“Almost Stable” Matchings in the Roommates Problem

Abstract

Abstract. An instance of the classical Stable Roommates problem (sr) need not admit a stable matching. This motivates the problem of finding a matching that is "as stable as possible", i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an sr instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within n 1 2 −ε , for any ε > 0, unless P=NP. If the preference lists contain ties, we improve this result to n 1−ε . Also, we show that, given an integer K and an sr instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an sr instance I.

The Stable Roommates Problem with Short Lists

“…egal | − |M| ≤ 0 The derived lower bound for |M (1,2) | is negative or zero in this case. Yet we know that at most half of the edges in M egal are (1,2)-pairs, and c(e) ≥ 4 for the rest of the edges in M egal .…”

“…Then M is a most-stable matching in I if |bp(M)| = bp(I). The problem of finding a most-stable matching was shown to be NP-hard and not approximable within n k−ε , for any ε > 0, unless P = NP, where k = 1 2 if I is an instance of SR and k = 1 if I is an instance of SRT [1]. To the best of our knowledge, there has not been any previous work published on either the problem of finding an egalitarian stable matching in a solvable instance of SRI with bounded-length preference lists or the solvability of SRTI with boundedlength preference lists.…”

Stable Marriage and Roommates Problems with Restricted Edges: Complexity and Approximability

“…Since sm is a restriction of sr, all established results on the NP-hardness and inapproximability of min bp sm restricted carry over to the non-bipartite sr case. As a matter of fact, more is true, since min bp sr restricted is NP-hard and difficult to approximate even if P = ∅ and Q = ∅ [20]. We summarise these observations as follows.…”

“…A natural goal is to find a matching minimising |bp(M )|; following the consensus in the literature, such a matching is called almost stable. This approach has a broad literature: almost stable matchings have been investigated in sm [17][18][19] and sr [20,21] instances.…”

“…Moreover Theorems 3.12 and 3.8 imply that each of min bp sr forbidden and min bp sr forced is NP-hard even if all preference lists are of length at most 3 or, in the latter case, |Q| = 1. Finally min bp sr restricted is NP-hard and not approximable within n 1 2 −ε , for any ε > 0, unless P = NP, even if P = ∅ and Q = ∅ [20].…”