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

TL;DR

This paper investigates the parameterized complexity of the NP-hard Stable Roommates problem with Ties and Incomplete Lists, providing both hardness results and fixed-parameter tractability findings based on various graph parameters.

Contribution

It extends previous work by analyzing the problem's complexity with respect to multiple graph parameters, showing hardness and tractability results.

Findings

W[1]-hardness for maximum stable matching with bounded treedepth, tree-cut width, and disjoint paths modulator

Fixed-parameter tractability for the existence of stable matchings with respect to tree-cut width

Fixed-parameter tractability results for feedback edge set number

Abstract

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 disjoint paths modulator 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.