Cycles in Generalized Networks

Abstract

Generalized networks specify two parameters for each arc, a cost and a gain. If x units enter an arc a, then x · g(a) exit. Arcs may generate or consume flow, i.e., they are gainy or lossy. The objective is a cheapest path of a unit flow from the source (SGSP) and the single-pair cheapest path (SPGSP).

There are several types of negative cycles. A lossy cycles decreases the gain. Then even a negative cost cycle has only bounded cost. A gainy cycle increases the flow. Then even a positive cost cycle may induce a total cost of minus infinity.

We solve SGSP by an extension of the Bellman-Ford algorithm. At the heart of the algorithm is a new and effective cycle detection strategy. The algorithm solves SGSP in O(nmlog n), which improves to O(nm) in lossless networks and to O(n log n + m) in a monotone setting. Our algorithm is simpler and at least a factor of O(n) faster than the previous algorithms using linear programming or complex parametric search and scaling techniques. This improvement is a big step for such a well-investigated problem.

To the contrary, the single-pair generalized shortest path problem SPGSP is NP-hard, even with nonnegative costs and uniformly lossy arcs.