Solutions for the Stable Roommates Problem with Payments

  • Péter Biró
  • Matthijs Bomhoff
  • Petr A. Golovach
  • Walter Kern
  • Daniël Paulusma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7551)


The stable roommates problem with payments has as input a graph G = (V,E) with an edge weighting w: E → ℝ +  and the problem is to find a stable solution. A solution is a matching M with a vector \(p\in{\mathbb R}^V_+\) that satisfies p u  + p v  = w(uv) for all uv ∈ M and p u  = 0 for all u unmatched in M. A solution is stable if it prevents blocking pairs, i.e., pairs of adjacent vertices u and v with p u  + p v  < w(uv). By pinpointing a relationship to the accessibility of the coalition structure core of matching games, we give a simple constructive proof for showing that every yes-instance of the stable roommates problem with payments allows a path of linear length that starts in an arbitrary unstable solution and that ends in a stable solution. This result generalizes a result of Chen, Fujishige and Yang for bipartite instances to general instances. We also show that the problems Blocking Pairs and Blocking Value, which are to find a solution with a minimum number of blocking pairs or a minimum total blocking value, respectively, are NP-hard. Finally, we prove that the variant of the first problem, in which the number of blocking pairs must be minimized with respect to some fixed matching, is NP-hard, whereas this variant of the second problem is polynomial-time solvable.


Stable Solution Weighted Graph Adjacent Vertex Stable Match Assignment Game 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Péter Biró
    • 1
  • Matthijs Bomhoff
    • 2
  • Petr A. Golovach
    • 3
  • Walter Kern
    • 2
  • Daniël Paulusma
    • 3
  1. 1.Institute of EconomicsHungarian Academy of SciencesBudapestHungary
  2. 2.Faculty of Electrical Engineering, Mathematics and Computer ScienceUniversity of TwenteEnschedeThe Netherlands
  3. 3.School of Engineering and Computing SciencesDurham University, Science LaboratoriesDurhamUK

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