The generalized independent set problem: Polyhedral analysis and solution approaches
•The Generalized Independent Set Problem is studied.•The polyhedron of an integer linear programming formulation is investigated.•Linear programming based heuristics are designed, implemented, and tested.•GRASP-tabu search using a 0–1 quadratic programming formulation is most effective. In the Gener...
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| Vydáno v: | European journal of operational research Ročník 260; číslo 1; s. 41 - 55 |
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| Jazyk: | angličtina |
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Elsevier B.V
01.07.2017
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| ISSN: | 0377-2217, 1872-6860 |
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| Abstract | •The Generalized Independent Set Problem is studied.•The polyhedron of an integer linear programming formulation is investigated.•Linear programming based heuristics are designed, implemented, and tested.•GRASP-tabu search using a 0–1 quadratic programming formulation is most effective.
In the Generalized Independent Set Problem, we are given a graph, a revenue for each vertex, and a set of removable edges with associated removal costs, and we seek to find an independent set that maximizes the net benefit, i.e., the difference between the revenues collected for the vertices in the independent set and the costs incurred for any removal of edges with both endpoints in the independent set. We study the polyhedron associated with a 0–1 linear programming formulation of the Generalized Independent Set Problem, deriving a number of facet-inducing inequalities, and we develop linear programming based heuristics to obtain high-quality solutions in a short amount of time. We also develop a heuristic method based on an unconstrained 0–1 quadratic programming formulation of the Generalized Independent Set Problem. In an extensive computational study, we assess the performance of these heuristics in terms of quality and efficiency. The best heuristic is then used to produce an initial solution for a branch-and-cut algorithm which uses some of the proposed facet-inducing inequalities. |
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| AbstractList | •The Generalized Independent Set Problem is studied.•The polyhedron of an integer linear programming formulation is investigated.•Linear programming based heuristics are designed, implemented, and tested.•GRASP-tabu search using a 0–1 quadratic programming formulation is most effective.
In the Generalized Independent Set Problem, we are given a graph, a revenue for each vertex, and a set of removable edges with associated removal costs, and we seek to find an independent set that maximizes the net benefit, i.e., the difference between the revenues collected for the vertices in the independent set and the costs incurred for any removal of edges with both endpoints in the independent set. We study the polyhedron associated with a 0–1 linear programming formulation of the Generalized Independent Set Problem, deriving a number of facet-inducing inequalities, and we develop linear programming based heuristics to obtain high-quality solutions in a short amount of time. We also develop a heuristic method based on an unconstrained 0–1 quadratic programming formulation of the Generalized Independent Set Problem. In an extensive computational study, we assess the performance of these heuristics in terms of quality and efficiency. The best heuristic is then used to produce an initial solution for a branch-and-cut algorithm which uses some of the proposed facet-inducing inequalities. |
| Author | Mansini, Renata Colombi, Marco Savelsbergh, Martin |
| Author_xml | – sequence: 1 givenname: Marco surname: Colombi fullname: Colombi, Marco email: m.colombi006@unibs.it organization: Department of Information Engineering, University of Brescia, Via Branze 38 - 25123 Brescia, Italy – sequence: 2 givenname: Renata surname: Mansini fullname: Mansini, Renata email: renata.mansini@unibs.it organization: Department of Information Engineering, University of Brescia, Via Branze 38 - 25123 Brescia, Italy – sequence: 3 givenname: Martin surname: Savelsbergh fullname: Savelsbergh, Martin email: martin.savelsbergh@isye.gatech.edu organization: H Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Drive, NW, Atlanta, GA 30332, USA |
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| Cites_doi | 10.1002/net.20088 10.1007/s10107-002-0356-4 10.1090/dimacs/026 10.1093/forestscience/43.4.544 10.1016/j.disopt.2005.03.004 10.1016/j.cor.2011.12.006 10.1287/mnsc.1040.0242 10.1007/s11590-006-0007-4 10.1007/s00291-003-0153-3 10.1287/opre.42.5.860 10.1007/BF01299447 10.1007/BF01580121 10.1007/s10107-005-0651-y 10.1007/BF01580222 10.1186/1748-7188-1-9 |
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| Keywords | Generalized independent set Facets Combinatorial optimization 0–1 quadratic programming GRASP tabu search |
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| SubjectTerms | 0–1 quadratic programming Combinatorial optimization Facets Generalized independent set GRASP tabu search |
| Title | The generalized independent set problem: Polyhedral analysis and solution approaches |
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