Online Learning for Min Sum Set Cover and Pandora's Box

TL;DR

This paper introduces an online learning algorithm for Min Sum Set Cover and Pandora's Box problems, achieving near-optimal performance in both full-information and bandit settings, with extensions to matroid constraints.

Contribution

It presents a computationally efficient, constant-competitive online algorithm for these problems, including generalizations to bandit feedback and matroid constraints.

Findings

The algorithm is constant-competitive against the optimal search order.

It extends to bandit settings with partial feedback.

Generalizations include matroid constraints.

Abstract

Two central problems in Stochastic Optimization are Min Sum Set Cover and Pandora's Box. In Pandora's Box, we are presented with $n$ boxes, each containing an unknown value and the goal is to open the boxes in some order to minimize the sum of the search cost and the smallest value found. Given a distribution of value vectors, we are asked to identify a near-optimal search order. Min Sum Set Cover corresponds to the case where values are either 0 or infinity. In this work, we study the case where the value vectors are not drawn from a distribution but are presented to a learner in an online fashion. We present a computationally efficient algorithm that is constant-competitive against the cost of the optimal search order. We extend our results to a bandit setting where only the values of the boxes opened are revealed to the learner after every round. We also generalize our results to other commonly studied variants of Pandora's Box and Min Sum Set Cover that involve selecting more than a single value subject to a matroid constraint.