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      Matchings with lower quotas: Algorithms and complexity

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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 \(G= (A \cup P, E)\) 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 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 \(u_{max}\) 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.

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          Author and article information

          Journal
          2014-11-30
          2016-03-27
          Article
          1412.0325
          5470ed7f-f4e4-44f0-8380-f9d94229c08f

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          05C85
          preliminary version has appeared at ISAAC 2015
          cs.DM cs.DS

          Data structures & Algorithms,Discrete mathematics & Graph theory
          Data structures & Algorithms, Discrete mathematics & Graph theory

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