A Note on the Approximation of a Minimum-Weight Maximal Independent Set |
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Authors: | Marc Demange |
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Institution: | (1) CERMSEM, Université Paris I, Maison des Sciences Economiques, 106-112, Boulevard de l'Hôpital, 75647 Paris Cedex 13, France and;(2) LAMSADE, Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 Paris Cedex 16, France |
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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 guaranteed by every algorithm while no polynomial time algorithm can guarantee the ratio (1 – ). 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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Keywords: | combinatorial problem polynomial-time approximation algorithms computational complexity independent set |
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