WALCOM 2018: WALCOM: Algorithms and Computation pp 240-251

# Complexity of the Maximum k-Path Vertex Cover Problem

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10755)

## Abstract

This paper introduces the maximum version of the k-path vertex cover problem, called the Maximum k-Path Vertex Cover problem ($$\mathsf{{Max}}{P_k}\mathsf{VC}$$ for short): A path consisting of k vertices, i.e., a path of length $$k-1$$ is called a k-path. If a k-path $$P_k$$ includes a vertex v in a vertex set S, then we say that S or v covers $$P_k$$. Given a graph $$G = (V, E)$$ and an integer s, the goal of $$\mathsf{{Max}}{P_k}\mathsf{VC}$$ is to find a vertex subset $$S\subseteq V$$ of at most s vertices such that the number of k-paths covered by S is maximized. $$\mathsf{{Max}}{P_k}\mathsf{VC}$$ is generally NP-hard. In this paper we consider the tractability/intractability of $$\mathsf{{Max}}{P_k}\mathsf{VC}$$ on subclasses of graphs: We prove that $$\mathsf{{Max}}{P_3}\mathsf{VC}$$ and $$\mathsf{{Max}}{P_4}\mathsf{VC}$$ remain NP-hard even for split graphs and for chordal graphs, respectively. Furthermore, if the input graph is restricted to graphs with constant bounded treewidth, then $$\mathsf{{Max}}{P_3}\mathsf{VC}$$ can be solved in polynomial time.

## References

1. 1.
Apollonio, N., Simeone, B.: The maximum vertex coverage problem on bipartite graphs. Dis. Appl. Math. 165, 37–48 (2014)
2. 2.
Betzler, N., Niedermeier, R., Uhlmann, J.: Tree decompositions of graphs: saving memory in dynamic programming. Dis. Optim. 3, 220–229 (2006)
3. 3.
Hans, L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)
4. 4.
Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: A $${O}(c^k n)$$ 5-approximation algorithm for treewidth. SIAM J. Comput. 45(2), 317–378 (2016)
5. 5.
Brešar, B., Kardoš, F., Katrenič, J., Semanišin, G.: Minimum k-path vertex cover. Dis. Appl. Math. 159(12), 1189–1195 (2011)
6. 6.
Camby, E.: Connecting hitting sets and hitting paths in graphs. Ph.D. thesis, Doctoral Thesis (2015)Google Scholar
7. 7.
Caskurlu, B., Mkrtchyan, V., Parekh, O., Subramani, K.: On partial vertex cover and budgeted maximum coverage problems in bipartite graphs. In: IFIP International Conference on Theoretical Computer Science, pp. 13–26. Springer, Heidelberg (2014)Google Scholar
8. 8.
Courcelle, B.: Graph rewriting: an algebraic and logic approach. In: Handbook of Theoretical Computer Science, vol. B, pp. 193–242 (1990)Google Scholar
9. 9.
Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)
10. 10.
Karp, R.: Reducibility among combinatorial problems. In: Compleixty of Computer Computations, pp. 85–103 (1972)Google Scholar
11. 11.
Katrenič, J.: A faster FPT algorithm for 3-path vertex cover. Inf. Process. Lett. 116(4), 273–278 (2016)
12. 12.
Li, X., Zhang, Z., Huang, X.: Approximation algorithms for minimum (weight) connected k-path vertex cover. Dis. Appl. Math. 205, 101–108 (2016)
13. 13.
Papadimitriou, C.H., Yannakakis, M.: The complexity of restricted spanning tree problems. J. ACM 29(2), 285–309 (1982)
14. 14.
Jianhua, T., Jin, Z.: An FPT algorithm for the vertex cover P4 problem. Dis. Appl. Math. 200, 186–190 (2016)
15. 15.
Jianhua, T., Zhou, W.: A factor 2 approximation algorithm for the vertex cover P3 problem. Inf. Process. Lett. 111(14), 683–686 (2011)
16. 16.
Jianhua, T., Zhou, W.: A primal-dual approximation algorithm for the vertex cover P3 problem. Theor. Comput. Sci. 412(50), 7044–7048 (2011)
17. 17.
Xiao, M., Kou, S.: Exact algorithms for the maximum dissociation set and minimum 3-path vertex cover problems. Theor. Comput. Sci. 657, 86–97 (2017)
18. 18.
Yannakakis, M.: Node-deletion problems on bipartite graphs. SIAM J. Comput. 10, 310–327 (1981)
19. 19.
Zhang, Z., Li, X., Shi, Y., Nie, H., Zhu, Y.: PTAS for minimum k-path vertex cover in ball graph. Inf. Process. Lett. 119, 9–13 (2017)