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A linear programming formulation for the maximum complete multipartite subgraph problem

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Abstract

Let G be a simple undirected graph with node set V(G) and edge set E(G). We call a subset independent if F is contained in the edge set of a complete multipartite (not necessarily induced) subgraph of G, F is dependent otherwise. In this paper we characterize the independents and the minimal dependents of G. We note that every minimal dependent of G has size two if and only if G is fan and prism-free. We give a 0-1 linear programming formulation of the following problem: find the maximum weight of a complete multipartite subgraph of G, where G has nonnegative edge weights. This formulation may have an exponential number of constraints with respect to |V(G)| but we show that the continuous relaxation of this 0-1 program can be solved in polynomial time.

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  1. Équipe Combinatoire et Optimisation, Case 189, UFR 921, Université Pierre et Marie Curie, 4 place Jussieu, 75252, Cedex Paris, France

    Denis Cornaz

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  1. Denis Cornaz
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Correspondence to Denis Cornaz.

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Cornaz, D. A linear programming formulation for the maximum complete multipartite subgraph problem. Math. Program. 105, 329–344 (2006). https://doi.org/10.1007/s10107-005-0656-6

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  • DOI: https://doi.org/10.1007/s10107-005-0656-6

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