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.
Santini, A., Malaguti, E. (2024). The min-Knapsack problem with compactness constraints and applications in statistics. EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 312(1), 385-397 [10.1016/j.ejor.2023.07.020].
The min-Knapsack problem with compactness constraints and applications in statistics
Malaguti, Enrico
2024
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.File | Dimensione | Formato | |
---|---|---|---|
1-s2.0-S0377221723005593-main.pdf
accesso aperto
Tipo:
Versione (PDF) editoriale
Licenza:
Creative commons
Dimensione
1.71 MB
Formato
Adobe PDF
|
1.71 MB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.