Perfect matching in bipartite hypergraphs subject to a demand graph

Motivated by the problem of assigning plots to tenants, we present a version of the bipartite hypergraph matching problem. This version deals with a hypergraph with a constraint on its hyperedges, defined by a demand graph. We study the complexity of the matching problem for different demand graphs....

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Published in	Annals of operations research Vol. 321; no. 1-2; pp. 39 - 48
Main Authors	Aronshtam, Lior, Ilani, Hagai, Shufan, Elad
Format	Journal Article
Language	English
Published	New York Springer US 01.02.2023 Springer Springer Nature B.V
Subjects	Assignment problem Business and Management Combinatorics Demand Demand (Economics) Graph matching Graph theory Graphs Matching theory Operations research Operations Research/Decision Theory Original Research Tenants Theory of Computation Israel Assignment Perfect matching Bipartite hypergraph
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Abstract	Motivated by the problem of assigning plots to tenants, we present a version of the bipartite hypergraph matching problem. This version deals with a hypergraph with a constraint on its hyperedges, defined by a demand graph. We study the complexity of the matching problem for different demand graphs. The matching problem for 3-uniform hypergraphs is polynomially solvable if the set of perfect matchings of the demand graph can be polynomially generated. On the other hand, when the number of disjoint even cycles in the demand graph is Ω ( n 1 / k ) , for some constant k , the matching problem is NP-complete. For non-uniform hypergraphs, we show that the problem is NP-complete, even for very simple demand graphs.
AbstractList	Motivated by the problem of assigning plots to tenants, we present a version of the bipartite hypergraph matching problem. This version deals with a hypergraph with a constraint on its hyperedges, defined by a demand graph. We study the complexity of the matching problem for different demand graphs. The matching problem for 3-uniform hypergraphs is polynomially solvable if the set of perfect matchings of the demand graph can be polynomially generated. On the other hand, when the number of disjoint even cycles in the demand graph is Ω(n1/k), for some constant k, the matching problem is NP-complete. For non-uniform hypergraphs, we show that the problem is NP-complete, even for very simple demand graphs. Motivated by the problem of assigning plots to tenants, we present a version of the bipartite hypergraph matching problem. This version deals with a hypergraph with a constraint on its hyperedges, defined by a demand graph. We study the complexity of the matching problem for different demand graphs. The matching problem for 3-uniform hypergraphs is polynomially solvable if the set of perfect matchings of the demand graph can be polynomially generated. On the other hand, when the number of disjoint even cycles in the demand graph is Ω ( n 1 / k ) , for some constant k , the matching problem is NP-complete. For non-uniform hypergraphs, we show that the problem is NP-complete, even for very simple demand graphs. Motivated by the problem of assigning plots to tenants, we present a version of the bipartite hypergraph matching problem. This version deals with a hypergraph with a constraint on its hyperedges, defined by a demand graph. We study the complexity of the matching problem for different demand graphs. The matching problem for 3-uniform hypergraphs is polynomially solvable if the set of perfect matchings of the demand graph can be polynomially generated. On the other hand, when the number of disjoint even cycles in the demand graph is [Formula omitted], for some constant [Formula omitted], the matching problem is NP-complete. For non-uniform hypergraphs, we show that the problem is NP-complete, even for very simple demand graphs.
Audience	Academic
Author	Ilani, Hagai Aronshtam, Lior Shufan, Elad
Author_xml	– sequence: 1 givenname: Lior orcidid: 0000-0002-7211-2795 surname: Aronshtam fullname: Aronshtam, Lior organization: Department of Software Engineering, SCE – Shamoon College of Engineering – sequence: 2 givenname: Hagai orcidid: 0000-0003-3548-1572 surname: Ilani fullname: Ilani, Hagai organization: Department of Industrial Engineering and Management, SCE – Shamoon College of Engineering – sequence: 3 givenname: Elad orcidid: 0000-0001-7960-5798 surname: Shufan fullname: Shufan, Elad email: elads@sce.ac.il organization: Physics Department, SCE – Shamoon College of Engineering
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Cites_doi	10.1145/2229163.2229168 10.1007/s00493-012-2660-9 10.1016/j.omega.2020.102386 10.1137/0210055 10.1016/0893-9659(94)90045-0 10.1016/j.aam.2013.12.001 10.1093/qje/qjt019 10.1016/j.dam.2011.11.003 10.1137/1.9781611972238 10.1006/jctb.1997.1798 10.1016/j.disopt.2014.11.002 10.1016/0196-6774(90)90007-2 10.1007/s00373-017-1873-5 10.1016/j.endm.2010.05.043 10.1137/1.9781611973105.25 10.1007/978-1-4684-2001-2_9 10.1016/j.disopt.2010.07.004 10.1137/1.9781611974331.ch126 10.1016/j.aim.2011.03.015 10.1002/nav.3800020109
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References	FukudaKMatsuiTFinding all the perfect matchings in bipartite graphsApplied Mathematics Letters199471151810.1016/0893-9659(94)90045-0 Lovász, L., & Plummer, M.D. (1986). Matching theory. In Annals of discrete mathematics, 29. Burkard, R., Dell’Amico, M., & Martello, S. (2012). Assignment problems: Revised reprint. SIAM. Biró, P., & McDermid, E. (2010). Matching with sizes (or scheduling with processing set restrictions). In Technical report TR-2010-307. EsperetLKardošFKingADKrálDNorineSExponentially many perfect matchings in cubic graphsAdvances in Mathematics201122741646166410.1016/j.aim.2011.03.015 Delorme, M., Garcıa, S., Gondzio, J., Kalcsics, J., Manlove, D., Pettersson, W. (2020). Stability in the the hospitals/residents problem with couples and ties: Mathematical models and computational studies. Technical report ERGO-20-003. BiróPMcDermidEMatching with sizes (or scheduling with processing set restrictions)Discrete Applied Mathematics2014164616710.1016/j.dam.2011.11.003 SchrijverACounting 1-factors in regular bipartite graphsJournal of Combinatorial Theory, Series B199872112213510.1006/jctb.1997.1798 Annamalai, C. (2016). Finding perfect matchings in bipartite hypergraphs. In Proceedings of the twenty-seventh annual acm-siam symposium on discrete algorithms (pp. 1814–1823). ChudnovskyMSeymourPPerfect matchings in planar cubic graphsCombinatorica201232440342410.1007/s00493-012-2660-9 Karp, R.M. (1972). Reducibility among combinatorial problems. Complexity of computer computations (pp. 85–103). Springer KuhnHWThe Hungarian method for the assignment problemNaval research logistics quarterly195521–2839710.1002/nav.3800020109 BorndörferRHeismannOThe hypergraph assignment problemDiscrete Optimization201515152510.1016/j.disopt.2014.11.002 AsadpourAFeigeUSaberiASanta Claus meets hypergraph matchingsACM Transactions on Algorithms (TALG)2012831910.1145/2229163.2229168 RonnENp-complete stable matching problemsJournal of Algorithms199011228530410.1016/0196-6774(90)90007-2 Cygan, M., Grandoni, F., Mastrolilli, M. (2013). How to sell hyperedges: The hypermatching assignment problem. In Proceedings of the twenty-fourth annual acm-siam symposium on discrete algorithms (pp. 342–351). ChibaSFujitaSKawarabayashiK-ISakumaTMinimum degree conditions for vertex-disjoint even cycles in large graphsAdvances in Applied Mathematics20145410512010.1016/j.aam.2013.12.001 KojimaFPathakPARothAEMatching with couples: Stability and incentives in large marketsThe Quarterly Journal of Economics201312841585163210.1093/qje/qjt019 Aronshtam, L., Ilani, H., Shufan, E. (2019). Satisfying allocation of plots to tenants. In Proceedings of the 9th multidisciplinary international conference on scheduling: Theory and applications (pp. 30–41). BronfmanSAlonNHassidimARommARedesigning the Israeli medical internship matchACM Transactions on Economics and Computation (TEAC)201863–4118 ChibaSYamashitaTDegree conditions for the existence of vertex-disjoint cycles and paths: A surveyGraphs and Combinatorics201834118310.1007/s00373-017-1873-5 MarxDSchlotterIStable assignment with couples: Parameterized complexity and local searchDiscrete Optimization201181254010.1016/j.disopt.2010.07.004 HolyerIThe np-completeness of edge-coloringSIAM Journal on computing198110471872010.1137/0210055 Martello, S., & Toth, P. (1990). Knapsack problems: Algorithms and computer implementations. John Wiley & Sons Inc. 5073_CR13 5073_CR4 5073_CR12 5073_CR1 I Holyer (5073_CR16) 1981; 10 5073_CR2 L Esperet (5073_CR14) 2011; 227 5073_CR17 D Marx (5073_CR22) 2011; 8 M Chudnovsky (5073_CR11) 2012; 32 5073_CR21 K Fukuda (5073_CR15) 1994; 7 5073_CR20 A Schrijver (5073_CR24) 1998; 72 A Asadpour (5073_CR3) 2012; 8 P Biró (5073_CR5) 2014; 164 E Ronn (5073_CR23) 1990; 11 S Chiba (5073_CR10) 2018; 34 F Kojima (5073_CR18) 2013; 128 R Borndörfer (5073_CR6) 2015; 15 S Bronfman (5073_CR7) 2018; 6 5073_CR8 S Chiba (5073_CR9) 2014; 54 HW Kuhn (5073_CR19) 1955; 2
References_xml	– reference: EsperetLKardošFKingADKrálDNorineSExponentially many perfect matchings in cubic graphsAdvances in Mathematics201122741646166410.1016/j.aim.2011.03.015 – reference: FukudaKMatsuiTFinding all the perfect matchings in bipartite graphsApplied Mathematics Letters199471151810.1016/0893-9659(94)90045-0 – reference: BiróPMcDermidEMatching with sizes (or scheduling with processing set restrictions)Discrete Applied Mathematics2014164616710.1016/j.dam.2011.11.003 – reference: KojimaFPathakPARothAEMatching with couples: Stability and incentives in large marketsThe Quarterly Journal of Economics201312841585163210.1093/qje/qjt019 – reference: Cygan, M., Grandoni, F., Mastrolilli, M. (2013). How to sell hyperedges: The hypermatching assignment problem. In Proceedings of the twenty-fourth annual acm-siam symposium on discrete algorithms (pp. 342–351). – reference: SchrijverACounting 1-factors in regular bipartite graphsJournal of Combinatorial Theory, Series B199872112213510.1006/jctb.1997.1798 – reference: Aronshtam, L., Ilani, H., Shufan, E. (2019). Satisfying allocation of plots to tenants. In Proceedings of the 9th multidisciplinary international conference on scheduling: Theory and applications (pp. 30–41). – reference: BorndörferRHeismannOThe hypergraph assignment problemDiscrete Optimization201515152510.1016/j.disopt.2014.11.002 – reference: BronfmanSAlonNHassidimARommARedesigning the Israeli medical internship matchACM Transactions on Economics and Computation (TEAC)201863–4118 – reference: Biró, P., & McDermid, E. (2010). Matching with sizes (or scheduling with processing set restrictions). In Technical report TR-2010-307. – reference: ChibaSYamashitaTDegree conditions for the existence of vertex-disjoint cycles and paths: A surveyGraphs and Combinatorics201834118310.1007/s00373-017-1873-5 – reference: MarxDSchlotterIStable assignment with couples: Parameterized complexity and local searchDiscrete Optimization201181254010.1016/j.disopt.2010.07.004 – reference: ChibaSFujitaSKawarabayashiK-ISakumaTMinimum degree conditions for vertex-disjoint even cycles in large graphsAdvances in Applied Mathematics20145410512010.1016/j.aam.2013.12.001 – reference: Karp, R.M. (1972). Reducibility among combinatorial problems. Complexity of computer computations (pp. 85–103). Springer – reference: Martello, S., & Toth, P. (1990). Knapsack problems: Algorithms and computer implementations. John Wiley & Sons Inc. – reference: Lovász, L., & Plummer, M.D. (1986). Matching theory. In Annals of discrete mathematics, 29. – reference: Delorme, M., Garcıa, S., Gondzio, J., Kalcsics, J., Manlove, D., Pettersson, W. (2020). Stability in the the hospitals/residents problem with couples and ties: Mathematical models and computational studies. Technical report ERGO-20-003. – reference: AsadpourAFeigeUSaberiASanta Claus meets hypergraph matchingsACM Transactions on Algorithms (TALG)2012831910.1145/2229163.2229168 – reference: Burkard, R., Dell’Amico, M., & Martello, S. (2012). Assignment problems: Revised reprint. SIAM. – reference: ChudnovskyMSeymourPPerfect matchings in planar cubic graphsCombinatorica201232440342410.1007/s00493-012-2660-9 – reference: KuhnHWThe Hungarian method for the assignment problemNaval research logistics quarterly195521–2839710.1002/nav.3800020109 – reference: Annamalai, C. (2016). Finding perfect matchings in bipartite hypergraphs. In Proceedings of the twenty-seventh annual acm-siam symposium on discrete algorithms (pp. 1814–1823). – reference: RonnENp-complete stable matching problemsJournal of Algorithms199011228530410.1016/0196-6774(90)90007-2 – reference: HolyerIThe np-completeness of edge-coloringSIAM Journal on computing198110471872010.1137/0210055 – volume: 8 start-page: 1 issue: 3 year: 2012 ident: 5073_CR3 publication-title: ACM Transactions on Algorithms (TALG) doi: 10.1145/2229163.2229168 – volume: 32 start-page: 403 issue: 4 year: 2012 ident: 5073_CR11 publication-title: Combinatorica doi: 10.1007/s00493-012-2660-9 – ident: 5073_CR13 doi: 10.1016/j.omega.2020.102386 – volume: 10 start-page: 718 issue: 4 year: 1981 ident: 5073_CR16 publication-title: SIAM Journal on computing doi: 10.1137/0210055 – volume: 7 start-page: 15 issue: 1 year: 1994 ident: 5073_CR15 publication-title: Applied Mathematics Letters doi: 10.1016/0893-9659(94)90045-0 – volume: 54 start-page: 105 year: 2014 ident: 5073_CR9 publication-title: Advances in Applied Mathematics doi: 10.1016/j.aam.2013.12.001 – volume: 128 start-page: 1585 issue: 4 year: 2013 ident: 5073_CR18 publication-title: The Quarterly Journal of Economics doi: 10.1093/qje/qjt019 – ident: 5073_CR2 – volume: 164 start-page: 61 year: 2014 ident: 5073_CR5 publication-title: Discrete Applied Mathematics doi: 10.1016/j.dam.2011.11.003 – ident: 5073_CR8 doi: 10.1137/1.9781611972238 – volume: 72 start-page: 122 issue: 1 year: 1998 ident: 5073_CR24 publication-title: Journal of Combinatorial Theory, Series B doi: 10.1006/jctb.1997.1798 – volume: 15 start-page: 15 year: 2015 ident: 5073_CR6 publication-title: Discrete Optimization doi: 10.1016/j.disopt.2014.11.002 – volume: 11 start-page: 285 issue: 2 year: 1990 ident: 5073_CR23 publication-title: Journal of Algorithms doi: 10.1016/0196-6774(90)90007-2 – volume: 34 start-page: 1 issue: 1 year: 2018 ident: 5073_CR10 publication-title: Graphs and Combinatorics doi: 10.1007/s00373-017-1873-5 – ident: 5073_CR4 doi: 10.1016/j.endm.2010.05.043 – ident: 5073_CR20 – ident: 5073_CR12 doi: 10.1137/1.9781611973105.25 – ident: 5073_CR21 – ident: 5073_CR17 doi: 10.1007/978-1-4684-2001-2_9 – volume: 8 start-page: 25 issue: 1 year: 2011 ident: 5073_CR22 publication-title: Discrete Optimization doi: 10.1016/j.disopt.2010.07.004 – ident: 5073_CR1 doi: 10.1137/1.9781611974331.ch126 – volume: 227 start-page: 1646 issue: 4 year: 2011 ident: 5073_CR14 publication-title: Advances in Mathematics doi: 10.1016/j.aim.2011.03.015 – volume: 2 start-page: 83 issue: 1–2 year: 1955 ident: 5073_CR19 publication-title: Naval research logistics quarterly doi: 10.1002/nav.3800020109 – volume: 6 start-page: 1 issue: 3–4 year: 2018 ident: 5073_CR7 publication-title: ACM Transactions on Economics and Computation (TEAC)
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SubjectTerms	Assignment problem Business and Management Combinatorics Demand Demand (Economics) Graph matching Graph theory Graphs Matching theory Operations research Operations Research/Decision Theory Original Research Tenants Theory of Computation
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Title	Perfect matching in bipartite hypergraphs subject to a demand graph
URI	https://link.springer.com/article/10.1007/s10479-022-05073-9 https://www.proquest.com/docview/2769445689
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