An improved derandomized approximation algorithm for the max-controlled set problem

Martinhon, Carlos; Protti, Fábio

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 45 (2011) no. 2, pp. 181-196.

A vertex i of a graph G = (V,E) is said to be controlled by $M\subseteq V$ if the majority of the elements of the neighborhood of i (including itself) belong to M. The set M is a monopoly in G if every vertex $i\in V$ is controlled by M. Given a set $M\subseteq V$ and two graphs G₁ = ($V,E_1$) and G₂ = ($V,E_2$) where $E_1\subseteq E_2$, the monopoly verification problem (mvp) consists of deciding whether there exists a sandwich graph G = (V,E) (i.e., a graph where $E_1\subseteq E\subseteq E_2$) such that M is a monopoly in G = (V,E). If the answer to the mvp is No, we then consider the max-controlled set problem (mcsp), whose objective is to find a sandwich graph G = (V,E) such that the number of vertices of G controlled by M is maximized. The mvp can be solved in polynomial time; the mcsp, however, is NP-hard. In this work, we present a deterministic polynomial time approximation algorithm for the mcsp with ratio $\frac{1}{2}$ + $\frac{1+\sqrt{n}}{2n-2}$, where n=|V|>4. (The case $n\le 4$ is solved exactly by considering the parameterized version of the mcsp.) The algorithm is obtained through the use of randomized rounding and derandomization techniques based on the method of conditional expectations. Additionally, we show how to improve this ratio if good estimates of expectation are obtained in advance.

Classification : 68W20, 68W25
Mots clés : derandomization, Monte Carlo method, randomized rounding, sandwich problems

[1] S. Arora and S. Safra, Probabilistic checking of proofs: A new characterization of NP. J. ACM 45 (1998) 70-122. | MR | Zbl

[2] J.-C. Bermond and D. Peleg, The power of small coalitions in graphs. Discrete Appl. Math. 127 (2003) 399-414. | MR | Zbl

[3] P. Dagum, R. Karp, M. Luby and S. Ross, An optimal algorithm for Monte Carlo estimation. SIAM J. Comput. 29 (2000) 1484-1496. | MR | Zbl

[4] R.G. Downey and M.R. Fellows, Fixed parameter tractability and completeness I: Basic results. SIAM J. Comput. 24 (1995) 873-921. | MR | Zbl

[5] D. Dubashi and D. Ranjan, Balls and bins: A study of negative dependence. Random Struct. Algorithms 13 (1998) 99-124. | MR | Zbl

[6] P. Erdös and J. Spencer, The Probabilistic Method in Combinatorics. Academic Press, San Diego (1974). | Zbl

[7] D. Fitoussi and M. Tennenholtz, Minimal social laws. Proc. AAAI'98 (1998) 26-31. | MR

[8] R. Gandhi, S. Khuler, S. Parthasarathy and A. Srinivasan, Dependent rounding and its applications to approximation algorithms. J. ACM 53 (2006) 324-360. | MR

[9] M.C. Golumbic, H. Kaplan and R. Shamir, Graph sandwich problems. J. Algorithms 19 (1994) 449-473. | MR | Zbl

[10] H. Kaplan and R. Shamir, Bounded degree interval sandwich problems. Algorithmica 24 (1999) 96-104. | MR | Zbl

[11] N. Karmarkar, A new polynomial time algorithm for linear programming. Combinatorica 4 (1984) 375-395. | MR | Zbl

[12] S. Khot, On the power of unique 2-prover 1-round games, in STOC '02: Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, NY, USA, ACM Press (2002) 767-775. | MR | Zbl

[13] N. Linial, D. Peleg, Y. Rabinovich and N. Saks, Sphere packing and local majorities in graphs. Proc. 2nd Israel Symposium on Theoretical Computer Science, IEEE Computer Society Press, Rockville, MD (1993) 141-149.

[14] K. Makino, M. Yamashita and T. Kameda, Max-and min-neighborhood monopolies. Algorithmica 34 (2002) 240-260. | MR | Zbl

[15] R. Motwani and P. Raghavan, Randomized Algorithms. Cambridge University Press, London, 1995. | MR | Zbl

[16] D. Peleg, Local majority voting, small coalitions and controlling monopolies in graphs: A review. Technical Report CS96-12, Weizmann Institute, Rehovot (1996).

[17] P. Raghavan and C.D. Thompson, Randomized rounding: A technique for provably good algorithms and algorithmic proofs. Combinatorica 7 (1987) 365-374. | MR | Zbl

[18] J.D. Rose, A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations, in Graph Theory and Computing, edited by R.C. Reed, Academic Press, New York (1972) 183-217. | MR | Zbl

[19] Y. Shoham and M. Tennenholtz, Emergent conventions in multi-agent systems: Initial experimental results and observations. Proc. International Conference on Principles of Knowledge Representation and Reasoning (1992) 225-231.

[20] Y. Shoham and M. Tennenholtz, On the systhesis of useful social laws for artificial agent societies. Proc. AAAI'92 (1992) 276-281.

[21] S.J. Wright, Primal-Dual Interior-Point Methods. SIAM (1997). | MR | Zbl

[22] M. Yannakakis, Computing the minimum fill-in is NP-complete. SIAM J. Algebr. Discrete Methods 2 (1981) 77-79. | MR | Zbl