Abstract
Let \(f:2^{N} \rightarrow \cal R^{+}\) be a non-decreasing submodular set function, and let \((N,\cal I)\) be a matroid. We consider the problem \(\max_{S \in \cal I} f(S)\). It is known that the greedy algorithm yields a 1/2-approximation [9] for this problem. It is also known, via a reduction from the max-k-cover problem, that there is no (1 − 1/e + ε)-approximation for any constant ε> 0, unless P = NP [6]. In this paper, we improve the 1/2-approximation to a (1 − 1/e)-approximation, when f is a sum of weighted rank functions of matroids. This class of functions captures a number of interesting problems including set coverage type problems. Our main tools are the pipage rounding technique of Ageev and Sviridenko [1] and a probabilistic lemma on monotone submodular functions that might be of independent interest.
We show that the generalized assignment problem (GAP) is a special case of our problem; although the reduction requires |N| to be exponential in the original problem size, we are able to interpret the recent (1 − 1/e)-approximation for GAP by Fleischer et al. [10] in our framework. This enables us to obtain a (1 − 1/e)-approximation for variants of GAP with more complex constraints.
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Calinescu, G., Chekuri, C., Pál, M., Vondrák, J. (2007). Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract). In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_15
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DOI: https://doi.org/10.1007/978-3-540-72792-7_15
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