[Paper Review] How Hard Is It to Satisfy (Almost) All Roommates?

Jiehua Chen, Danny Hermelin+2 more|arXiv (Cornell University)|Jul 13, 2017

Game Theory and Voting Systems43 references5 citations

TL;DR

This paper investigates two optimization variants of the Stable Roommates problem: Egalitarian Stable Roommates, which minimizes the sum of agents' dissatisfaction (egalitarian cost γ), and Min-Block-Pair Stable Roommates, which minimizes the number of blocking pairs β. It proves that Egalitarian Stable Roommates is fixed-parameter tractable when parameterized by γ, while Min-Block-Pair Stable Roommates is W[1]-hard even with preference lists of length at most five, highlighting a fundamental complexity dichotomy between the two variants.

ABSTRACT

The classic Stable Roommates problem (the non-bipartite generalization of the well-known Stable Marriage problem) asks whether there is a stable matching for a given set of agents, i.e. a partitioning of the agents into disjoint pairs such that no two agents induce a blocking pair. Herein, each agent has a preference list denoting who it prefers to have as a partner, and two agents are blocking if they prefer to be with each other rather than with their assigned partners. Since stable matchings may not be unique, we study an NP-hard optimization variant of Stable Roommates, called Egal Stable Roommates, which seeks to find a stable matching with a minimum egalitarian cost gamma, i.e. the sum of the dissatisfaction of the agents is minimum. The dissatisfaction of an agent is the number of agents that this agent prefers over its partner if it is matched; otherwise it is the length of its preference list. We also study almost stable matchings, called Min-Block-Pair Stable Roommates, which seeks to find a matching with a minimum number beta of blocking pairs. Our main result is that Egal Stable Roommates parameterized by gamma is fixed-parameter tractable, while Min-Block-Pair Stable Roommates parameterized by beta is W[1]-hard, even if the length of each preference list is at most five.

Motivation & Objective

• To study the parameterized complexity of Egalitarian Stable Roommates, an optimization variant of the Stable Roommates problem that minimizes the sum of agents' dissatisfaction.
• To investigate the complexity of Min-Block-Pair Stable Roommates, which seeks matchings with the fewest blocking pairs when no stable matching exists.
• To clarify the parameterized tractability frontier for these two optimization variants, especially under bounded preference list length.
• To resolve open questions on the complexity of related problems, such as Min-Block-Agents Stable Roommates, by providing tight hardness results and algorithms.

Proposed method

• Proposes a fixed-parameter tractable (FPT) algorithm for Egalitarian Stable Roommates parameterized by the egalitarian cost γ, using kernelization to reduce the instance size to O(γ²).
• Employs a reduction from Multi-Colored Independent Set to prove W[1]-hardness of Min-Block-Pair Stable Roommates parameterized by β, even with preference lists of length at most five.
• Designs specialized vertex-selection and verification gadgets in the reduction to simulate independent set solutions in the Stable Roommates setting.
• Uses a guessing-based algorithm for Min-Block-Agents Stable Roommates that enumerates possible blocking agents and checks candidate blocking pairs via a subroutine from Abraham et al. [6].
• Applies the Exponential Time Hypothesis (ETH) to argue that no f(β)·no(β)-time algorithm exists for Min-Block-Pair Stable Roommates unless ETH fails.
• Leverages structural properties of preference lists and blocking pair definitions to ensure correctness of the reduction and hardness proof.

Experimental results

Research questions

• RQ1 Is Egalitarian Stable Roommates fixed-parameter tractable when parameterized by the egalitarian cost γ?
• RQ2 Is Min-Block-Pair Stable Roommates W[1]-hard when parameterized by the number of blocking pairs β, even with bounded preference list length?
• RQ3 Can the parameterized complexity of Egalitarian Stable Roommates be extended to cases with ties in preference lists?
• RQ4 What is the tightest possible running time for solving Egalitarian Stable Roommates, and is the current FPT algorithm optimal?
• RQ5 Does the Min-Block-Agents variant of Stable Roommates admit a polynomial kernel or is it also W[1]-hard?

Key findings

• Egalitarian Stable Roommates is fixed-parameter tractable when parameterized by the egalitarian cost γ, with a kernel of size O(γ²).
• Min-Block-Pair Stable Roommates is W[1]-hard when parameterized by β, even when each preference list has length at most five.
• The W[1]-hardness result for Min-Block-Pair Stable Roommates is tight, as the lower bound matches the best-known approximation and parameterized complexity bounds.
• The reduction from Multi-Colored Independent Set to Min-Block-Pair Stable Roommates shows that the lower bound on β given by Abraham et al. [6, Lemma 4] is tight.
• Min-Block-Agents Stable Roommates is NP-hard and W[1]-hard with respect to the number of blocking agents η, even with preference lists of length at most five.
• A polynomial-time algorithm for Min-Block-Agents Stable Roommates exists with running time O(2η² · nη+2), based on guessing blocking agents and pairs, and verifying via a subroutine from Abraham et al. [6].