Pseudo-polynomial algorithms for solving the Knapsack Problem with dependencies between items
We consider a new variant of the Knapsack Problem with dependencies between items. In this variant, the set of items is partitioned into subsets with dependencies among them, and an item can be selected from a subset only if at least one item is selected from each of its dependent subsets. We develo...
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| Vydáno v: | Computers & operations research Ročník 158; s. 106281 |
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01.10.2023
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| Abstract | We consider a new variant of the Knapsack Problem with dependencies between items. In this variant, the set of items is partitioned into subsets with dependencies among them, and an item can be selected from a subset only if at least one item is selected from each of its dependent subsets. We develop pseudo-polynomial algorithms to solve this new constrained version in the cases where the dependencies (between the subsets of items rather than items) are represented by out-trees, in-trees, and directed acyclic graphs. We consider both cases, when the weight and profit of each item are similar, which is the classical Subset Sum problem, and the case when they take arbitrary non-negative values. The proposed algorithms run in O(nW) times and spaces for out-trees, while for in-trees and acyclic digraphs it runs in O(nW+mW2) and O(nW+max{mW2,m(nW)}), respectively, where n is the number of items, W is the knapsack capacity, and m is the number of nodes. Experiments on randomly generated knapsack instances with different graphs of dependency are carried out to assess algorithm efficiency, and show the running dependency on different instance parameters.
•An Integer Linear Programming model for the handled problem.•Dynamic programming algorithms for solving the problem.•Considering dependencies represented by in-trees, out-tree, and DAG.•The implementation of all the proposed approaches.•The experimental validation on different benchmarks. |
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| AbstractList | We consider a new variant of the Knapsack Problem with dependencies between items. In this variant, the set of items is partitioned into subsets with dependencies among them, and an item can be selected from a subset only if at least one item is selected from each of its dependent subsets. We develop pseudo-polynomial algorithms to solve this new constrained version in the cases where the dependencies (between the subsets of items rather than items) are represented by out-trees, in-trees, and directed acyclic graphs. We consider both cases, when the weight and profit of each item are similar, which is the classical Subset Sum problem, and the case when they take arbitrary non-negative values. The proposed algorithms run in O(nW) times and spaces for out-trees, while for in-trees and acyclic digraphs it runs in O(nW+mW2) and O(nW+max{mW2,m(nW)}), respectively, where n is the number of items, W is the knapsack capacity, and m is the number of nodes. Experiments on randomly generated knapsack instances with different graphs of dependency are carried out to assess algorithm efficiency, and show the running dependency on different instance parameters.
•An Integer Linear Programming model for the handled problem.•Dynamic programming algorithms for solving the problem.•Considering dependencies represented by in-trees, out-tree, and DAG.•The implementation of all the proposed approaches.•The experimental validation on different benchmarks. |
| ArticleNumber | 106281 |
| Author | Kheddouci, Hamamache Lalou, Mohammed |
| Author_xml | – sequence: 1 givenname: Mohammed orcidid: 0000-0002-8837-4351 surname: Lalou fullname: Lalou, Mohammed email: mohammed.lalou@u-bourgogne.fr organization: Universite de Bourgogne, UFR-Science et Techniques, Lab LIB, 09 Avenue Alain Savary, 21000 Dijon, France – sequence: 2 givenname: Hamamache surname: Kheddouci fullname: Kheddouci, Hamamache organization: Université Claude Bernard Lyon 1, UFR-Informatique, Lab LIRIS, 43 bd du 11 Novembre 1918, F-69622, 69100, Villeurbanne, France |
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| Cites_doi | 10.1051/ro/2016049 10.1016/j.cosrev.2018.02.002 10.1016/j.cor.2017.09.019 10.1287/opre.27.3.503 10.1007/s10878-016-0035-7 10.1007/s00500-016-2465-7 10.1016/j.jda.2012.04.011 10.1007/978-3-030-48439-2_41 10.1287/ijoc.2016.0742 10.1287/ijoc.1090.0344 10.1016/j.eswa.2019.113077 10.1111/itor.12381 10.1007/s10878-018-0262-1 10.1016/j.cor.2004.03.002 10.1016/j.ejor.2021.11.004 10.1007/s00186-019-00664-y 10.1287/opre.46.1.17 10.1287/moor.8.1.1 10.1016/j.cor.2016.11.015 10.1007/978-1-4684-2001-2_9 10.1016/j.cor.2015.05.005 10.1007/978-3-319-00080-0 10.1007/s00186-021-00767-5 10.1016/j.dam.2006.08.006 10.1007/s11590-018-1371-6 10.1137/0201008 |
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| Keywords | Constrained knapsack problem In-trees Out-trees Dynamic programming Directed acyclic graphs Complexity |
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| Snippet | We consider a new variant of the Knapsack Problem with dependencies between items. In this variant, the set of items is partitioned into subsets with... |
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| SubjectTerms | Complexity Computer Science Constrained knapsack problem Directed acyclic graphs Dynamic programming In-trees Out-trees |
| Title | Pseudo-polynomial algorithms for solving the Knapsack Problem with dependencies between items |
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