Cardinality-Constrained Continuous Knapsack Problem with Concave Piecewise-Linear Utilities

TL;DR

This paper introduces algorithms for a complex resource allocation problem with concave utilities, providing approximation schemes and competitive algorithms for offline and online settings, supported by empirical evaluation.

Contribution

It develops the first polynomial-time approximation scheme and competitive algorithms for the cardinality-constrained knapsack problem with concave piecewise-linear utilities, including online variants.

Findings

The offline problem admits a fully polynomial-time approximation scheme.

A greedy $(1 - 1/e)$-approximation algorithm is derived for the offline problem.

The online problem has a $rac{10.427}{eta}$-competitive algorithm with improved guarantees for small or large capacities.

Abstract

We study an extension of the cardinality-constrained knapsack problem wherein each item has a concave piecewise linear utility structure (CCKP), which is motivated by applications such as resource management problems in monitoring and surveillance tasks. Our main contributions are combinatorial algorithms for the offline CCKP and an online version of the CCKP. For the offline problem, we present a fully polynomial-time approximation scheme and show that it can be cast as the maximization of a submodular function with cardinality constraints; the latter property allows us to derive a greedy $(1 - \frac{1}{e})$-approximation algorithm. For the online CCKP in the random order model, we derive a $\frac{10.427}{\alpha}$-competitive algorithm based on $\alpha$-approximation algorithms for the offline CCKP; moreover, we derive stronger guarantees for the cases wherein the cardinality capacity is very small or relatively large. Finally, we investigate the empirical performance of the proposed algorithms in numerical experiments.