# Vertex Cover in Conflict Graphs: Complexity and a Near Optimal Approximation

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9486)

## Abstract

Given finite number of forests of complete multipartite graph, conflict graph is a sum graph of them. Graph of this class can model many natural problems, such as in database application and others. We show that this property is non-trivial if limiting the number of forests of complete multipartite graph, then study the problem of vertex cover on conflict graph in this paper. The complexity results list as follow,
• If the number of forests of complete multipartite graph is fixed, conflict graph is non-trivial property, but finding 1.36-approximation algorithms is NP-hard.

• Given 2 forests of complete multipartite graph and maximum degree less than 7, vertex cover problem of conflict graph is NP-complete. Without the degree restriction, it is shown to be NP-hard to find an algorithm for vertex cover of conflict graph within $$\frac{17}{16}-\varepsilon$$, for any $$\varepsilon >0$$.

Given conflict graph consists of r forests of complete multipartite graph, we design an approximation algorithm and show that the approximation ratio can be bounded by $$2-\frac{1}{2^r}$$. Furthermore, under the assumption of UGC, the approximation algorithm is shown to be near optimal by proving that, it is hard to improve the ratio with a factor independent of the size (number of vertex) of conflict graph.

## Keywords

Approximation algorithm Vertex cover Complete multipartite graph

## References

1. 1.
Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Boston (1995)
2. 2.
Ásgeirsson, E.I., Stein, C.: Divide-and-conquer approximation algorithm for vertex cover. SIAM J. Discret. Math. 23(3), 1261–1280 (2009)
3. 3.
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Ann. Discrete Math. 25, 27–46 (1985)
4. 4.
Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162, 439–485 (2005)
5. 5.
Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Comput. 1(2), 180–187 (1972)
6. 6.
Håstad, J.: Some optimal inapproximability results. J. ACM (JACM) 48(4), 798–859 (2001)
7. 7.
Karakostas, G.: A better approximation ratio for the vertex cover problem. ACM Trans. Algorithms 5(4), 41:1–41:8 (2009)
8. 8.
Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2 - $$\epsilon$$. J. Comput. Syst. Sci. 74(3), 335–349 (2008)
9. 9.
Kuhn, F., Mastrolilli, M.: Vertex cover in graphs with locally few colors. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 498–509. Springer, Heidelberg (2011)
10. 10.
Monien, B., Speckenmeyer, E.: Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Inform. 22(1), 115–123 (1985)
11. 11.
Nemhauser, G.L., Trotter Jr, L.E.: Vertex packings: structural properties and algorithms. Math. Program. 8(1), 232–248 (1975)
12. 12.
Sitton, D.: Maximum matchings in complete multipartite graphs. Furman Univ. Electron. J. Undergraduate Math. 2, 6–16 (1996)Google Scholar
13. 13.
Williamson, D.P., Shmoys, D.B.: The design of approximation algorithms. Cambridge University Press, New York (2011)

© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Dongjing Miao
• 1
Email author
• Jianzhong Li
• 1
• Xianmin Liu
• 1
• Hong Gao
• 1
1. 1.Harbin Institute of TechnologyHarbinChina