# Many-to-one matchings with lower quotas: algorithms and complexity

Arulselvan, A., Cseh, A., Groß, M., and Matuschke, J. (2015) Many-to-one matchings with lower quotas: algorithms and complexity. In: ISAAC 2015: the 26th International Symposium on Algorithms, Nagoya, Japan, 9-11 Dec 2015, pp. 176-187. ISBN 9783662489710

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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-toone 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 classic 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]. Finally, we also present an approximation algorithm for the general case with performance guarantee umax + 1, which is asymptotically best possible unless P = NP.

Item Type: Conference Proceedings Published Yes Manlove, Professor David Arulselvan, A., Cseh, A., Groß, M., Manlove, D. F., and Matuschke, J. College of Science and Engineering > School of Computing Science 0302-9743 9783662489710 Copyright © 2015 The Authors First published in Lecture Notes in Computer Science 9472: 176-187 Reproduced under a Creative Commons License Organisation

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
607071Efficient Algorithms for Mechanism Design Without Monetary Transfer.David ManloveEngineering & Physical Sciences Research Council (EPSRC)EP/K010042/1COM - COMPUTING SCIENCE