Solving minimum k-cardinality cut problems in planar graphs

The present work tackles a recent problem in the class of cardinality constrained combinatorial optimization problems for the planar graph case: the minimum k-cardinality cut problem. Given an undirected edge-weighted connected graph the min k-cardinality cut problem consists in finding a partition of the vertex set V in two sets V-1, V-2 such that the number of the edges between V-1 and V-2 is exactly k and the sum of the weights of these edges is minimal. Although for general graphs the problem is already strongly NP-hard, we have found a pseudopolynomial algorithm for the planar graph case. This algorithm is based on the fact that the min k-cardinality cut problem in the original graph is equivalent to a bi-weighted exact perfect matching problem in a suitable transformation of the geometric dual graph. Because the Lagrangian relaxation of cardinality constraint yields a max cut problem and max cut is polynomially solvable in planar graphs, we also develop a Lagrangian heuristic for the min k-cardinality cut in planar graphs. We compare the performance of this heuristic with the performance of a more general heuristic based on a Semidefinite Programming relaxation and on the Goemans and Williamson's random hyperplane technique. (C) 2006 Wiley Periodicals, Inc.