On the Complexity of Detecting Constrained Negative Cost Cycles

Given a positive integer $k$ and a directed graph with a cost on each edge, the $k$-length negative cost cycle ($k$\emph{LNCC}) problem is to determine whether there exists a negative cost cycle with at least $k$ edges, and the fixed-point \emph{$k$-}length negative cost cycle \emph{trail (FP$k$LNCCT)} 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 $k$\emph{LNCC} problem first emerged in deadlock avoidance in synchronized streaming computing network \cite{spaa10}, generalizing two famous problems: negative cycle detection and the $k$-cycle problem. As a warmup by-production, the paper first shows that \emph{FP$k$LNCCT is }${\cal NP}$-complete in multigraph\emph{ }even for\emph{ $k=3$} by reducing from the \emph{3SAT} problem. Then as the main result, we prove the ${\cal NP}$-completeness of $k$\emph{LNCC} by giving a sophisticated reduction from the 3 Occurrence 3-Satisfiability (\emph{3O3SAT}) problem, a known ${\cal NP}$-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 $2$\emph{LNCC} (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 \cite{spaa10} whether $k$\emph{LNCC} admits polynomial-time algorithms.