Solving minimum k-cardinality cut problems in planar graphs

被引:1
|
作者
Bruglieri, Maurizio
Maffioli, Francesco
Trubian, Marco
机构
[1] Politecn Milan, Dipartimento Elettron & Informaz, I-20133 Milan, Italy
[2] Univ Milan, Dipartimento Sci Informaz, I-20100 Milan, Italy
关键词
cut problems; minimum k-cardinality cut; Lagrangian relaxation; exact perfect matching; planar graphs; semidefrnite programming;
D O I
10.1002/net.20129
中图分类号
TP3 [计算技术、计算机技术];
学科分类号
0812 ;
摘要
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.
引用
收藏
页码:195 / 208
页数:14
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