A NEW SCALING ALGORITHM FOR THE MAXIMUM MEAN CUT PROBLEM

We present a new scaling algorithm for the maximum mean cut problem. T he mean of a cut is defined by the cut capacity divided by the number of area crossing the cut. The algorithm uses an approximate binary sea rch and solves the circulation feasibility problem with relaxed capaci ty bounds. The maximum mean cut problem has recently been studied as a dual analogue of the minimum mean cycle problem in the framework of t he minimum cost flow problem by Ervolina and McCormick. A network N = (G, lower, upper) with lower and upper are capacities is said to be de lta-feasible if N has a feasible circulation when we relax the capacit y bounds by delta; that is, we use (lower(a) - delta, upper(a) + delta ) bounds instead of(lower(a), upper(a)) bounds for each are a is an el ement of A. During an approximate binary search we maintain two bounds , LB and UB, such that N is LB-infeasible and UB-feasible, and we redu ce the interval size (LB, UB) by at least one-third at each iteration. For a graph with n vertices, m arcs, and integer capacities bounded b y U, the running time of this algorithm is O(mn log(nU)). This time bo und is better than the time achieved by McCormick and Ervolina under t he similarity condition (that is, U = O(nO(0(1)))). Our algorithm can be naturally used for the circulation feasibility problem, and thus pr ovides a new scaling algorithm for the minimum cut problem.