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 edgeweighted connected graph the min k -cardinality cut problem consists in finding a partition of the vertex set V in two sets V1, V2 such that the number of the edges between V1 and V2 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. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(4), 195–208 2006

[1]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[2]  Mauro Dell'Amico,et al.  The k-cardinality Assignment Problem , 1997, Discret. Appl. Math..

[3]  Yazid M. Sharaiha,et al.  Heuristics for cardinality constrained portfolio optimisation , 2000, Comput. Oper. Res..

[4]  Jason Cong,et al.  FlowMap: an optimal technology mapping algorithm for delay optimization in lookup-table based FPGA designs , 1994, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[5]  L. Lovász,et al.  Annals of Discrete Mathematics , 1986 .

[6]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[7]  Horst W. Hamacher,et al.  Fixed Cardinality Combinatorial Optimization Problems - A Survey , 1999 .

[8]  Horst W. Hamacher,et al.  Integer programming approaches tofacilities layout models with forbidden areas , 1998 .

[9]  Vijay V. Vazirani,et al.  Matching is as easy as matrix inversion , 1987, STOC.

[10]  Giovanni Rinaldi,et al.  An efficient algorithm for the minimum capacity cut problem , 1990, Math. Program..

[11]  John N. Tsitsiklis,et al.  Introduction to linear optimization , 1997, Athena scientific optimization and computation series.

[12]  William R. Pulleyblank,et al.  Exact arborescences, matchings and cycles , 1987, Discret. Appl. Math..

[13]  Mechthild Stoer,et al.  A simple min-cut algorithm , 1997, JACM.

[14]  T. C. Hu,et al.  Multi-Terminal Network Flows , 1961 .

[15]  Francesco Maffioli,et al.  Random Pseudo-Polynomial Algorithms for Exact Matroid Problems , 1992, J. Algorithms.

[16]  J. Edmonds.  Systems of distinct representatives and linear algebra , 1967 .

[17]  Martin Grötschel,et al.  An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design , 1988, Oper. Res..

[18]  Francisco Barahona,et al.  On cuts and matchings in planar graphs , 1993, Math. Program..

[19]  Joachim von zur Gathen,et al.  Parallel algorithms for algebraic problems , 1983, SIAM J. Comput..

[20]  Francesco Maffioli,et al.  on the Computation of Pfaffians , 1994, Discret. Appl. Math..

[21]  Francesco Maffioli,et al.  Cardinality constrained minimum cut problems: complexity and algorithms , 2004, Discret. Appl. Math..

[22]  M. Ehrgott,et al.  Heuristics for the K-Cardinality Tree and Subgraph Problems , 1996 .

[23]  Frank Harary,et al.  Finite Graphs and Networks, An Introduction with Applications. , 1967 .

[24]  Francesco Maffioli,et al.  Randomized Algorithms over Finite Fields for the Exact Parity Base Problem , 1999, J. Complex..

[25]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .