# On the complexity of and algorithms for detecting **k**-length negative cost cycles

**k**

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## Abstract

Let *G* be a directed graph with an integral cost on each edge. For a given positive integer *k*, the *k*-length negative cost cycle (*kLNCC*) problem is to determine whether *G* contains a negative cost cycle with at least *k* edges. Because of its applications in deadlock avoidance in synchronized streaming computing network, *kLNCC* was first studied in paper (Li et al. in Proceedings of the 22nd ACM symposium on parallelism in algorithms and architectures, pp 243–252, 2010), but remains open whether the problem is \({\mathcal{NP}}\)-hard. In this paper, we first show that an even harder problem, the fixed-point *k*-length negative cost cycle *trail (FPkLNCCT)* problem that is to determine whether *G* contains 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 a multigraph even when \(k=3\) by reducing from the *3SAT* problem. Then, we prove the \({\mathcal{NP}}\)-completeness of *kLNCC* by giving a more sophisticated reduction from the 3 occurrence 3-satisfiability (*3O3SAT*) problem which is known \({\mathcal{NP}}\)-complete. The complexity result for *kLNCC* is interesting since polynomial-time algorithms are known for both *2LNCC,* which is actually equivalent to negative cycle detection, and the *k*-cycle problem, which is to determine whether *G* contains a cycle with of length at least *k*. Thus, this paper closes the open problem proposed by Li et al. (2010) whether *kLNCC* admits polynomial-time algorithms. Last but not the least, we present for *3LNCC* a randomized algorithm that, if *G* contains a negative cycle of length at most *L*, can find a solution with a probability \(1-\epsilon \) for any \(\epsilon \in (0,\,1]\) within runtime \(O(2^{\min \{L,\,h\}}mn\left\lceil \ln \frac{1}{\epsilon }\right\rceil )\), where *m*, *n* and *h* are respectively the numbers of edges, vertices and length 2 negative cost cycles in *G*.

## Keywords

*k*-Length negative cost cycle \({\mathcal{NP}}\)-Complete 3 occurrence 3-satisfiability 3-Satisfiability Randomized algorithm

## Notes

### Acknowledgements

This research work is supported by Natural Science Foundation of China (Nos. 61772005, 61300025) and Natural Science Foundation of Fujian Province (No. 2017J01753). Part of the work was done when Peng Li was with Washington University in St. Louis as a PhD student. The authors would like to thank the anonymous reviewers of COCOA 2017 for their insightful comments which helped us significantly improve the quality of the paper.

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