Fractionally Subadditive Maximization under an Incremental Knapsack Constraint

TL;DR

This paper studies the problem of incrementally maximizing fractionally subadditive functions under a growing knapsack constraint, providing algorithms with competitive ratios and establishing bounds for flow problems.

Contribution

It introduces an algorithm with a competitive ratio of at most max{3.293√M, 2M} for the problem and provides lower bounds, also capturing flow potential-based models.

Findings

Algorithm achieves competitive ratio at most max{3.293√M, 2M}.

Lower bound of max{2.618, M} on competitive ratio.

Framework applies to potential-based flows with bounds between 2 and 2M.

Abstract

We consider the problem of maximizing a fractionally subadditive function under a knapsack constraint that grows over time. An incremental solution to this problem is given by an order in which to include the elements of the ground set, and the competitive ratio of an incremental solution is defined by the worst ratio over all capacities relative to an optimum solution of the corresponding capacity. We present an algorithm that finds an incremental solution of competitive ratio at most $\max\{3.293\sqrt{M},2M\}$, under the assumption that the values of singleton sets are in the range $[1,M]$, and we give a lower bound of $\max\{2.618,M\}$ on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a lower bound of $\max\{2,M\}$ and an upper bound of $2M$ for the incremental maximization of classical flows with capacities in $[1,M]$ which is tight for the unit capacity case.