Biró, P. and Bomhoff, M. and Golovach, P.A. and Kern, W. and Paulusma, Daniel (2012) 'Solutions for the stable rommates problem with payments.', in Graph-theoretic concepts in computer science : 38th International Workshop, WG 2012, Jerusalem, Israel, 26-28 June 2012 ; revised selected papers. Berlin ; Heidelberg: Springer, 69-80 . Lecture notes in computer science., 7551 (9551).
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∈R V + that satisfies pu + pv = w(uv) for all uv ∈ M and pu = 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 pu + pv < 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.
|Item Type:||Book chapter|
|Full text:||Full text not available from this repository.|
|Publisher Web site:||http://dx.doi.org/10.1007/978-3-642-34611-8_10|
|Date accepted:||No date available|
|Date deposited:||No date available|
|Date of first online publication:||2012|
|Date first made open access:||No date available|
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