Branch-and-Cut algorithm for the connected-cut problem

International audienceLet G=(V,E) be an undirected connected graph. Let W be a subset of V, distinct from V. The set W is said proper when it is non-empty. We denote by δ(W), the set of edges of E having exactly one endnode in W. A set F is called a cut of G, if there exists a proper subset W of V such that F=δ(W). Note that removing from a graph the edges of a cut can leave more than 2 connected components. Consequently, a cut is said to be connected if both subgraphs G[W] =(W,E(W)) and G[V\W] =(V\W,E(V\W)) are connected. Let c be a cost function defined on the edges of E, the maximum cut problem (Max-C) consists in finding a cut of maximum weight (where the weight of a cut is given by the summation of the costs of all the edges composing the cut). The particular case where the cost function is non-positive is called the minimum cut problem (Min-C). In a similar way, we can define the maximum (resp. minimum) connected cut problem Max-CC (resp. Min-CC) as finding a maximum cost connected cut. It is easily seen that an optimal solution of the Min-C will be a connected cut and thus an optimal solution of the Min-CC. Consequently the Min-CC can be solved in polynomial time. The Max-C is strongly NP-hard. However, when the graph is planar, the Max-C problem can be solved in polynomial time [1], whereas the Max-CC is still NP-hard [2]. In this talk, we present integer formulations for the Max-CC together with preliminary results on the associated polytope. We also propose a Branch-and-Cut algorithm in order to solve instances generated from the TSPLIB. [1] F. Hadlock, Finding a maximum cut of a planar graph in polynomial time, SIAM Journal of Computing vol. 4 (1975) 221-225.[2] Haglin, Venkatesan, Approximation and Intractability results for the maximum cut problem and its variants, IEEE Trans. on computers 40 (1991) 110-113.