Relaxed Most Negative Cycle and Most Positive Cut Canceling Algorithms for Minimum Cost Flow

Abstract

This paper presents two new scaling algorithms for the minimum cost network flow problem, one a primal cycle canceling algorithm, the other a dual cut canceling algorithm. Both algorithms scale a relaxed optimality parameter, and create a second, inner relaxation. The primal algorithm uses the inner relaxation to cancel a most negative node-disjoint family of cycles w.r.t. the scaled parameter, the dual algorithm uses it to cancel most positive cuts w.r.t. the scaled parameter. We show that in a network with n nodes and m arcs, both algorithms need to cancel only O(mn) objects per scaling phase. Furthermore, we show how to efficiently implement both algorithms to yield weakly polynomial running times that are as fast as any other cycle or cut canceling algorithms. Our algorithms have potential practical advantages compared to some other canceling algorithms as well. Along the way, we give a comprehensive survey of cycle and cut canceling algorithms for min-cost flow. We also clarify the formal duality between cycles and cuts.

Fast Cycle Canceling Algorithms for Minimum Cost Submodular Flow*

“…The first notion is a generalization of an idea from the relaxed cycle canceling algorithm from our min cost flow paper [33]: Use an assignment problem with scaled and rounded costs that is a relaxation of the (NP Hard) problem of finding a most negative augmenting cycle to find a most negative family of cycles, and cancel the resulting cycles. Since each canceled cycle consists of tight arcs, we call this relaxation Tight Arc Cycle Canceling.…”

“…The min cost flow versions of the Tight Arc Cycle Canceling [33] include variants which do not need to round costs. Unfortunately, we have not been able to extend these variants to submodular flow.…”

“…This algorithm is, however, the first polynomial algorithm for submodular flow with separable convex costs. In [23] we develop a cut canceling algorithm that is the dual of Tight Arc Cycle Canceling algorithm of this paper, in the sense that they both come from extensions of a dual pair of min cost flow algorithms from [33]. The paper [6] shows how to improve the capacity scaling algorithm to an O(n 4 h log U ) algorithm, which can also be seen as an improvement of the algorithm in [23].…”

“…The strategy of this paper is to extend cycle canceling algorithms for min cost flow (see [33] for a survey) to submodular flow. Cui and Fujishige [2] extended the minimum-mean cycle canceling algorithm of Goldberg and Tarjan [15], but the running time is only pseudopolynomial.…”

“…Section 3 develops two fast polynomial cycle canceling algorithms for submodular flow. The first is Tight Arc Cycle Canceling, which is an extension of our relaxed most negative disjoint family of cycles canceling algorithm for minimum cost flow [33]. The second is Negative Arc Cycle Canceling, which is an extension of Goldberg's cycle canceling algorithm [14].…”