The cycle roommates problem: a hard case of kidney exchange

Recently, a number of interesting algorithmic problems have arisen from the emergence, in a number of countries, of kidney exchange schemes, whereby live donors are matched with recipients according to compatibility and other considerations. One such problem can be modeled by a variant of the well-known stable roommates problem in which blocking cycles, as well as the normal blocking pairs, are significant. We show here that this variant of the stable roommates problem is NP-complete, thus solving an open question posed by Cechlárová and Lacko. 1 Background and problem statement An instance of the stable roommates problem (SR) consists of a set of participants {p1, . . . , pn}, and for each participant, a preference list, which is a total order over a subset of the others. We say that pj is acceptable to pi if pj appears on pi’s preference list, and that pi prefers pj to pk if pj and pk are both acceptable to pi and pj precedes pk on pi’s list. An acceptable pair is a pair {pi, pj} each of whom is acceptable to the other. A matching is a set M of acceptable pairs so that each participant appears in at most one pair of M . If M is a matching and {pi, pj} ∈ M , we say that pi and pj are both matched in M and are partners of each other, and we write M(pi) = pj , M(pj) = pi. A blocking pair for a matching M is an acceptable pair {pi, pj} such that pi is either unmatched in M or prefers pj to M(pi) and pj is either unmatched or prefers pi to M(pj). A matching that admits no blocking pair is called stable. The stable roommates problem was introduced by Gale and Shapley [3] as a non-bipartite variant of the classical stable marriage problem (SM). They showed that, in contrast to the case of SM, there are SR instances for which no stable matching exists. Knuth [7] asked whether the stable roommates problem is solvable in polynomial time, and Irving [6] answered this question in the affirmative with a polynomial-time algorithm that determines whether a stable matching for a roommates instance exists, and if so finds one such matching. Subsequently Gusfield [4] presented an analysis of the structure of SR instances, and exploited this structure to solve variants of the problem. The monograph of Gusfield and Irving [5] includes a comprehensive treatment of these results.