Efficient Submodular Function Maximization under Linear Packing Constraints

TL;DR

This paper presents efficient algorithms with provable approximation guarantees for maximizing monotone submodular functions under linear packing constraints, extending results to large width and sparse matrix settings.

Contribution

The authors develop new combinatorial algorithms with approximation ratios matching theoretical bounds for submodular maximization under linear packing constraints.

Findings

Achieves an approximation ratio of Ω(1 / m^{1/W}) for general packing constraints.

Provides a near-optimal approximation of (1 - ε)(1 - 1/e) when the width W is large.

Designs a fast algorithm with approximation ratio depending on sparsity and width for binary, k-column sparse matrices.

Abstract

We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix $A \in [0,1]^{m \times n}$, a vector $b \in [1,\infty)^m$, and a monotone submodular set function $f: 2^{[n]} \rightarrow \bbR_+$. The objective is to find a set $S$ that maximizes $f(S)$ subject to $A x_{S} \leq b$, where $x_S$ stands for the characteristic vector of the set $S$. A well-studied special case of this problem is when $f$ is linear. This special case captures the class of packing integer programs. Our main contribution is an efficient combinatorial algorithm that achieves an approximation ratio of $\Omega(1 / m^{1/W})$, where $W = \min\{b_i / A_{ij} : A_{ij} > 0\}$ is the width of the packing constraints. This result matches the best known performance guarantee for the linear case. One immediate corollary of this result is that the algorithm under consideration achieves constant factor approximation when the number of constraints is constant or when the width of the constraints is sufficiently large. This motivates us to study the large width setting, trying to determine its exact approximability. We develop an algorithm that has an approximation ratio of $(1 - \epsilon)(1 - 1/e)$ when $W = \Omega(\ln m / \epsilon^2)$. This result essentially matches the theoretical lower bound of $1 - 1/e$. We also study the special setting in which the matrix $A$ is binary and $k$-column sparse. A $k$-column sparse matrix has at most $k$ non-zero entries in each of its column. We design a fast combinatorial algorithm that achieves an approximation ratio of $\Omega(1 / (Wk^{1/W}))$, that is, its performance guarantee only depends on the sparsity and width parameters.