Approximation Algorithms for Min-k-Overlap Problems Using the Principal Lattice of Partitions Approach

Abstract

In this paper we discuss strategies for constructing approximation algorithms for solving theMin-k-cut problem, theMin-k-vertex sharing problem, and their generalization theMin-k-overlap problem. In each case, we first formulate an appropriate submodular function and construct its principal lattice of partitions (PLP) (H. Narayanan,Linear Algebra Appl.144(1991), 179–216). Applying an elementary strategy on an appropriate subinterval of the PLP yields the approximation algorithms. We give two such strategies. For convenience and greater generality we present our results in terms of bipartite graphs (which may be regarded as representing hypergraphs). We observe that the Min-k-cut problem isNP-Hard(O. Goldschmidt and D. S. Hochbaum,in“Proceedings 29th Annual Symposium on the Foundations of Computer Science,” pp. 444–451, 1988). Approximation algorithms for the Min-k-cut problem were first given in (H. Saran and V. V. Vazirani,in“Proceedings 32nd Annual Symposium on the Foundations of Computer Science, 1991”). Their algorithms yield a Min-k-cut which is at most 2(1−(1/k)) times worse than the optimal. Our algorithm using the first strategy has a 2(1−(1/n)) factor (times worse than optimal) for both Min-k-cut and Min-k-vertex sharing (1) problems (nis |V(G)|, |E(G)| respectively). For Min-k-vertex sharing (2) this strategy yields aq(1−(1/|E(G)|)) factor, whereqis the maximum degree of a vertex in the graphG. Using a second strategy we are able to obtain a (2−(k′/n′)) factor for the Min-k-cut problem, wherek′=k−k0,n′=n−k0,k0being an appropriate integer defined in the paper. In our approach the chief labor consists in finding a sequence of increasingly coarser partitions (starting with the partition into singleton blocks and ending with the single block partition ofS) called theprincipal sequence. Once this is done the remaining effort is essentiallyO(|S|logS).