# Coloring *k*-Colorable Semirandom Graphs in Polynomial Expected Time via Semidefinite Programming

## Abstract

The analysis of algorithms on semirandom graph instances intermediates smoothly between the analysis in the worst case and the average case. The aim of this paper is to present an algorithm for finding a large independent set in a semirandom graph in polynomial expected time, thereby extending the work of Feige and Kilian [4]. In order to preprocess the input graph, the algorithm makes use of SDP-based techniques. The analysis of the algorithm shows that not only is the expected running time polynomial, but even arbitrary moments of the running time are polynomial in the number of vertices of the input graph. The algorithm for the independent set problem yields an algorithm for *k*-coloring semirandom *k*-colorable graphs, and for the latter algorithm a similar result concerning its running time holds, as also for the independent set algorithm. The results on both problems are essentially best-possible, by the hardness results obtained in [4].

## Keywords

Polynomial Time Algorithm Input Graph Hardness Result Coin Toss Expected Running Time## Preview

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## References

- 1.Alon, N., Kahale, N.: Approximating the independence number via the ϑ-function. Math. Programming
**80**(1998) 253–264.MathSciNetGoogle Scholar - 2.Blum, A., Spencer, J.: Coloring random and semirandom
*k*-colorable graphs. J. of Algorithms**19(2)**(1995) 203–234MathSciNetCrossRefGoogle Scholar - 3.Coja-Oghlan, A.: Zum Färben
*k*-färbbarer semizufälliger Graphen in erwarteter Polynomzeit mittels Semidefiniter Programmierung. Technical Report**141**, Fachbereich Mathematik der Universität Hamburg (2002)Google Scholar - 4.Feige, U., Kilian, J.: Heuristics for semirandom graph problems. preprint (2000)Google Scholar
- 5.Garey, M.R., Johnson, D.S.: Computers and intractability. W.H. Freeman and Company 1979Google Scholar
- 6.Håstad, J.: Clique is hard to approximate within n
^{1-ε}. Proc. 37th Annual Symp. on Foundations of Computer Science (1996) 627–636Google Scholar - 7.Karger, D., Motwani, R., Sudan, M.: Approximate graph coloring by semidefinite programming. Proc. of the 35th. IEEE Symp. on Foundations of Computer Science (1994) 2–13Google Scholar
- 8.Khanna, S., Linial, N., Safra, S.: On the hardness of approximating the chromatic number, Proc. 2nd Israeli Symp. on Theory and Computing Systems (1992) 250–260Google Scholar
- 9.Mahajan, S., Ramesh, H.: Derandomizing semidefinite programming based approximation algorithms. Proc. 36th IEEE Symp. on Foundations of Computer Science (1995) 162–169Google Scholar
- 10.Subramanian, C.: Minimum coloring random and semirandom graphs in polynomial expected time. Proc. 36th Annual Symp. on Foundations of Computer Science (1995) 463–472Google Scholar