045) 2002; 138 Butenko, Wilhelm (b0050) 2006; 173 Alba (b0005) 1973; 3 Blażewicz, Formanowicz, Kasprak (b0025) 2005; 161 Boginski, Butenko, Pardalos (b0030) 2006; 33 Martins (10.1016/j.ejor.2010.11.023_b0065) 2010; 37 10.1016/j.ejor.2010.11.023_b0020 Blażewicz (10.1016/j.ejor.2010.11.023_b0025) 2005; 161 Mokken (10.1016/j.ejor.2010.11.023_b0070) 1979; 13 Boginski (10.1016/j.ejor.2010.11.023_b0030) 2006; 33 Alba (10.1016/j.ejor.2010.11.023_b0005) 1973; 3 Alidaee (10.1016/j.ejor.2010.11.023_b0010) 2007; 181 Bourjolly (10.1016/j.ejor.2010.11.023_b0045) 2002; 138 Sorensen (10.1016/j.ejor.2010.11.023_b0085) 2004; 154 Macambira (10.1016/j.ejor.2010.11.023_b0060) 2000; 123 Bourjolly (10.1016/j.ejor.2010.11.023_b0040) 2000; 27 Park (10.1016/j.ejor.2010.11.023_b0080) 1996; 95 Bomze (10.1016/j.ejor.2010.11.023_b0035) 1999 Nemhauser (10.1016/j.ejor.2010.11.023_b0075) 1988 Balasundaram (10.1016/j.ejor.2010.11.023_b0015) 2005; 10 Wasserman (10.1016/j.ejor.2010.11.023_b0090) 1994 Butenko (10.1016/j.ejor.2010.11.023_b0050) 2006; 173 10.1016/j.ejor.2010.11.023_b0055
References_xml	– reference: Carvalho, F.D., Almeida, M.T., 2008. Strong valid inequalities for the 2-club problem. Centro de Investigação Operacional, Working Paper 2/2008. Available at – volume: 173 start-page: 1 year: 2006 end-page: 7 ident: b0050 article-title: Clique detection models in computational biochemistry and genomics publication-title: European Journal of Operational Research – start-page: 1 year: 1999 end-page: 74 ident: b0035 article-title: The maximum clique problem publication-title: Handbook of Combinatorial Optimization – volume: 27 start-page: 559 year: 2000 end-page: 569 ident: b0040 article-title: Heuristics for finding k-clubs in an undirected graph publication-title: Computers & Operations Research – volume: 3 start-page: 113 year: 1973 end-page: 126 ident: b0005 article-title: A graph-theoretic definition of a sociometric clique publication-title: Journal of Mathematical Sociology – volume: 181 start-page: 592 year: 2007 end-page: 597 ident: b0010 article-title: Solving the maximum weight clique problem via unconstrained quadratic programming publication-title: European Journal of Operational Research – volume: 154 start-page: 57 year: 2004 end-page: 70 ident: b0085 article-title: New facets and branch and cut algorithm for the weighted clique problem publication-title: European Journal of Operational Research – volume: 13 start-page: 161 year: 1979 end-page: 173 ident: b0070 article-title: Cliques, clubs and clans publication-title: Quality and Quantity – year: 1988 ident: b0075 article-title: Integer and Combinatorial Optimization – volume: 95 start-page: 671 year: 1996 end-page: 682 ident: b0080 article-title: An extended formulation approach to the edge-weighted maximal clique problem publication-title: European Journal of Operational Research – volume: 161 start-page: 585 year: 2005 end-page: 597 ident: b0025 article-title: Selected combinatorial problems of computational biology publication-title: European Journal of Operational Research – volume: 33 start-page: 3171 year: 2006 end-page: 3184 ident: b0030 article-title: Mining market data: a network approach publication-title: Computers & Operations Research – volume: 123 start-page: 346 year: 2000 end-page: 371 ident: b0060 article-title: The edge-weighted clique problem: valid inequalities facets, and polyhedral computations publication-title: European Journal of Operational Research – volume: 138 start-page: 21 year: 2002 end-page: 28 ident: b0045 article-title: An exact algorithm for the maximum k-club problem in an undirected graph publication-title: European Journal of Operational Research – reference: (accessed December 2009). – reference: Balasundaram, B., Butenko, S., Hicks, I.V., 2009. Clique Relaxations in Social Network Analysis: The Maximum k-plex Problem. Available at – volume: 10 start-page: 23 year: 2005 end-page: 39 ident: b0015 article-title: Novel approaches for analyzing biological networks publication-title: Journal of Combinatorial Optimization – volume: 37 start-page: 1348 year: 2010 end-page: 1358 ident: b0065 article-title: Extended and discretized formulations for the maximum clique problem publication-title: Computers & Operations Research – year: 1994 ident: b0090 article-title: Social Network Analysis – volume: 181 start-page: 592 year: 2007 ident: 10.1016/j.ejor.2010.11.023_b0010 article-title: Solving the maximum weight clique problem via unconstrained quadratic programming publication-title: European Journal of Operational Research doi: 10.1016/j.ejor.2006.06.035 – volume: 10 start-page: 23 year: 2005 ident: 10.1016/j.ejor.2010.11.023_b0015 article-title: Novel approaches for analyzing biological networks publication-title: Journal of Combinatorial Optimization doi: 10.1007/s10878-005-1857-x – year: 1988 ident: 10.1016/j.ejor.2010.11.023_b0075 – year: 1994 ident: 10.1016/j.ejor.2010.11.023_b0090 – volume: 27 start-page: 559 year: 2000 ident: 10.1016/j.ejor.2010.11.023_b0040 article-title: Heuristics for finding k-clubs in an undirected graph publication-title: Computers & Operations Research doi: 10.1016/S0305-0548(99)00047-7 – volume: 154 start-page: 57 year: 2004 ident: 10.1016/j.ejor.2010.11.023_b0085 article-title: New facets and branch and cut algorithm for the weighted clique problem publication-title: European Journal of Operational Research doi: 10.1016/S0377-2217(02)00852-4 – volume: 123 start-page: 346 year: 2000 ident: 10.1016/j.ejor.2010.11.023_b0060 article-title: The edge-weighted clique problem: valid inequalities facets, and polyhedral computations publication-title: European Journal of Operational Research doi: 10.1016/S0377-2217(99)00262-3 – volume: 138 start-page: 21 year: 2002 ident: 10.1016/j.ejor.2010.11.023_b0045 article-title: An exact algorithm for the maximum k-club problem in an undirected graph publication-title: European Journal of Operational Research doi: 10.1016/S0377-2217(01)00133-3 – ident: 10.1016/j.ejor.2010.11.023_b0055 – start-page: 1 year: 1999 ident: 10.1016/j.ejor.2010.11.023_b0035 article-title: The maximum clique problem – volume: 3 start-page: 113 year: 1973 ident: 10.1016/j.ejor.2010.11.023_b0005 article-title: A graph-theoretic definition of a sociometric clique publication-title: Journal of Mathematical Sociology doi: 10.1080/0022250X.1973.9989826 – volume: 37 start-page: 1348 year: 2010 ident: 10.1016/j.ejor.2010.11.023_b0065 article-title: Extended and discretized formulations for the maximum clique problem publication-title: Computers & Operations Research doi: 10.1016/j.cor.2009.10.010 – volume: 173 start-page: 1 year: 2006 ident: 10.1016/j.ejor.2010.11.023_b0050 article-title: Clique detection models in computational biochemistry and genomics publication-title: European Journal of Operational Research doi: 10.1016/j.ejor.2005.05.026 – ident: 10.1016/j.ejor.2010.11.023_b0020 – volume: 13 start-page: 161 year: 1979 ident: 10.1016/j.ejor.2010.11.023_b0070 article-title: Cliques, clubs and clans publication-title: Quality and Quantity doi: 10.1007/BF00139635 – volume: 161 start-page: 585 year: 2005 ident: 10.1016/j.ejor.2010.11.023_b0025 article-title: Selected combinatorial problems of computational biology publication-title: European Journal of Operational Research doi: 10.1016/j.ejor.2003.10.054 – volume: 95 start-page: 671 year: 1996 ident: 10.1016/j.ejor.2010.11.023_b0080 article-title: An extended formulation approach to the edge-weighted maximal clique problem publication-title: European Journal of Operational Research doi: 10.1016/0377-2217(95)00299-5 – volume: 33 start-page: 3171 year: 2006 ident: 10.1016/j.ejor.2010.11.023_b0030 article-title: Mining market data: a network approach publication-title: Computers & Operations Research doi: 10.1016/j.cor.2005.01.027
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Snippet	Given an undirected graph G = ( V, E), a k-club is a subset of V that induces a subgraph of diameter at most k. The k-club problem is that of finding the... Given an undirected graph G = (V, E), a k-club is a subset of V that induces a subgraph of diameter at most k. The k-club problem is that of finding the...
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SubjectTerms	Applied sciences Combinatorial optimization Combinatorial optimization Integer programming k-club problem Heuristics Computer science; control theory; systems Cutting Density Exact sciences and technology Flows in networks. Combinatorial problems Graph theory Graphs Heuristic Heuristics Inequalities Information retrieval. Graph Integer programming k-club problem Linear programming Mathematical models Mathematical programming Operational research Operational research and scientific management Operational research. Management science Optimization Optimization algorithms Studies Theoretical computing Upper bounds
Title	Upper bounds and heuristics for the 2-club problem
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