Efficiency in the Roommates Problem

TL;DR

This paper studies the problem of verifying efficiency of a given matching in the roommates problem, a generalization of stable matching where some agents may remain unmatched. It extends the $O(n^2)$ approach of Morrill (2010) to settings with unmatched agents by introducing a modified graph $G'$ with virtual vertices and a relation to irrational pairs. A central result is that a matching is efficient iff the base graph $G$ contains no alternating path or alternating cycle, and, when no irrational pairs are present, this corresponds to the absence of an alternating cycle in $G'$. The Iterative Biconnected Component Decomposition Algorithm (IBCDA) determines efficiency in $O(n^2)$ time by locating a non-trivial block in $G'$ that contains an alternating cycle, providing a practical polynomial-time decision procedure with broad applicability in economics and related areas.

Abstract

We propose an $O(n^2)$-time algorithm to determine whether a given matching is efficient in the roommates problem.

Figures (3)

• Figure 1: The figure on the left illustrates an alternating path, whereas the figure on the right illustrates an alternating cycle. Special edges are depicted in black, and normal edges are depicted in white.
• Figure 2: An example of biconnected component decomposition. It consists of three blocks.
• Figure 3: This represents the case where vertices 1 and 4 are removed from the block containing vertices 1, 4, 2, and 8.