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

34 Pages Posted: 9 Feb 2023 Last revised: 8 Feb 2024

See all articles by Miao Bai

Miao Bai

Department of Operations and Information Management, University of Connecticut

Carlos Cardonha

University of Connecticut, School of Business

Date Written: February 6, 2024

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 − 1/e)-approximation algorithm. For the online CCKP in the random order model, we derive a 10.427/α -competitive algorithm based on α-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.

Keywords: continuous knapsack, nonlinear knapsack, approximation algorithms, online algorithms

Suggested Citation

Bai, Miao and Cardonha, Carlos, Cardinality-Constrained Continuous Knapsack Problem with Concave Piecewise-Linear Utilities (February 6, 2024). University of Connecticut School of Business Research Paper No. 23-05, Available at SSRN: https://ssrn.com/abstract=4350988 or http://dx.doi.org/10.2139/ssrn.4350988

Miao Bai

Department of Operations and Information Management, University of Connecticut ( email )

OPIM Dept.
2100 Hillside Road, U1041
Storrs, CT CT 06269-1041
United States

Carlos Cardonha (Contact Author)

University of Connecticut, School of Business ( email )

2100A Hillside Rd Unit 1041
Storrs, CT 06269-1041
United States
8604864885 (Phone)

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