Parameterized Complexity of Stable Roommates with Ties and Incomplete Lists Through the Lens of Graph Parameters

Authors Robert Bredereck , Klaus Heeger , Dušan Knop , Rolf Niedermeier

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Robert Bredereck
  • Technische Universität Berlin, Chair of Algorithmics and Computational Complexity, Germany
Klaus Heeger
  • Technische Universität Berlin, Chair of Algorithmics and Computational Complexity, Germany
Dušan Knop
  • Technische Universität Berlin, Chair of Algorithmics and Computational Complexity, Germany
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
Rolf Niedermeier
  • Technische Universität Berlin, Chair of Algorithmics and Computational Complexity, Germany

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Robert Bredereck, Klaus Heeger, Dušan Knop, and Rolf Niedermeier. Parameterized Complexity of Stable Roommates with Ties and Incomplete Lists Through the Lens of Graph Parameters. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We continue and extend previous work on the parameterized complexity analysis of the NP-hard Stable Roommates with Ties and Incomplete Lists problem, thereby strengthening earlier results both on the side of parameterized hardness as well as on the side of fixed-parameter tractability. Other than for its famous sister problem Stable Marriage which focuses on a bipartite scenario, Stable Roommates with Incomplete Lists allows for arbitrary acceptability graphs whose edges specify the possible matchings of each two agents (agents are represented by graph vertices). Herein, incomplete lists and ties reflect the fact that in realistic application scenarios the agents cannot bring all other agents into a linear order. Among our main contributions is to show that it is W[1]-hard to compute a maximum-cardinality stable matching for acceptability graphs of bounded treedepth, bounded tree-cut width, and bounded feedback vertex number (these are each time the respective parameters). However, if we "only" ask for perfect stable matchings or the mere existence of a stable matching, then we obtain fixed-parameter tractability with respect to tree-cut width but not with respect to treedepth. On the positive side, we also provide fixed-parameter tractability results for the parameter feedback edge set number.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Matchings and factors
  • Stable matching
  • acceptability graph
  • fixed-parameter tractability
  • W[1]-hardness
  • treewidth
  • treedepth
  • tree-cut width
  • feedback set numbers


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