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

by   Robert Bredereck, et al.

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.


page 1

page 2

page 3

page 4


Parameterized Complexity of Geodetic Set

A vertex set S of a graph G is geodetic if every vertex of G lies on a s...

Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems

When computing stable matchings, it is usually assumed that the preferen...

Algorithmic Applications of Tree-Cut Width

The recently introduced graph parameter tree-cut width plays a similar r...

Approximating Stable Matchings with Ties of Bounded Size

Finding a stable matching is one of the central problems in algorithmic ...

A note on the complexity of Feedback Vertex Set parameterized by mim-width

We complement the recent algorithmic result that Feedback Vertex Set is ...

Two-sided popular matchings in bipartite graphs with forbidden/forced elements and weights

Two-sided popular matchings in bipartite graphs are a well-known general...

On the fixed-parameter tractability of the maximum connectivity improvement problem

In the Maximum Connectivity Improvement (MCI) problem, we are given a di...