Submodular maximization with uncertain knapsack capacity

TL;DR

This paper studies maximizing monotone submodular functions with uncertain knapsack capacity, proposing policies with provable robustness ratios and algorithms for expected value maximization under capacity uncertainty.

Contribution

It introduces new adaptive and universal policies with proven robustness ratios for uncertain knapsack capacity and provides approximation algorithms for expected value maximization.

Findings

Randomized policy with robustness ratio (1-1/e)/2

Deterministic policy with robustness ratio 2(1-1/e)/21

Polynomial-time randomized algorithm with approximation ratio (1-1/√e)/4-ε

Abstract

We consider the maximization problem of monotone submodular functions under an uncertain knapsack constraint. Specifically, the problem is discussed in the situation that the knapsack capacity is not given explicitly and can be accessed only through an oracle that answers whether or not the current solution is feasible when an item is added to the solution. Assuming that cancellation of the last item is allowed when it overflows the knapsack capacity, we discuss the robustness ratios of adaptive policies for this problem, which are the worst case ratios of the objective values achieved by the output solutions to the optimal objective values. We present a randomized policy of robustness ratio $(1-1/e)/2$, and a deterministic policy of robustness ratio $2(1-1/e)/21$. We also consider a universal policy that chooses items following a precomputed sequence. We present a randomized universal policy of robustness ratio $(1-1/\sqrt[4]{e})/2$. When the cancellation is not allowed, no randomized adaptive policy achieves a constant robustness ratio. Because of this hardness, we assume that a probability distribution of the knapsack capacity is given, and consider computing a sequence of items that maximizes the expected objective value. We present a polynomial-time randomized algorithm of approximation ratio $(1-1/\sqrt[4]{e})/4-\epsilon$ for any small constant $\epsilon >0$.