Multi-Agent Submodular Optimization

• Published in: arXiv.org
• Main Authors: Santiago, Richard, Shepherd, F Bruce
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 04.11.2018
• Subjects: Approximation; Computer Science - Data Structures and Algorithms; Graph theory; Mathematical analysis; Multiagent systems; Optimization
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1803.03767

Recent years have seen many algorithmic advances in the area of submodular optimization: (SO) \(\min/\max~f(S): S \in \mathcal{F}\), where \(\mathcal{F}\) is a given family of feasible sets over a ground set \(V\) and \(f:2^V \rightarrow \mathbb{R}\) is submodular. This progress has been coupled with a wealth of new applications for these models. Our focus is on a more general class of \emph{multi-agent submodular optimization} (MASO) which was introduced by Goel et al. in the minimization setting: \(\min \sum_i f_i(S_i): S_1 \uplus S_2 \uplus \cdots \uplus S_k \in \mathcal{F}\). Here we use \(\uplus\) to denote disjoint union and hence this model is attractive where resources are being allocated across \(k\) agents, each with its own submodular cost function \(f_i()\). In this paper we explore the extent to which the approximability of the multi-agent problems are linked to their single-agent {\em primitives}, referred to informally as the {\em multi-agent gap}. We present different reductions that transform a multi-agent problem into a single-agent one. For maximization we show that (MASO) admits an \(O(\alpha)\)-approximation whenever (SO) admits an \(\alpha\)-approximation over the multilinear formulation, and thus substantially expanding the family of tractable models. We also discuss several family classes (such as spanning trees, matroids, and \(p\)-systems) that have a provable multi-agent gap of 1. In the minimization setting we show that (MASO) has an \(O(\alpha \cdot \min \{k, \log^2 (n)\})\)-approximation whenever (SO) admits an \(\alpha\)-approximation over the convex formulation. In addition, we discuss the class of "bounded blocker" families where there is a provably tight O\((\log n)\) gap between (MASO) and (SO).