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arXiv · 2011.06939

The Submodular Santa Claus Problem in the Restricted Assignment Case

Etienne Bamas·Paritosh Garg·Lars Rohwedder
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Abstract

The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application of their structural result. In the mentioned problem $n$ unsplittable resources have to be assigned to $m$ players, each with a monotone submodular utility function $f_i$. The goal is to maximize $\min_i f_i(S_i)$ where $S_1,\dotsc,S_m$ is a partition of the resources. The result by Goemans et al. implies a polynomial time $O(n^{1/2 +\varepsilon})$-approximation algorithm. Since then progress on this problem was limited to the linear case, that is, all $f_i$ are linear functions. In particular, a line of research has shown that there is a polynomial time constant approximation algorithm for linear valuation functions in the restricted assignment case. This is the special case where each player is given a set of desired resources $\Gamma_i$ and the individual valuation functions are defined as $f_i(S) = f(S \cap \Gamma_i)$ for a global linear function $f$. This can also be interpreted as maximizing $\min_i f(S_i)$ with additional assignment restrictions, i.e., resources can only be assigned to certain players. In this paper we make comparable progress for the submodular variant. Namely, if $f$ is a monotone submodular function, we can in polynomial time compute an $O(\log\log(n))$-approximate solution.

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Etienne Bamas, Paritosh Garg, Lars Rohwedder. 2020-11-13. The Submodular Santa Claus Problem in the Restricted Assignment Case. https://arxiv.org/abs/2011.06939

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