Abstract
We consider the polynomial approximation behavior of the problem of finding, in a graph with weighted vertices, a maximal independent set minimizing the sum of the weights. In the spirit of a work of Halldórson dealing with the unweighted case, we extend it and perform approximation hardness results by using a reduction from the minimum coloring problem. In particular, a consequence of our main result is that there does not exist any polynomial time algorithm approximating this problem within a ratio independent of the weights, unless P = NP. We bring also to the fore a very simple ratio ρ guaranteed by every algorithm while no polynomial time algorithm can guarantee the ratio (1 − ∈)ρ. The known hardness results for the unweighted case can be deduced. We finally discuss approximation results for both weighted and unweighted cases: we perform an approximation ratio that is valid for any algorithm for the former and propose an analysis of a greedy algorithm for the latter.
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References
M. Bellare, O. Goldreich, and M. Sudan, Free bits, PCPs and non-approximability: towards tight results, in Proc. of 36th Ann. IEEE Symp. on Foundations of Comput. Sci., IEEE Computer Society, pp. 422–431, 1995.
C. Berge, Graphs and Hypergraphs, North Holland: Amsterdam, 1973.
V. Chvátal, A Greedy Heuristic for the Set Covering Problem, Mathematics of Operations Research, vol. 4, pp. 233–235, 1979.
P. Crescenzi and V. Kann, A Compendium of NP Optimization Problems, available in electronic form at http://www.nada.kth.se/~viggo/problemlist/compendium.html, 1996.
M. Demange and V.T. Paschos, Improved Approximations for Maximum Independent Set via Approximation Chain, Appl. Math. Lett., vol. 10, pp. 105–110, 1997.
U. Feige and J. Kilian, Zero Knowledge and the Chromatic Number, in Proc. 11th Annual IEEE Conf. Comp. Complexity, pp. 278–287, 1996.
M.R. Garey and D.S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness, Freeman: San Francisco, 1979.
M.M. Halldórsson, A Still Better Performance Guarantee for Approximate Graph Coloring, Information Processing Letters, vol. 45, pp. 19–23, 1993.
M.M. Halldórsson, Approximating the Minimum Maximal Independence Number, Information Processing Letters, vol. 46, pp. 169–172, 1993.
M.M. Halldórsson, Approximation via Partitioning, Technical Report IS-RR-95–0003F, School of Information Science, Japan Advanced Institute of Science and Technology, Hokuriku, 1995.
M.M. Halldórsson and J. Radhakrishnan, Greed is Good: Approximation Independent Set in Sparse and Bounded Degree Graphs, in Proc. 26th Ann. ACM Symp. on Theory of Computing, pp. 439–448, 1994.
J. Håstad, Clique is Hard to Approximate within n 1−ε, in Proc. 37th Annual IEEE Symp. Found. Comput. Sci., pp. 627–636, 1996.
R.W. Irving, On Approximating the Minimum Independent Dominating Set, Information Processing Letters, vol. 37, pp. 197–200, 1991.
D. S. Johnson, Worst-case Behaviour of Graph Coloring Algorithms, in Proc. 5th Sountheastern Conf. on Combinatorics, Graph Theory and Computing, Utilitas Mathematica Publ.: Winnipeg, Ont., pp. 513–527, 1974.
C. Lund and M. Yannakakis, On the Hardness of Approximating Minimization Problems, Proc. STOC'93, pp. 286–293, 1993.
H.U. Simon, On Approximate Solutions for Combinatorial Optimization Problems, SIAM J. Disc. Math., vol. 3, pp. 294–310, 1990.
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Demange, M. A Note on the Approximation of a Minimum-Weight Maximal Independent Set. Computational Optimization and Applications 14, 157–169 (1999). https://doi.org/10.1023/A:1008765214400
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DOI: https://doi.org/10.1023/A:1008765214400