Finding minimum-quotient cuts in planar graphs

Abstract

Given a graph G = ( V, E) where each vertex v € V is assigned a weight and each edge e £ E is assigned a cost c(e), the quotient of a cut partitioning the vertices of V into sets S and S is c(S, S)/ min{u/(S), ti(5)}, where c(5,5) is the sum of the costs of the edges crossing the cut and w(S) and w(S) are the sum of the weights of the vertices in S and S, respectively. The problem of finding a cut whose quotient is minimum for a graph has attracted considerable attention in recent years, due in large part to the work of Rao [14, 15] and Leighton and Rao [8]. They have shown that an algorithm (exact or approximation) for the minimum-quotient-cut prob-lem can be used to obtain an approximation algorithm for the more famous iriinimum-6-balanced-cut problem, which requires finding a cut (5, S) minimizing c(S. 5) subject to the constraint bW (5) < (1 - b)W, where W is the total vertex weight and b is some fixed balance in the range 0 < b < 1/2. Unfortunately, the minimum-quotient-cut problem is strongly NP-hard for general graphs, and the best polynomial-Time approximation algorithm known for the general problem (due to Leighton and Rao [8]) may produce a cut whose quotient is 6>(lgrj) times optimal, where n is the size of the graph. However, for planar graphs, the minimum- quoticnt-cut problem appears more tractable. In particular. Rao [14,15] has developed several efficient approximation algorithms for the planar version of the problem, each of which always produces a cut whose quotient is at most some constant times optimal. In this paper, we improve Rao's results, both in terms of their accuracy and their speed. We begin with two pseudopolynomial-Time exact algorithms for the planar minirnum-quotient-cut problem. As Rao's most accurate approximation algorithm for the problem -Also a pseudopolynomial-Time algorithm - guarantees only a 1.5-Times-optirnal cut, our algorithms represent a significant advance. They are also the first polynomial-Time algorithms known for the unweighted version of the planar minimum-quotient-cut problem (assigning weight 1 to all vertices). We also show how to speed up Rao's other two approximation algorithms, which run in polynomial time and have somewhat weaker accuracy bounds. Finally, we prove that the minimum-quotient-cut problem is weakly NP-hard for 2-outerplanar and series-parallel graphs, even if all edge costs are 1, and that the rninirnum-6-balanced-cut problem is weakly NP-hard for trees of bandwidth 2. This last result is the first intractability result known for the planar minimunv6-balanced-cut problem, and thus it partially resolves the longstanding open question of this problem's true complexity.