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Knapsack with compactness: a semidefinite approach

Hubert Villuendas, Mathieu Besançon, Jérôme Malick

TL;DR

The paper tackles the min-knapsack with a compactness constraint by developing semidefinite programming formulations that lift binary decisions to a rank-one matrix $\mathbf{X}=xx^\top$, enabling tight relaxations. It introduces two SDP approaches: an explicit-compactness SDP and a penalized SDP with a tunable hyperparameter $\lambda$ to balance compactness against accuracy, complemented by strengthening techniques such as valid inequalities and maximal insufficient subset cuts. Computational results on challenging instances show that strengthened SDP and MIS cuts improve bounds and reduce fractionality, while the penalized variants offer competitive runtime with a controllable trade-off between compactness and accuracy. The framework yields high-quality heuristics for difficult instances and holds practical promise for applications like change-point detection in time series, where compact, interpretable solution sets are valuable.

Abstract

The min-knapsack problem with compactness constraints extends the classical knapsack problem, in the case of ordered items, by introducing a restriction ensuring that they cannot be too far apart. This problem has applications in statistics, particularly in the detection of change-points in time series. In this paper, we propose a semidefinite programming approach for this problem, incorporating compactness in constraints or in objective. We study and compare the different relaxations, and argue that our method provides high-quality heuristics and tight bounds. In particular, the single hyperparameter of our penalized semidefinite models naturally balances the trade-off between compactness and accuracy of the computed solutions. Numerical experiments illustrate, on the hardest instances, the effectiveness and versatility of our approach compared to the existing mixed-integer programming formulation.

Knapsack with compactness: a semidefinite approach

TL;DR

The paper tackles the min-knapsack with a compactness constraint by developing semidefinite programming formulations that lift binary decisions to a rank-one matrix , enabling tight relaxations. It introduces two SDP approaches: an explicit-compactness SDP and a penalized SDP with a tunable hyperparameter to balance compactness against accuracy, complemented by strengthening techniques such as valid inequalities and maximal insufficient subset cuts. Computational results on challenging instances show that strengthened SDP and MIS cuts improve bounds and reduce fractionality, while the penalized variants offer competitive runtime with a controllable trade-off between compactness and accuracy. The framework yields high-quality heuristics for difficult instances and holds practical promise for applications like change-point detection in time series, where compact, interpretable solution sets are valuable.

Abstract

The min-knapsack problem with compactness constraints extends the classical knapsack problem, in the case of ordered items, by introducing a restriction ensuring that they cannot be too far apart. This problem has applications in statistics, particularly in the detection of change-points in time series. In this paper, we propose a semidefinite programming approach for this problem, incorporating compactness in constraints or in objective. We study and compare the different relaxations, and argue that our method provides high-quality heuristics and tight bounds. In particular, the single hyperparameter of our penalized semidefinite models naturally balances the trade-off between compactness and accuracy of the computed solutions. Numerical experiments illustrate, on the hardest instances, the effectiveness and versatility of our approach compared to the existing mixed-integer programming formulation.

Paper Structure

This paper contains 24 sections, 6 theorems, 54 equations, 20 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Knapsack and compactness
  3. Example of application
  4. Contributions and outline
  5. Modeling as a semidefinite program
  6. Equivalent SDP reformulations
  7. A first SDP relaxation
  8. Penalized SDP relaxation
  9. Strengthening the semidefinite relaxation
  10. Valid inequalities
  11. Maximal insufficient subset cuts
  12. Computationnal results
  13. Experimental setup
  14. Computational experiments
  15. Strenghening valid inequalities \ref{['eq: inegalites quadratiques 1']} to \ref{['eq: inegalites quadratiques 4']}
  16. ...and 9 more sections

Key Result

Proposition 1

Let $\mathcal{X}\subseteq\mathop{\mathrm{\mathbb{S}}}\nolimits^n$ denote the set of all symmetric matrices $\mathbf{X}$ that verify the linear inequalities where $\mathop{\mathrm{diag}}\nolimits:\mathop{\mathrm{\mathbb{S}}}\nolimits^n\rightarrow\mathop{\mathrm{\mathbb{R}}}\nolimits^n$ is the operator taking a matrix $\mathbf{X}$ and associates the vector of its diagonal entries. Then both eq: ori

Figures (20)

  • Figure 1: a non-compact selection with $\Delta =2$ and $n=7$.
  • Figure 2: a compact selection with $\Delta =2$ and $n=7$.
  • Figure 3: a time serie and its possible changes in variance.
  • Figure 4: probabilities associated to each point of being the first change point of the time serie.
  • Figure 5: selected items for the \ref{['eq: mKP sans compactness']}, building a non-compact$0.75$-credible set, with a compactness parameter $\Delta = 1$.
  • ...and 15 more figures

Theorems & Definitions (15)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Conjecture 1
  • Remark 1
  • Remark 2: Alternative penalization approaches
  • Definition 1: Maximal insufficient subset
  • Proposition 3: Adapted from the classical knapsack litterature, see e.g. Proposition 7.1 in conforti2014integer
  • Proposition 4
  • ...and 5 more