NP-Completeness of the Negative k-Cycle Problem

Given a positive integer k and a directed graph with a cost on each edge, the k-length negative cost cycle (kLNCC ) problem is to determine whether there exists a negative cost cycle with at least k edges, and the fixed-point k-length negative cost cycle trail (FPkLNCCT) problem is to determine whether there exists a negative trail enrouting a given vertex (as the fixed point) and containing only cycles with at least k edges. The kLNCC problem first emerged in deadlock avoidance in synchronized streaming computing network [10], generalizing two famous problems: negative cycle detection and the k-cycle problem. As a warmup by-production, the paper first shows that FPkLNCCT is NP-complete in multigraph even for k = 3 by reducing from the 3SAT problem. Then as the main result, we prove the NP-completeness of kLNCC by giving a sophisticated reduction from the 3 Occurrence 3-Satisfiability (3O3SAT ) problem, a known NP-complete special case of 3SAT in which a variable occurs at most three times. The complexity result is interesting, since polynomial time algorithms are known for both 2LNCC (essentially no restriction on the value of k) and the k-cycle problem of fixed k. This paper closes the open problem proposed by Li et al. in [10] whether kLNCC admits polynomial-time algorithms.