Algorithmica

, Volume 80, Issue 8, pp 2221–2239

# Approximability of Clique Transversal in Perfect Graphs

• Samuel Fiorini
• R. Krithika
• N. S. Narayanaswamy
• Venkatesh Raman
Article

## Abstract

Given an undirected simple graph G, a set of vertices is an r-clique transversal if it has at least one vertex from every r-clique. Such sets generalize vertex covers as a vertex cover is a 2-clique transversal. Perfect graphs are a well-studied class of graphs on which a minimum weight vertex cover can be obtained in polynomial time. Further, an r-clique transversal in a perfect graph is also a set of vertices whose deletion results in an $$(r-1)$$-colorable graph. In this work, we study the problem of finding a minimum weight r-clique transversal in a perfect graph. This problem is known to be $$\mathsf {NP}$$-hard for $$r \ge 3$$ and admits a straightforward r-approximation algorithm. We describe two different $$\frac{r+1}{2}$$-approximation algorithms for the problem. Both the algorithms are based on (different) linear programming relaxations. The first algorithm employs the primal–dual method while the second uses rounding based on a threshold value. We also show that the problem is APX-hard and describe hardness results in the context of parameterized algorithms and kernelization.

## Keywords

r-Clique transversal Odd cycle transversal Perfect graphs Approximation algorithms Linear programming

## References

1. 1.
Abu-Khzam, F.N.: A kernelization algorithm for $$d$$-hitting set. J. Comput. Syst. Sci. 76(7), 524–531 (2010)
2. 2.
Abu-Khzam, F.N., Fernau, H.: Kernels: annotated, proper and induced. In: Parameterized and Exact Computation, Volume 4169 of Lecture Notes in Computer Science, pp. 264–275. Springer, Berlin (2006)Google Scholar
3. 3.
Berge, C.: Färbung von graphen, deren sämtliche bzw. deren ungerade kreise starr sind (zusammenfassung). Wissenschaftliche Zeitschrift, Martin Luther Universität Halle-Wittenberg Mathematisch-Naturwissenschaftliche Reihe 10, 114–115 (1961)Google Scholar
4. 4.
Berry, L.A., Kennedy, W.S., King, A.D., Li, Z., Reed, B.A.: Finding a maximum-weight induced $$k$$-partite subgraph of an $$i$$-triangulated graph. Discret. Appl. Math. 158(7), 765–770 (2010)
5. 5.
Chudnovsky, M., Cornuéjols, G., Liu, X., Seymour, P.D., Vusković, K.: Recognizing berge graphs. Combinatorica 25(2), 143–186 (2005)
6. 6.
Cygan, M., Fomin, F .V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)
7. 7.
Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. J. Comput. Syst. Sci. 67, 789–807 (2003)
8. 8.
Chudnovsky, M., Robertson, N., Seymour, P.D., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164(1), 51–229 (2006)
9. 9.
Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162(1), 439–485 (2005)
10. 10.
Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM 61(4), 23:1–23:27 (2014)
11. 11.
Guruswami, V., Lee, E.: Inapproximability of feedback vertex set for bounded length cycles. Electron. Colloq. Comput. Complex. 21, 6 (2014)Google Scholar
12. 12.
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)
13. 13.
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 2nd edn. Elsevier, Amsterdam (2004)
14. 14.
Hochbaum, D .S.: Approximation Algorithms for NP-Hard Problems. PWS Publishing Co, Boston (1997)
15. 15.
Iwata, Y., Oka, K., Yuichi, Y.: Linear-time FPT algorithms via network flow. In: Proceedings of the ACM–SIAM Symposium on Discrete Algorithms, pp. 1749–1761 (2014)Google Scholar
16. 16.
Impagliazzo, R., Paturi, R.: Complexity of k-SAT. In: Proceedings of IEEE Conference on Computational Complexity, pp. 237–240 (1999)Google Scholar
17. 17.
Iwata, Y., Wahlström, M., Yoshida, Y.: Half-integrality, LP-branching, and FPT algorithms. SIAM J. Comput. 45(4), 1377–1411 (2016)
18. 18.
Krithika, R., Narayanaswamy, N.S.: Another disjoint compression algorithm for odd cycle transversal. Inf. Process. Lett. 113(22–24), 849–851 (2013)
19. 19.
Krithika, R., Narayanaswamy, N.S.: Parameterized algorithms for $$(r, l)$$-partization. J. Graph Algorithms Appl. 17(2), 129–146 (2013)
20. 20.
Knuth, D.E.: The sandwich theorem. Electron. J. Comb. 1(A1), 1–48 (1994)
21. 21.
Khot, S., Regev, O.: Vertex cover might be hard to approximate to within $$2-\epsilon$$. J. Comput. Syst. Sci. 74(3), 335–349 (2008)
22. 22.
Lokshtanov, D., Narayanaswamy, N .S., Raman, V., Ramanujan, M .S., Saurabh, S.: Faster parameterized algorithms using linear programming. ACM Trans. Algorithms 11(2), 15:1–15:31 (2014)
23. 23.
Lovász, L.: Normal hypergraphs and the perfect graph conjecture. Discret. Math. 2(3), 253–267 (1972)
24. 24.
Lovász, L.: On Minimax Theorems of Combinatorics. Ph.D thesis, Matemathikai Lapok, vol. 26, pp. 209–264 (1975)Google Scholar
25. 25.
Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980)
26. 26.
Nemhauser, G.L., Trotter, L.E.: Properties of vertex packing and independence system polyhedra. Math. Program. 6(1), 48–61 (1974)
27. 27.
Nemhauser, G.L., Trotter, L.E.: Vertex packings: structural properties and algorithms. Math. Program. 8(1), 232–248 (1975)
28. 28.
Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32, 299–301 (2004)
29. 29.
Wagler, A.: Critical and anticritical edges in perfect graphs. In: Graph-Theoretic Concepts in Computer Science, Volume 2204 of Lecture Notes in Computer Science, pp. 317–327 (2001)Google Scholar
30. 30.
Williamson, D .P., Shmoys, D .B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)

## Authors and Affiliations

• Samuel Fiorini
• 1
• R. Krithika
• 2
• N. S. Narayanaswamy
• 3
• Venkatesh Raman
• 2
1. 1.Département de MathématiqueUniversité libre de BruxellesBrusselsBelgium
2. 2.The Institute of Mathematical SciencesHBNIChennaiIndia
3. 3.Department of Computer Science and EngineeringIndian Institute of Technology MadrasChennaiIndia