Finding all minimum cost flows and a faster algorithm for the K best flow problem

TL;DR

This paper presents a new, faster algorithm for finding all optimal integer flows in a network, improving efficiency over previous methods and providing bounds on the number of solutions.

Contribution

The authors introduce an improved algorithm for the all optimal integer flow problem, replacing shortest path computations with a more efficient cycle detection method.

Findings

Algorithm runs in O(F (m + n) + mn + M) time for all optimal flows.

Enhanced method for K best integer flows with O(Kn^3 + M) complexity.

Derived bounds on the number of optimal and feasible integer solutions.

Abstract

This paper addresses the problem of determining all optimal integer solutions of a linear integer network flow problem, which we call the all optimal integer flow (AOF) problem. We derive an O(F (m + n) + mn + M ) time algorithm to determine all F many optimal integer flows in a directed network with n nodes and m arcs, where M is the best time needed to find one minimum cost flow. We remark that stopping Hamacher's well-known method for the determination of the K best integer flows at the first sub-optimal flow results in an algorithm with a running time of O(F m(n log n + m) + M ) for solving the AOF problem. Our improvement is essentially made possible by replacing the shortest path sub-problem with a more efficient way to determine a so called proper zero cost cycle using a modified depth-first search technique. As a byproduct, our analysis yields an enhanced algorithm to determine the K best integer flows that runs in O(Kn3 + M ). Besides, we give lower and upper bounds for the number of all optimal integer and feasible integer solutions. Our bounds are based on the fact that any optimal solution can be obtained by an initial optimal tree solution plus a conical combination of incidence vectors of all induced cycles with bounded coefficients.