Bounds on Longest Simple Cycles in Weighted Directed Graphs via Optimum Cycle Means
Ali Dasdan
TL;DR
The paper tackles the challenge of bounding the weight and length of the longest simple cycle in directed weighted graphs, a problem that is NP-hard to solve exactly or approximate. It leverages Optimum Cycle Means, namely $\lambda_{min}$ and $\lambda_{max}$, which are computable in strongly polynomial time, to derive both strict algebraic bounds and heuristic estimators for longest cycles. The authors introduce mean-based inequalities, distinguish between max-weight and max-length cycles, and provide dual results for shortest cycles, along with two bounded-error estimators, $\lambda_{avg}$ and $\lambda_{geo}$, whose performance depends on weight distributions. Experimental results on ISCAS circuits show that strict bounds are often loose, but the heuristics offer bounded, distribution-dependent accuracy (best with uniform weights for $\lambda_{avg}$ and with skewed weights for $\lambda_{geo}$), enabling practical use in branch-and-bound and related problems. The work highlights potential for integrating these bounds into solvers and clarifies limitations, especially under highly skewed weight distributions and varying graph structures.
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.