Shinji Misono

A new scaling algorithm for the maximum mean cut problem

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(no(1))). Our algorithm can be naturally used for the circulation feasibility problem, and thus provides a new scaling algorithm for the minimum cut problem.

New TSP Construction Heuristics and Their Relationshipsto the 2-Opt

Correction heuristics for the traveling salesman problem (TSP), with the 2-Opt applied as a postprocess, are studied with respect to their tour lengths and computation times. This study analyzes the “2-Opt dependency,” which indicates how the performance of the 2-Opt depends on the initial tours built by the construction heuristics. In accordance with the analysis, we devise a new construction heuristic, the recursive-selection with long-edge preference (RSL) method, which runs faster than the multiple-fragment method and produces a comparable tour when they are combined with the 2-Opt.