A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints

TL;DR

This paper introduces a tight approximation algorithm for maximizing monotone submodular functions under mixed packing and covering constraints, with extensions to matroid and multi-objective scenarios, and offers a novel combinatorial dynamic programming method.

Contribution

The paper presents a new enumeration-based approximation algorithm that handles both packing and covering constraints, extending to matroid and multi-objective cases, and introduces a fast, deterministic combinatorial approach.

Findings

Achieves a $1 - 1/e - \\epsilon$ approximation ratio with minimal constraint violation.

First deterministic non-trivial approximation for pure packing constraints.

Develops a versatile combinatorial dynamic programming technique.

Abstract

Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering constraints. We present a tight approximation algorithm that for any constant $\epsilon >0$ achieves a guarantee of $1-\frac{1}{\mathrm{e}}-\epsilon$ while violating only the covering constraints by a multiplicative factor of $1-\epsilon$. Our algorithm is based on a novel enumeration method, which unlike previous known enumeration techniques, can handle both packing and covering constraints. We extend the above main result by additionally handling a matroid independence constraints as well as finding (approximate) pareto set optimal solutions when multiple submodular objectives are present. Finally, we propose a novel and purely combinatorial dynamic programming approach that can be applied to several special cases of the problem yielding not only {\em deterministic} but also considerably faster algorithms. For example, for the well studied special case of only packing constraints (Kulik {\em et. al.} [Math. Oper. Res. `13] and Chekuri {\em et. al.} [FOCS `10]), we are able to present the first deterministic non-trivial approximation algorithm. We believe our new combinatorial approach might be of independent interest.