A Zero-Space algorithm for Negative Cost Cycle Detection in networks

Abstract

This paper is concerned with the problem of checking whether a network with positive and negative costs on its arcs contains a negative cost cycle. The Negative Cost Cycle Detection (NCCD) problem is one of the more fundamental problems in network design and finds applications in a number of domains ranging from Network Optimization and Operations Research to Constraint Programming and System Verification. As per the literature, approaches to this problem have been either Relaxation-based or Contraction-based. We introduce a fundamentally new approach for negative cost cycle detection; our approach, which we term as the Stressing Algorithm, is based on exploiting the connections between the NCCD problem and the problem of checking whether a system of difference constraints is feasible. The Stressing Algorithm is an incremental, comparison-based procedure which is as efficient as the fastest known comparison-based algorithm for this problem. In particular, on a network with n vertices and m edges, the Stressing Algorithm takes O(m · n) time to detect the presence of a negative cost cycle or to report that none exists. A very important feature of the Stressing Algorithm is that it uses zero extra space; this is in marked contrast to all known algorithms that require (n) extra space. It is well known that the NCCD problem is closely related to the Single Source Shortest Paths (SSSP) problem, i.e., the problem of determining the shortest path distances of all the vertices in a network, from a specified source; indeed most algorithms in the literature for the NCCD problem are modifications of approaches to the SSSP problem. At this juncture, it is not clear whether the Stressing Algorithm could be extended to solve the SSSP problem, even if O(n) extra space is available.

A mechanical verification of the stressing algorithm for negative cost cycle detection in networks

“…The principal motivation for our work is that by mechanizing the proof of correctness, we have developed the background needed for verifying graph algorithms related to the NCCD problem. We also note that the proof outlined here is significantly simpler than the proof of correctness described in [12], which makes use of mathematical induction over polyhedral spaces. The mechanization described here led us to an even simpler proof for an algorithm that identifies the negative cycle using only O(n) extra space, essentially to store the pointer to the edge along which a vertex was most recently stressed.…”

“…In this paper, we focus on the correctness analysis of a theoretically efficient algorithm for NCCD called the stressing algorithm. This algorithm is essentially greedy and was first proposed as a distributed, space-efficient algorithm for the problem of detecting inconsistencies in sensor network arrangements [11] and subsequently proved correct [12]. Here, we focus on proving the correctness of an implementation of the Stressing Algorithm using SRI's Prototype Verification System (PVS) [8].…”

“…In Section 2, a brief introduction to the Stressing Algorithm is provided; a comprehensive exposition can be found in [12]. Section 3 details the verification of the Stressing Algorithm in PVS including the explicit construction of the negative cycle.…”

On the Design, Analysis, and Implementation of Algorithms for Selected Problems in Graphs and Networks

“…The problem of finding a negative cost cycle in a graph is called the negative cost cycle detection problem (NCCD). For directed graphs, the problem has been widely studied [9,10,11,12,13,14,15,16]. In Part II of this thesis, we focus on undirected graphs.…”