On the Complexity of Detecting k-Length Negative Cost Cycles

Given a positive integer k and a directed graph G with a real number cost on each edge, the k-length negative cost cycle (k LNCC) problem that first emerged in deadlock avoidance in synchronized streaming computing network [14] is to determine whether G contains a negative cost cycle of at least k edges. The paper first shows a related problem of k LNCC, namely the fixed-point trail with k-length negative cost cycle (FPT k LNCC) problem which is to determine whether there exists a negative closed trail enrouting a given vertex as the fixed point and containing only cycles with at least k edges, is \(\mathcal{NP}\)-complete in multigraphs even when \(k=3\) by reducing from the 3SAT problem. Then as the main result, we prove the \(\mathcal{NP}\)-completeness of k LNCC by giving a more sophisticated reduction from the 3 Occurrence 3-Satisfiability (3O3SAT) problem, a known \(\mathcal{NP}\)-complete special case of 3SAT in which a variable occurs at most three times. The complexity result is surprising, since polynomial time algorithms are known for both 2LNCC (essentially no restriction on the value of k) and the k-cycle problem with k being fixed which is to determine whether there exists a cycle of at least length k in a given graph. This closes the open problem proposed by Li et al. in [14, 15] whether k LNCC admits polynomial-time algorithms.