Bounds on Longest Simple Cycles in Weighted Directed Graphs via Optimum Cycle Means

TL;DR

This paper introduces a novel method using optimum cycle means to efficiently estimate bounds and heuristics for the longest simple cycle in weighted directed graphs, improving over existing approaches.

Contribution

It presents a new approach leveraging optimum cycle means to derive bounds and heuristics for the longest cycle problem, applicable to general graphs.

Findings

Heuristic estimates are significantly tighter than strict bounds.

Arithmetic mean is better for symmetric weight distributions.

Geometric mean performs better for skewed weight distributions.

Abstract

The problem of finding the longest simple cycle in a directed graph is NP-hard, with critical applications in computational biology, scheduling, and network analysis. Existing approaches include exact algorithms with exponential runtimes, approximation algorithms limited to specific graph classes, and heuristics with no formal guarantees. In this paper, we exploit optimum cycle means (minimum and maximum cycle means), computable in strongly polynomial time, to derive both strict bounds and heuristic estimates for the weight and length of the longest simple cycle in general graphs. The strict bounds can prune search spaces in exact algorithms while the heuristic estimates (the arithmetic mean and geometric mean of the optimum cycle means) guarantee bounded approximation error. Crucially, a single computation of optimum cycle means yields both the bounds and the heuristic estimates. Experimental evaluation on ISCAS benchmark circuits demonstrates that, compared to true values, the strict algebraic lower bounds are loose (median 80--99% below) while the heuristic estimates are much tighter: the arithmetic mean and the geometric mean have median errors of 6--13% vs. 11--21% for symmetric (uniform) weights and 41--92% vs. 25--35% for skewed (log-normal) weights, favoring the arithmetic mean for symmetric distributions and the geometric mean for skewed distributions.