Maximizing a Submodular Function with Bounded Curvature under an Unknown Knapsack Constraint

TL;DR

This paper presents a new policy for maximizing monotone submodular functions under unknown knapsack constraints, achieving improved robustness factors that depend on the function's curvature.

Contribution

It introduces a policy with a robustness factor decreasing with curvature, improving previous bounds, and extends Wolsey's greedy algorithm analysis to unknown constraints.

Findings

Robustness factor of 1/2 for c=0 matches the best known.

Robustness factor of approximately 0.35 for c=1 improves over previous 0.06.

Provides tight approximation guarantees for the proposed algorithms.

Abstract

This paper studies the problem of maximizing a monotone submodular function under an unknown knapsack constraint. A solution to this problem is a policy that decides which item to pack next based on the past packing history. The robustness factor of a policy is the worst case ratio of the solution obtained by following the policy and an optimal solution that knows the knapsack capacity. We develop a policy with a robustness factor that is decreasing in the curvature $c$ of the submodular function. For the extreme cases $c=0$ corresponding to an additive objective function, it matches a previously known and best possible robustness factor of $1/2$. For the other extreme case of $c=1$ it yields a robustness factor of $\approx 0.35$ improving over the best previously known robustness factor of $\approx 0.06$. The analysis of our policy relies on a greedy algorithm that is a slight modification of Wolsey's greedy algorithm for the submodular knapsack problem with a known knapsack constraint. We obtain tight approximation guarantees for both of these algorithms in the setting of a submodular objective function with curvature $c$.