A New Algorithm for Solving a Special Matching Problem with a General Form Value Function under Constraints
We consider the assignment problem with a special structure with a general form value function and constraints prohibiting certain matchings. In this case, the matching cost may be undefined until some permutation is found. We formulate the problem in terms of graph theory and reduce it to finding a...
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| Published in | Automation and remote control Vol. 80; no. 1; pp. 81 - 92 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Moscow
Pleiades Publishing
01.01.2019
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0005-1179 1608-3032 |
| DOI | 10.1134/S0005117919010077 |
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| Abstract | We consider the assignment problem with a special structure with a general form value function and constraints prohibiting certain matchings. In this case, the matching cost may be undefined until some permutation is found. We formulate the problem in terms of graph theory and reduce it to finding a minimal cost path in a graph with nonlocal edge weights. The proposed method for solving the problem is a modification of the Dijkstra’s shortest path algorithm in a weighted directed graph. This research is motivated by well drilling applications. We also show the analysis of our numerical experiments. |
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| AbstractList | We consider the assignment problem with a special structure with a general form value function and constraints prohibiting certain matchings. In this case, the matching cost may be undefined until some permutation is found. We formulate the problem in terms of graph theory and reduce it to finding a minimal cost path in a graph with nonlocal edge weights. The proposed method for solving the problem is a modification of the Dijkstra’s shortest path algorithm in a weighted directed graph. This research is motivated by well drilling applications. We also show the analysis of our numerical experiments. |
| Author | Lomovitskii, P. V. Uzhegov, D. V. Khlyupin, A. N. Anan’ev, A. A. |
| Author_xml | – sequence: 1 givenname: D. V. surname: Uzhegov fullname: Uzhegov, D. V. email: denis.uzhegov@phystech.edu organization: MIPhT Engineering Center for Hard to Recover Minerals, Moscow Institute of Physics and Technology (State University) – sequence: 2 givenname: A. A. surname: Anan’ev fullname: Anan’ev, A. A. organization: MIPhT Engineering Center for Hard to Recover Minerals, Moscow Institute of Physics and Technology (State University) – sequence: 3 givenname: P. V. surname: Lomovitskii fullname: Lomovitskii, P. V. organization: MIPhT Engineering Center for Hard to Recover Minerals, Moscow Institute of Physics and Technology (State University) – sequence: 4 givenname: A. N. surname: Khlyupin fullname: Khlyupin, A. N. organization: MIPhT Engineering Center for Hard to Recover Minerals, Moscow Institute of Physics and Technology (State University) |
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| Keywords | shortest path in a graph quadratic assignment problem invalid matchings Dijkstra’s algorithm |
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| References_xml | – reference: BurkardR.E.CelaE.Linear Assignment Problems and Extensions, in Handbook Combinat. Optim.1999 – reference: KoopmansT.C.BeckmannM.Assignment Problems and the Location of Economic ActivitiesEconometrica: J. Econometric Soc.19572515376891060098.1220310.2307/1907742 – reference: SahniS.GonzalezT.P-Complete Approximation ProblemsJ. ACM (JACM)19762335555654083130348.9015210.1145/321958.321975 – reference: KuhnH.W.The Hungarian Method for the Assignment ProblemNaval Res. Logist. (NRL)1955 – reference: BurkardR.E.CelaE.RoteG.WoegingerG.J.The Quadratic Assignment Problem with a Monotone Anti-Monge and a Symmetric Toeplitz Matrix: Easy and Hard CasesMath. Program.199882112515816276320949.90077 – reference: JüngerM.KaibelV.A Basic Study of the QAP Polytope, Techincal Report 96.215, Institut für Informatik1996 – reference: BurkardR.Dell’AmicoM.MartelloS.Assignment Problems20121196.9000210.1137/1.9781611972238 – reference: ÇelaE.SchmuckN.S.WimerS.WoegingerG.J.The Wiener Maximum Quadratic Assignment Problem [Online], arXiv.org20111233.90282 – reference: HardyG.H.LittlewoodJ.E.PolyaG.The Maximum of a Certain Bilinear FormProc. London Math. Soc.192621265282157528452.0206.0310.1112/plms/s2-25.1.265 – reference: BertsekasD.P.A New Algorithm for the Assignment ProblemMath. Program.19812111521716238350461.9006910.1007/BF01584237 – reference: ManiezzoV.ColorniA.The Ant System Applied to the Quadratic Assignment ProblemIEEE Transact. Knowledge Data Eng.199911576977810.1109/69.806935 – reference: HamdyA.T.Operations Research: An Introduction20020227.90002 – reference: TateD.M.SmithA.E.A Genetic Approach to the Quadratic Assignment ProblemComput. Oper. Res.199522173830812.9009910.1016/0305-0548(93)E0020-T – reference: BarrR.S.GloverF.KlingmanD.The Alternating Basis Algorithm for Assignment ProblemsMath. Program.19771311134440390378.9009710.1007/BF01584319 – reference: TaillardE.Robust Taboo Search for the Quadratic Assignment ProblemParallel Comput.1991174443455112301510.1016/S0167-8191(05)80147-4 – reference: LovászL.PlummerM.Matching Theory, Amsterdam: Elsevier1986 – reference: DantzigG.B.Linear Programming and Extensions19630108.3310310.7249/R366 – reference: DemidenkoV.M.FinkeG.GordonV.S.Well Solvable Cases of the Quadratic Assignment Problem with Monotone and Bimonotone MatricesJ. Math. Modell. Algorithms20065216718722315631121.9010710.1007/s10852-005-9013-2 – volume-title: Linear Assignment Problems and Extensions, in Handbook Combinat. Optim. year: 1999 ident: 912_CR7 – volume-title: A Basic Study of the QAP Polytope, Techincal Report 96.215, Institut für Informatik year: 1996 ident: 912_CR17 – volume: 25 start-page: 53 issue: 1 year: 1957 ident: 912_CR8 publication-title: Econometrica: J. Econometric Soc. doi: 10.2307/1907742 – volume-title: The Wiener Maximum Quadratic Assignment Problem [Online], arXiv.org year: 2011 ident: 912_CR18 – volume-title: Operations Research: An Introduction year: 2002 ident: 912_CR6 – volume-title: Assignment Problems year: 2012 ident: 912_CR13 doi: 10.1137/1.9781611972238 – volume: 21 start-page: 152 issue: 1 year: 1981 ident: 912_CR5 publication-title: Math. Program. doi: 10.1007/BF01584237 – volume-title: Linear Programming and Extensions year: 1963 ident: 912_CR2 doi: 10.7249/R366 – volume: 13 start-page: 1 issue: 1 year: 1977 ident: 912_CR3 publication-title: Math. Program. doi: 10.1007/BF01584319 – volume-title: Naval Res. Logist. (NRL) year: 1955 ident: 912_CR4 – volume: 2 start-page: 265 issue: 1 year: 1926 ident: 912_CR10 publication-title: Proc. London Math. Soc. doi: 10.1112/plms/s2-25.1.265 – volume: 5 start-page: 167 issue: 2 year: 2006 ident: 912_CR12 publication-title: J. Math. Modell. Algorithms doi: 10.1007/s10852-005-9013-2 – volume-title: Matching Theory, Amsterdam: Elsevier year: 1986 ident: 912_CR1 – volume: 22 start-page: 73 issue: 1 year: 1995 ident: 912_CR14 publication-title: Comput. Oper. Res. doi: 10.1016/0305-0548(93)E0020-T – volume: 11 start-page: 769 issue: 5 year: 1999 ident: 912_CR15 publication-title: IEEE Transact. Knowledge Data Eng. doi: 10.1109/69.806935 – volume: 17 start-page: 443 issue: 4 year: 1991 ident: 912_CR16 publication-title: Parallel Comput. doi: 10.1016/S0167-8191(05)80147-4 – volume: 82 start-page: 125 issue: 1 year: 1998 ident: 912_CR11 publication-title: Math. Program. – volume: 23 start-page: 555 issue: 3 year: 1976 ident: 912_CR9 publication-title: J. ACM (JACM) doi: 10.1145/321958.321975 |
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| SubjectTerms | Algorithms CAE) and Design Calculus of Variations and Optimal Control; Optimization Computer-Aided Engineering (CAD Control Control in Technical Systems Graph theory Matching Mathematics Mathematics and Statistics Mechanical Engineering Mechatronics Operations research Permutations Robotics Shortest-path problems Systems Theory Well drilling |
| Title | A New Algorithm for Solving a Special Matching Problem with a General Form Value Function under Constraints |
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