Parameterized Complexity of Stable Roommates with Ties and Incomplete Lists Through the Lens of Graph Parameters
Abstract
We continue and extend previous work on the parameterized complexity analysis of the NPhard 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 fixedparameter 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 maximumcardinality stable matching for acceptability graphs of bounded treedepth, bounded treecut 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 fixedparameter tractability with respect to treecut width but not with respect to treedepth. On the positive side, we also provide fixedparameter tractability results for the parameter feedback edge set number.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.09379
 Bibcode:
 2019arXiv191109379B
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics
 EPrint:
 An extended abstract of this paper will appear at ISAAC 2019