A new scaling algorithm for the maximum mean cut problem

Abstract

We present a new scaling algorithm for the maximum mean cut problem. The mean of a cut is defined by the cut capacity divided by the number of arcs crossing the cut. The algorithm uses an approximate binary search and solves the circulation feasibility problem with relaxed capacity bounds. The maximum mean cut problem has recently been studied as a dual analogue of the minimum mean cycle problem in the framework of the minimum cost flow problem by Ervolina and McCormick. A networkN=(G, lower, upper) with lower and upper arc capacities is said to be δ-feasible ifN has a feasible circulation when we relax the capacity bounds by δ; that is, we use (lower(a)- δ, upper(a)+δ) bounds instead of (lower(a), upper(a)) bounds for each arca εA. During an approximate binary search we maintain two bounds,LB andUB, such thatN is LB-infeasible andUB-feasible, and we reduce the interval size (LB, UB) by at least one-third at each iteration. For a graph withn vertices, m arcs, and integer capacities bounded byU, the running time of this algorithm is O(mn log(nU). This time bound is better than the time achieved by McCormick and Ervolina under thesimilarity condition (that is,U=O(n^o(1))). Our algorithm can be naturally used for the circulation feasibility problem, and thus provides a new scaling algorithm for the minimum cut problem.