The min-Knapsack problem with compactness constraints and applications in statistics

Alberto Santini and Enrico Malaguti

European Journal of Operational Research, 2024, vol. 312, issue 1, 385-397

Abstract: In the min-Knapsack problem, one is given a set of items, each having a certain cost and weight. The objective is to select a subset with minimum cost, such that the sum of the weights is not smaller than a given constant. In this paper, we introduce an extension of the min-Knapsack problem with additional “compactness constraints” (mKPC), stating that selected items cannot lie too far apart. This extension has applications in statistics, including in algorithms for change-point detection in time series. We propose three solution methods for the mKPC. The first two methods use the same Mixed-Integer Programming (MIP) formulation but with two different approaches: passing the complete model with a quadratic number of constraints to a black-box MIP solver or dynamically separating the constraints using a branch-and-cut algorithm. Numerical experiments highlight the advantages of this dynamic separation. The third approach is a dynamic programming labelling algorithm. Finally, we focus on the particular case of the unit-cost mKPC (1c-mKPC), which has a specific interpretation in the context of the statistical applications mentioned above. We prove that the 1c-mKPC is solvable in polynomial time with a different ad-hoc dynamic programming algorithm. Experimental results show that this algorithm vastly outperforms both generic approaches for the mKPC and a simple greedy heuristic from the literature.

Keywords: Cutting; Knapsack problems; Applications in statistics; Dynamic programming (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:ejores:v:312:y:2024:i:1:p:385-397

DOI: 10.1016/j.ejor.2023.07.020

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