Matchings with Lower Quotas: Algorithms and Complexity

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G = ( A ∪ ˙ P , E ) with weights on the edges in E , and with lower and upper quotas on the vertices in P . We seek a maximum weight many-to-one matching satisfying two s...

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Vydáno v:	Algorithmica Ročník 80; číslo 1; s. 185 - 208
Hlavní autoři:	Arulselvan, Ashwin, Cseh, Ágnes, Groß, Martin, Manlove, David F., Matuschke, Jannik
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.01.2018 Springer Nature B.V
Témata:	Algorithm Analysis and Problem Complexity Algorithms Colleges & universities Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Graphs Matching Mathematics of Computing Polynomials Quotas Students Theory of Computation Maximum matching Bounded treewidth Many-to-one matching Inapproximability Project allocation
ISSN:	0178-4617, 1432-0541
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Abstract	We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G = ( A ∪ ˙ P , E ) with weights on the edges in E , and with lower and upper quotas on the vertices in P . We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NP -hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota u max as basis, and we prove that this dependence is necessary unless FPT = W [ 1 ] . The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee u max + 1 , which is asymptotically best possible unless P = NP . Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas.
AbstractList	We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G=(A∪˙P,E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NP-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota umax as basis, and we prove that this dependence is necessary unless FPT=W[1]. The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee umax+1, which is asymptotically best possible unless P=NP. Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas. We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G = ( A ∪ ˙ P , E ) with weights on the edges in E , and with lower and upper quotas on the vertices in P . We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NP -hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota u max as basis, and we prove that this dependence is necessary unless FPT = W [ 1 ] . The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee u max + 1 , which is asymptotically best possible unless P = NP . Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas.
Author	Manlove, David F. Arulselvan, Ashwin Matuschke, Jannik Groß, Martin Cseh, Ágnes
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Snippet	We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G = ( A ∪ ˙ P , E ) with... We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G=(A∪˙P,E) with weights on the...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Colleges & universities Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Graphs Matching Mathematics of Computing Polynomials Quotas Students Theory of Computation
Title	Matchings with Lower Quotas: Algorithms and Complexity
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