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

Miao Bai, Carlos Cardonha

TL;DR

This work studies the cardinality-constrained continuous knapsack problem with concave piecewise-linear utilities (CCKP), combining a knapsack constraint $W$ with a cardinality cap $C$ and item-specific concave utilities $R_j(\cdot)$. The offline problem admits an exact dynamic-programming based approach that can be discretized to yield a fully polynomial-time approximation scheme (FPTAS) and also a greedy $(1-1/e)$-approximation by exploiting the induced submodularity of a component-based objective. For the online version in the random-order model, the authors design a three-phase algorithm with a sampling, secretary, and knapsack phase, achieving a competitive ratio of $\frac{10.427}{\alpha}$ where $\alpha$ is the offline approximation factor; stronger guarantees are derived for extreme values of $C$. Theoretical results are complemented by numerical experiments showing strong empirical performance of the greedy and online algorithms, and the framework extends to continuous Lipschitz utilities. Overall, the paper advances approximation and online algorithms for nonlinear, cardinality-bounded knapsack problems with practical relevance to resource management in monitoring applications.

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 $\frac{10.427}α$-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.

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

TL;DR

This work studies the cardinality-constrained continuous knapsack problem with concave piecewise-linear utilities (CCKP), combining a knapsack constraint with a cardinality cap and item-specific concave utilities . The offline problem admits an exact dynamic-programming based approach that can be discretized to yield a fully polynomial-time approximation scheme (FPTAS) and also a greedy -approximation by exploiting the induced submodularity of a component-based objective. For the online version in the random-order model, the authors design a three-phase algorithm with a sampling, secretary, and knapsack phase, achieving a competitive ratio of where is the offline approximation factor; stronger guarantees are derived for extreme values of . Theoretical results are complemented by numerical experiments showing strong empirical performance of the greedy and online algorithms, and the framework extends to continuous Lipschitz utilities. Overall, the paper advances approximation and online algorithms for nonlinear, cardinality-bounded knapsack problems with practical relevance to resource management in monitoring applications.

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 -approximation algorithm. For the online CCKP in the random order model, we derive a -competitive algorithm based on -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.
Paper Structure (33 sections, 22 theorems, 38 equations, 3 figures, 3 algorithms)

This paper contains 33 sections, 22 theorems, 38 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions to the literature
  3. Organization
  4. Definition and Notation
  5. Offline CCKP and Approximation Algorithms
  6. Utility Decomposition and Discreteness of Optimal Solutions
  7. FPTAS for the $\texttt{CCKP}{}$
  8. Dynamic programming algorithm
  9. Approximation scheme based on state space discretization:
  10. Greedy Algorithm for the $\texttt{CCKP}{}$
  11. Online CCKP in the Random Order Model
  12. Additional Notation and Assumptions
  13. Description of the Algorithm
  14. Sampling phase (lines 2-3):
  15. Secretary phase (lines 4-6):
  16. ...and 18 more sections

Key Result

Theorem 1

The $\texttt{CCKP}$ admits an FPTAS with running time $O(|\mathscr{T}\xspace| \frac{m^2}{\epsilon} C)$, where $m \coloneqq \sum\limits_{j \in \mathscr{T}\xspace}n_{j}$.

Figures (3)

  • Figure 1: Performance of Algorithm \ref{['alg:greedy']} in all instances of dataset A.
  • Figure 2: Performance of Algorithm \ref{['alg:online']} in all instances of datasets A and B.
  • Figure 3: The histogram of empirical ratios of Algorithm \ref{['alg:online']} for datasets A and B.

Theorems & Definitions (46)

  • Definition 1: Competitive Ratio
  • Remark 1
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • proof
  • Remark 2
  • Lemma 1: Lemma 1 in albers2021improved
  • Lemma 2: Lemma 7 in albers2021improved
  • ...and 36 more