A lifted-space dynamic programming algorithm for the Quadratic Knapsack Problem
The Quadratic Knapsack Problem (QKP) is a well-known combinatorial optimization problem which amounts to maximizing a quadratic function of binary variables, subject to a single linear constraint. It has many applications in finance, logistics, telecommunications, facility location, etc. The QKP is...
Saved in:
| Published in | Discrete Applied Mathematics Vol. 335; pp. 52 - 68 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
15.08.2023
|
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | The Quadratic Knapsack Problem (QKP) is a well-known combinatorial optimization problem which amounts to maximizing a quadratic function of binary variables, subject to a single linear constraint. It has many applications in finance, logistics, telecommunications, facility location, etc. The QKP is NP-hard in the strong sense and the state-of-the-art algorithm for solving the QKP can only handle problems of small and moderate sizes. In this paper, we present a novel deterministic heuristic algorithm for finding good QKP feasible solutions. This algorithm consists of combining the dynamic programming approach with a local search procedure, with the novelty that both are adapted and implemented in the space of lifted variables of the QKP. The algorithm runs in O(n3c) time and is able to find optimal solutions to more than 97% of standard instances, about 80% of some well-known hard QKP instances, as well as optimality gaps of 0.1% or less for other instances. |
|---|---|
| AbstractList | The Quadratic Knapsack Problem (QKP) is a well-known combinatorial optimization problem which amounts to maximizing a quadratic function of binary variables, subject to a single linear constraint. It has many applications in finance, logistics, telecommunications, facility location, etc. The QKP is NP-hard in the strong sense and the state-of-the-art algorithm for solving the QKP can only handle problems of small and moderate sizes. In this paper, we present a novel deterministic heuristic algorithm for finding good QKP feasible solutions. This algorithm consists of combining the dynamic programming approach with a local search procedure, with the novelty that both are adapted and implemented in the space of lifted variables of the QKP. The algorithm runs in O(n3c) time and is able to find optimal solutions to more than 97% of standard instances, about 80% of some well-known hard QKP instances, as well as optimality gaps of 0.1% or less for other instances. |
| Author | Djeumou Fomeni, Franklin |
| Author_xml | – sequence: 1 givenname: Franklin surname: Djeumou Fomeni fullname: Djeumou Fomeni, Franklin email: Djeumou_Fomeni.Franklin@uqam.ca organization: GERAD, CIRRELT & Department of Analytics, Operations and Information Technologies (AOTI), University of Quebec in Montreal, Canada |
| BookMark | eNp9kNtKAzEQhoNUsK0-gHd5gV1z6J7wqhRPWKiCgnchm0za1N3NkkShb2-KXnnRq2GY_xtmvhmaDG4AhK4pySmh5c0-17LPGWE8JywnhJ-hKa0rlpVVRSdomjJlxmj9cYFmIewJITR1U7RZ4s6aCDoLo1SA9WGQvVV49G7rZd_bYYtlt3Xexl2PjfM47gC_fkntZUy550GOQapP_OJd20F_ic6N7AJc_dU5er-_e1s9ZuvNw9Nquc4Ua6qYAV1IqplJtze8qReStQXn2kimirZoNLC6pbwmBeElJQraumC6roxpIA0043NU_e5V3oXgwQhlY7rIDdFL2wlKxNGL2IvkRRy9CMJE8pJI-o8cve2lP5xkbn8ZSC99W_AiKAuDAm09qCi0syfoH_IKfX4 |
| CitedBy_id | crossref_primary_10_1016_j_ejor_2024_12_032 crossref_primary_10_3390_electronics12214547 crossref_primary_10_1016_j_ejor_2024_12_019 crossref_primary_10_1016_j_ejor_2024_06_034 |
| Cites_doi | 10.1007/s10589-015-9763-3 10.1287/opre.22.1.180 10.1287/opre.28.5.1130 10.1016/S0167-6377(02)00122-0 10.1287/ijoc.2015.0678 10.1016/j.cor.2016.08.006 10.1016/j.orl.2016.05.005 10.1016/0377-2217(94)00229-0 10.1016/S0377-2217(03)00244-3 10.1023/A:1009898604624 10.1287/ijoc.1030.0029 10.1007/BFb0120892 10.1016/j.ejor.2016.06.013 10.1016/j.swevo.2015.09.005 10.1002/(SICI)1098-2418(199810/12)13:3/4<457::AID-RSA14>3.0.CO;2-W 10.1023/A:1009841107478 10.1137/S0097539705447037 10.1007/s00186-020-00702-0 10.1287/ijoc.2013.0555 10.1287/ijoc.1050.0172 10.1287/ijoc.11.2.125 10.1016/j.dam.2006.08.007 10.1016/j.ejor.2019.08.027 10.1007/s00453-008-9248-1 10.1016/j.endm.2014.11.035 |
| ContentType | Journal Article |
| Copyright | 2023 Elsevier B.V. |
| Copyright_xml | – notice: 2023 Elsevier B.V. |
| DBID | AAYXX CITATION |
| DOI | 10.1016/j.dam.2023.02.003 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1872-6771 |
| EndPage | 68 |
| ExternalDocumentID | 10_1016_j_dam_2023_02_003 S0166218X23000422 |
| GroupedDBID | -~X ADEZE AFTJW ALMA_UNASSIGNED_HOLDINGS FDB OAUVE AAYXX AI. CITATION FA8 VH1 WUQ |
| ID | FETCH-LOGICAL-c297t-e14a1d2f10193984a2b533dfa2c5b59de28b1380503610ceb852d87ff9e8b1d23 |
| IEDL.DBID | .~1 |
| ISSN | 0166-218X |
| IngestDate | Tue Jul 01 01:43:14 EDT 2025 Thu Apr 24 23:07:17 EDT 2025 Sat May 20 23:43:24 EDT 2023 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Integer programming Local search Knapsack problems Binary quadratic optimization Dynamic programming |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c297t-e14a1d2f10193984a2b533dfa2c5b59de28b1380503610ceb852d87ff9e8b1d23 |
| PageCount | 17 |
| ParticipantIDs | crossref_citationtrail_10_1016_j_dam_2023_02_003 crossref_primary_10_1016_j_dam_2023_02_003 elsevier_sciencedirect_doi_10_1016_j_dam_2023_02_003 |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 2023-08-15 |
| PublicationDateYYYYMMDD | 2023-08-15 |
| PublicationDate_xml | – month: 08 year: 2023 text: 2023-08-15 day: 15 |
| PublicationDecade | 2020 |
| PublicationTitle | Discrete Applied Mathematics |
| PublicationYear | 2023 |
| Publisher | Elsevier B.V |
| Publisher_xml | – name: Elsevier B.V |
| References | Kellerer, Strusevich (b24) 2010; 57 Glover, Kochenberger (b18) 2002 Fomeni, Letchford (b16) 2014; 26 Alon, Krivelevich, Sudakov (b2) 1998; 13 Fomeni, Kaparis, Letchford (b15) 2022; 44 Kellerer, Pferschy, Pisinger (b23) 2004 Billionet, Soutif (b8) 2004; 16 Fampa, Lubke, Wang, Wolkowicz (b14) 2020; 281 Taylor (b35) 2016; 44 Pisinger, Rasmussen, Sandvik (b30) 2007; 19 Balas, Zemel (b3) 1980; 28 Billionet, Calmels (b6) 1996; 92 P. Chaillou, P. Hansen, Y. Mahieu, Best network flow bound for the quadratic knapsack problem, in: Presented at the International Workshop on Network Flow Optimization, NETFLOW, Pisa, Italy, 1983. Khot (b25) 2006; 36 Billionet, Soutif (b7) 2004; 157 Pisinger (b29) 2007; 155 Gu, Nemhauser, Savelsbergh (b20) 2000; 4 Karp (b22) 1972 Caprara, Pisinger, Toth (b10) 1998; 11 Chen, Hao (b12) 2017; 77 Bellman (b4) 1957 Cacchiani, Iori, Locatelli, Martello (b9) 2022 Cunha, Lucena (b13) 2016; 63 Gallo, Hammer, Simeone (b17) 1980; 12 Helmberg, Rendl, Weismantel (b21) 2000; 4 Toumi, Cheikh, Jarboui (b36) 2015; 47 Glover, Woolsey (b19) 1974; 22 Rader, Woeginger (b32) 2002; 30 Schauer (b33) 2016; 255 Pferschy, Schauer (b28) 2016; 28 Bhaskara, Charikar, Chlamtac, Feige, Vijayaraghavan (b5) 2010 Patvardhan, Bansal, Srivastav (b27) 2016; 26 Schulze, Stiglmayr, Paquete, Fonseca, Willems, Ruzika (b34) 2020; 92 Alon, Arora, Manokaran, Moshkovitz, Weinstein (b1) 2011 Martello, Toth (b26) 1990 Rader (b31) 1997 Pisinger (10.1016/j.dam.2023.02.003_b29) 2007; 155 Gu (10.1016/j.dam.2023.02.003_b20) 2000; 4 Helmberg (10.1016/j.dam.2023.02.003_b21) 2000; 4 Alon (10.1016/j.dam.2023.02.003_b2) 1998; 13 Schulze (10.1016/j.dam.2023.02.003_b34) 2020; 92 Cunha (10.1016/j.dam.2023.02.003_b13) 2016; 63 Billionet (10.1016/j.dam.2023.02.003_b6) 1996; 92 Bellman (10.1016/j.dam.2023.02.003_b4) 1957 Bhaskara (10.1016/j.dam.2023.02.003_b5) 2010 Karp (10.1016/j.dam.2023.02.003_b22) 1972 Patvardhan (10.1016/j.dam.2023.02.003_b27) 2016; 26 Kellerer (10.1016/j.dam.2023.02.003_b24) 2010; 57 Fomeni (10.1016/j.dam.2023.02.003_b15) 2022; 44 Pisinger (10.1016/j.dam.2023.02.003_b30) 2007; 19 Balas (10.1016/j.dam.2023.02.003_b3) 1980; 28 Caprara (10.1016/j.dam.2023.02.003_b10) 1998; 11 10.1016/j.dam.2023.02.003_b11 Glover (10.1016/j.dam.2023.02.003_b19) 1974; 22 Fomeni (10.1016/j.dam.2023.02.003_b16) 2014; 26 Gallo (10.1016/j.dam.2023.02.003_b17) 1980; 12 Schauer (10.1016/j.dam.2023.02.003_b33) 2016; 255 Pferschy (10.1016/j.dam.2023.02.003_b28) 2016; 28 Chen (10.1016/j.dam.2023.02.003_b12) 2017; 77 Khot (10.1016/j.dam.2023.02.003_b25) 2006; 36 Billionet (10.1016/j.dam.2023.02.003_b7) 2004; 157 Toumi (10.1016/j.dam.2023.02.003_b36) 2015; 47 Kellerer (10.1016/j.dam.2023.02.003_b23) 2004 Taylor (10.1016/j.dam.2023.02.003_b35) 2016; 44 Glover (10.1016/j.dam.2023.02.003_b18) 2002 Rader (10.1016/j.dam.2023.02.003_b31) 1997 Rader (10.1016/j.dam.2023.02.003_b32) 2002; 30 Alon (10.1016/j.dam.2023.02.003_b1) 2011 Billionet (10.1016/j.dam.2023.02.003_b8) 2004; 16 Cacchiani (10.1016/j.dam.2023.02.003_b9) 2022 Martello (10.1016/j.dam.2023.02.003_b26) 1990 Fampa (10.1016/j.dam.2023.02.003_b14) 2020; 281 |
| References_xml | – volume: 26 start-page: 173 year: 2014 end-page: 183 ident: b16 article-title: A dynamic programming heuristic for the quadratic knapsack problem publication-title: INFORMS J. Comput. – volume: 4 start-page: 197 year: 2000 end-page: 215 ident: b21 article-title: A semidefinite programming approach to the quadratic knapsack problem publication-title: J. Comb. Optim. – volume: 155 start-page: 623 year: 2007 end-page: 648 ident: b29 article-title: The quadratic knapsack problem–a survey publication-title: Discrete and Applied Mathematics – volume: 36 start-page: 1025 year: 2006 end-page: 1071 ident: b25 article-title: Ruling out PTAS for graph min-bisection, dense publication-title: SIAM J. Comput. – volume: 281 start-page: 36 year: 2020 end-page: 49 ident: b14 article-title: Parametric convex quadratic relaxation of the quadratic knapsack publication-title: European J. Oper. Res. – volume: 92 start-page: 107 year: 2020 end-page: 132 ident: b34 article-title: On the rectangular knapsack problem: approximation of a specific quadratic knapsack problem publication-title: Math. Methods Oper. Res. – year: 2011 ident: b1 article-title: Inapproximabilty of Densest k-Subgraph from Average Case Hardness – volume: 77 start-page: 226 year: 2017 end-page: 239 ident: b12 article-title: An iterated “hyperplane exploration” approach for the quadratic knapsack problem publication-title: Comput. Oper. Res. – volume: 44 year: 2022 ident: b15 article-title: A cut-and-branch algorithm for the quadratic knapsack problem publication-title: Discrete Optim. – year: 2022 ident: b9 article-title: Knapsack problems–An overview of recent advances, Part II: Multiple, multidimensional, and quadratic knapsack problems publication-title: Comput. Oper. Res. – volume: 28 start-page: 1130 year: 1980 end-page: 1154 ident: b3 article-title: An algorithm for large zero-one knapsack problems publication-title: Oper. Res. – volume: 47 start-page: 269 year: 2015 end-page: 276 ident: b36 article-title: 0–1 quadratic knapsack problem solved with VNS algorithm publication-title: Electron. Notes Discrete Math. – volume: 92 start-page: 310 year: 1996 end-page: 325 ident: b6 article-title: Linear programming for the quadratic knapsack problem publication-title: European J. Oper. Res. – volume: 63 start-page: 97 year: 2016 end-page: 120 ident: b13 article-title: Lagrangian heuristics for the quadratic knapsack problem publication-title: Comput. Optim. Appl. – volume: 28 start-page: 308 year: 2016 end-page: 318 ident: b28 article-title: Approximation of the quadratic knapsack problem publication-title: INFORMS J. Comput. – volume: 57 start-page: 769 year: 2010 end-page: 795 ident: b24 article-title: Fully polynomial approximation schemes for a symmetric quadratic knapsack problem and its scheduling applications publication-title: Algorithmica – start-page: 85 year: 1972 end-page: 103 ident: b22 article-title: Reducibility among combinatorial problems publication-title: Complexity of Computer Computations – year: 2010 ident: b5 article-title: Detecting high log-densities: an publication-title: Proceedings of the 42nd ACM Symposium on Theory of Computing – volume: 4 start-page: 109 year: 2000 end-page: 129 ident: b20 article-title: Sequence–independent lifting in mixed integer programming publication-title: J. Comb. Optim. – volume: 255 start-page: 357 year: 2016 end-page: 363 ident: b33 article-title: Asymptotic behavior of the quadratic knapsack problem publication-title: European J. Oper. Res. – year: 1957 ident: b4 article-title: Dynamic Programming – year: 1997 ident: b31 article-title: Lifting Results for the Quadratic 0–1 Knapsack Polytope – volume: 44 start-page: 495 year: 2016 end-page: 497 ident: b35 article-title: Approximation of the quadratic knapsack problem publication-title: Oper. Res. Lett. – volume: 11 start-page: 125 year: 1998 end-page: 137 ident: b10 article-title: Exact solution of the quadratic knapsack problem publication-title: INFORMS J. Comput. – volume: 13 start-page: 457 year: 1998 end-page: 466 ident: b2 article-title: Finding a large hidden clique in a random graph publication-title: Random Struct. Algorithms – volume: 12 start-page: 132 year: 1980 end-page: 149 ident: b17 article-title: Quadratic knapsack problems publication-title: Math. Program. Stud. – year: 2002 ident: b18 article-title: Solving quadratic knapsack problems by reformulation and tabu search, single constraint case publication-title: Combinatorial and Global Optimization, Vol. 14 – volume: 22 start-page: 180 year: 1974 end-page: 182 ident: b19 article-title: Converting the 0–1 polynomial programming problem to a 0–1 linear program publication-title: Oper. Res. – volume: 26 start-page: 175 year: 2016 end-page: 190 ident: b27 article-title: Parallel improved quantum inspired evolutionary algorithm to solve large size quadratic knapsack problems publication-title: Swarm Evol. Comput. – volume: 157 start-page: 565 year: 2004 end-page: 575 ident: b7 article-title: An exact method based on Lagrangian decompositionfor the 0–1 quadratic knapsack problem publication-title: European J. Oper. Res. – volume: 16 start-page: 188 year: 2004 end-page: 197 ident: b8 article-title: Using a mixed integer programming tool for solving the 0–1 quadratic knapsack problem publication-title: INFORMS J. Comput. – year: 2004 ident: b23 article-title: Knapsack Problems – year: 1990 ident: b26 article-title: Knapsack Problems: Algorithms and Computer Implementations – volume: 30 start-page: 159 year: 2002 end-page: 166 ident: b32 article-title: The quadratic 0–1 knapsack problem with series–parallel support publication-title: Oper. Res. Lett. – reference: P. Chaillou, P. Hansen, Y. Mahieu, Best network flow bound for the quadratic knapsack problem, in: Presented at the International Workshop on Network Flow Optimization, NETFLOW, Pisa, Italy, 1983. – volume: 19 start-page: 280 year: 2007 end-page: 290 ident: b30 article-title: Solution of large quadratic knapsack problems through aggressive reduction publication-title: INFORMS J. Comput. – year: 1957 ident: 10.1016/j.dam.2023.02.003_b4 – year: 2010 ident: 10.1016/j.dam.2023.02.003_b5 article-title: Detecting high log-densities: an O(n1/4) approximation for densest k-subgraph – volume: 63 start-page: 97 year: 2016 ident: 10.1016/j.dam.2023.02.003_b13 article-title: Lagrangian heuristics for the quadratic knapsack problem publication-title: Comput. Optim. Appl. doi: 10.1007/s10589-015-9763-3 – year: 2011 ident: 10.1016/j.dam.2023.02.003_b1 – volume: 22 start-page: 180 year: 1974 ident: 10.1016/j.dam.2023.02.003_b19 article-title: Converting the 0–1 polynomial programming problem to a 0–1 linear program publication-title: Oper. Res. doi: 10.1287/opre.22.1.180 – volume: 28 start-page: 1130 year: 1980 ident: 10.1016/j.dam.2023.02.003_b3 article-title: An algorithm for large zero-one knapsack problems publication-title: Oper. Res. doi: 10.1287/opre.28.5.1130 – volume: 30 start-page: 159 year: 2002 ident: 10.1016/j.dam.2023.02.003_b32 article-title: The quadratic 0–1 knapsack problem with series–parallel support publication-title: Oper. Res. Lett. doi: 10.1016/S0167-6377(02)00122-0 – ident: 10.1016/j.dam.2023.02.003_b11 – volume: 28 start-page: 308 year: 2016 ident: 10.1016/j.dam.2023.02.003_b28 article-title: Approximation of the quadratic knapsack problem publication-title: INFORMS J. Comput. doi: 10.1287/ijoc.2015.0678 – year: 1997 ident: 10.1016/j.dam.2023.02.003_b31 – year: 2002 ident: 10.1016/j.dam.2023.02.003_b18 article-title: Solving quadratic knapsack problems by reformulation and tabu search, single constraint case – year: 1990 ident: 10.1016/j.dam.2023.02.003_b26 – year: 2022 ident: 10.1016/j.dam.2023.02.003_b9 article-title: Knapsack problems–An overview of recent advances, Part II: Multiple, multidimensional, and quadratic knapsack problems publication-title: Comput. Oper. Res. – volume: 77 start-page: 226 year: 2017 ident: 10.1016/j.dam.2023.02.003_b12 article-title: An iterated “hyperplane exploration” approach for the quadratic knapsack problem publication-title: Comput. Oper. Res. doi: 10.1016/j.cor.2016.08.006 – volume: 44 start-page: 495 year: 2016 ident: 10.1016/j.dam.2023.02.003_b35 article-title: Approximation of the quadratic knapsack problem publication-title: Oper. Res. Lett. doi: 10.1016/j.orl.2016.05.005 – volume: 92 start-page: 310 year: 1996 ident: 10.1016/j.dam.2023.02.003_b6 article-title: Linear programming for the quadratic knapsack problem publication-title: European J. Oper. Res. doi: 10.1016/0377-2217(94)00229-0 – volume: 157 start-page: 565 year: 2004 ident: 10.1016/j.dam.2023.02.003_b7 article-title: An exact method based on Lagrangian decompositionfor the 0–1 quadratic knapsack problem publication-title: European J. Oper. Res. doi: 10.1016/S0377-2217(03)00244-3 – volume: 44 year: 2022 ident: 10.1016/j.dam.2023.02.003_b15 article-title: A cut-and-branch algorithm for the quadratic knapsack problem publication-title: Discrete Optim. – volume: 4 start-page: 197 year: 2000 ident: 10.1016/j.dam.2023.02.003_b21 article-title: A semidefinite programming approach to the quadratic knapsack problem publication-title: J. Comb. Optim. doi: 10.1023/A:1009898604624 – year: 2004 ident: 10.1016/j.dam.2023.02.003_b23 – volume: 16 start-page: 188 year: 2004 ident: 10.1016/j.dam.2023.02.003_b8 article-title: Using a mixed integer programming tool for solving the 0–1 quadratic knapsack problem publication-title: INFORMS J. Comput. doi: 10.1287/ijoc.1030.0029 – volume: 12 start-page: 132 year: 1980 ident: 10.1016/j.dam.2023.02.003_b17 article-title: Quadratic knapsack problems publication-title: Math. Program. Stud. doi: 10.1007/BFb0120892 – volume: 255 start-page: 357 year: 2016 ident: 10.1016/j.dam.2023.02.003_b33 article-title: Asymptotic behavior of the quadratic knapsack problem publication-title: European J. Oper. Res. doi: 10.1016/j.ejor.2016.06.013 – volume: 26 start-page: 175 year: 2016 ident: 10.1016/j.dam.2023.02.003_b27 article-title: Parallel improved quantum inspired evolutionary algorithm to solve large size quadratic knapsack problems publication-title: Swarm Evol. Comput. doi: 10.1016/j.swevo.2015.09.005 – volume: 13 start-page: 457 year: 1998 ident: 10.1016/j.dam.2023.02.003_b2 article-title: Finding a large hidden clique in a random graph publication-title: Random Struct. Algorithms doi: 10.1002/(SICI)1098-2418(199810/12)13:3/4<457::AID-RSA14>3.0.CO;2-W – volume: 4 start-page: 109 year: 2000 ident: 10.1016/j.dam.2023.02.003_b20 article-title: Sequence–independent lifting in mixed integer programming publication-title: J. Comb. Optim. doi: 10.1023/A:1009841107478 – start-page: 85 year: 1972 ident: 10.1016/j.dam.2023.02.003_b22 article-title: Reducibility among combinatorial problems – volume: 36 start-page: 1025 year: 2006 ident: 10.1016/j.dam.2023.02.003_b25 article-title: Ruling out PTAS for graph min-bisection, dense k-subgraph, and bipartite clique publication-title: SIAM J. Comput. doi: 10.1137/S0097539705447037 – volume: 92 start-page: 107 year: 2020 ident: 10.1016/j.dam.2023.02.003_b34 article-title: On the rectangular knapsack problem: approximation of a specific quadratic knapsack problem publication-title: Math. Methods Oper. Res. doi: 10.1007/s00186-020-00702-0 – volume: 26 start-page: 173 year: 2014 ident: 10.1016/j.dam.2023.02.003_b16 article-title: A dynamic programming heuristic for the quadratic knapsack problem publication-title: INFORMS J. Comput. doi: 10.1287/ijoc.2013.0555 – volume: 19 start-page: 280 year: 2007 ident: 10.1016/j.dam.2023.02.003_b30 article-title: Solution of large quadratic knapsack problems through aggressive reduction publication-title: INFORMS J. Comput. doi: 10.1287/ijoc.1050.0172 – volume: 11 start-page: 125 year: 1998 ident: 10.1016/j.dam.2023.02.003_b10 article-title: Exact solution of the quadratic knapsack problem publication-title: INFORMS J. Comput. doi: 10.1287/ijoc.11.2.125 – volume: 155 start-page: 623 year: 2007 ident: 10.1016/j.dam.2023.02.003_b29 article-title: The quadratic knapsack problem–a survey publication-title: Discrete and Applied Mathematics doi: 10.1016/j.dam.2006.08.007 – volume: 281 start-page: 36 year: 2020 ident: 10.1016/j.dam.2023.02.003_b14 article-title: Parametric convex quadratic relaxation of the quadratic knapsack publication-title: European J. Oper. Res. doi: 10.1016/j.ejor.2019.08.027 – volume: 57 start-page: 769 year: 2010 ident: 10.1016/j.dam.2023.02.003_b24 article-title: Fully polynomial approximation schemes for a symmetric quadratic knapsack problem and its scheduling applications publication-title: Algorithmica doi: 10.1007/s00453-008-9248-1 – volume: 47 start-page: 269 year: 2015 ident: 10.1016/j.dam.2023.02.003_b36 article-title: 0–1 quadratic knapsack problem solved with VNS algorithm publication-title: Electron. Notes Discrete Math. doi: 10.1016/j.endm.2014.11.035 |
| SSID | ssj0001218 ssj0000186 ssj0006644 |
| Score | 2.4061065 |
| Snippet | The Quadratic Knapsack Problem (QKP) is a well-known combinatorial optimization problem which amounts to maximizing a quadratic function of binary variables,... |
| SourceID | crossref elsevier |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 52 |
| SubjectTerms | Binary quadratic optimization Dynamic programming Integer programming Knapsack problems Local search |
| Title | A lifted-space dynamic programming algorithm for the Quadratic Knapsack Problem |
| URI | https://dx.doi.org/10.1016/j.dam.2023.02.003 |
| Volume | 335 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LT8JAEN4QvODB-IxPsgdPJivt7pbuHgmRoATUKAm3ZtvtKgqFQLn6253tAzFRDx77mKb9Mpn5ujvzDUKX0gg_UowSxZVDuNSUSJ6VWBk74sRoru06ZH_Q7A753cgbVVC77IWxZZVF7M9jehatizONAs3GfDxuPAFZaUKCGgGJzpSsbAc7962XX3-4GxJSVh-tVi66fO0xQK7lhfJ3k9jnlHueWfWXVrZNnbJc0JP9nLU2MlFnF-0UFBK38rfcQ5U42Ufb_bX-6vIA3bfwZGyATBIIGFGMdT53HhfVWFPIV1hNXmaLcfo6xcBbMRjjx5XS1iEi3EvUfKmid_yQz5s5RMPOzXO7S4rRCSSi0k9J7HLlamrgaySTgisaAq_TRtHICz2pYypClwkrBgP8KYpD4VEtfGNkDBc0ZUeomsyS-BhhZRSwOs2o1ACYJwVTVBghhWMEg7_PE-SU0ARRoStux1tMgrKA7C0ANAOLZuBQK0Z6gq7WJvNcVOOvm3mJd_DNMwII-r-bnf7P7AzV7JFdNXa9c1RNF6v4AmhHGtYzv6qjrdZtrzv4BHsk0b4 |
| linkProvider | Elsevier |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3JTsMwEB1BewAOiFWU1QdOSFFb22ntY1VRBboAAqTeIieOodBNXf6fceIUkIAD1zgTJU-jmZfx-A3ApTSiHitGPcVVxeNSU0_ytMXK2BEnRnNt65DdXi145rd9v78GzfwsjG2rdLE_i-lptHZXyg7N8nQwKD8iWalhguojiU6VrNahaNWp_AIUGzftoPdFRcpKpG3mdZfPbQZMt9yJf9c8-6h82zNtANPKnlSnLNP0ZD8nri_JqLUD245Fkkb2oruwloz3YKu7kmCd78NdgwwHBvmkhzEjTojORs8T15A1wpRF1PBlMhssXkcEqStBY_KwVNr6REzaYzWdq_id3GcjZw7guXX91Aw8Nz3Bi6msL7ykylVVU4NfI5kUXNEIqZ02isZ-5EudUBFVmbB6MEih4iQSPtWiboxMcEFTdgiF8WScHAFRRiGx04xKjYD5UjBFhRFSVIxg-ANagkoOTRg7aXE74WIY5j1kbyGiGVo0wwq1eqQluFqZTDNdjb9u5jne4TfnCDHu_252_D-zC9gInrqdsHPTa5_Apl2xReSqfwqFxWyZnCELWUTnzss-AMav1G8 |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+lifted-space+dynamic+programming+algorithm+for+the+Quadratic+Knapsack+Problem&rft.jtitle=Discrete+Applied+Mathematics&rft.au=Djeumou+Fomeni%2C+Franklin&rft.date=2023-08-15&rft.issn=0166-218X&rft.volume=335&rft.spage=52&rft.epage=68&rft_id=info:doi/10.1016%2Fj.dam.2023.02.003&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_dam_2023_02_003 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0166-218X&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0166-218X&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0166-218X&client=summon |