Algorithms and Complexity of Strongly Stable Non-crossing Matchings

Abstract

A matching is called stable if it has no blocking pair, where a blocking pair is a man-woman pair, say (m, w), such that m and w are not matched with each other in the matching but if they get matched with each other, then both of them become better off. A matching is called non-crossing if it does not admit any pair of edges that cross each other when all men and women are arranged in two parallel vertical lines with men on one line and women on the other. Two notions of matchings that are stable as well as non-crossing have been identified in the literature, namely (i) weakly stable non-crossing matching (WSNM) and (ii) strongly stable non-crossing matching (SSNM). An SSNM is a non-crossing matching which is stable in the classical sense, whereas in a WSNM, a blocking pair satisfies an extra condition that it must not cross any matching edge. It is known that the problem of finding a WSNM, which always exists in an SMI instance, is polynomial time solvable. However, the problem of determining the existence of an SSNM in SMTI is known to be NP-complete. We show that this problem is fixed-parameter tractable (FPT) when parameterized by total length of ties. We introduce a new notion of stable non-crossing matching, namely semi-strongly stable non-crossing matching (SSSNM). We prove that the problem of determining the existence of an SSSNM in SMI is NP-complete even if size of every man’s preference list is at most two. On the positive side, we show that this problem is polynomial time solvable if every man’s preference list contains at most one woman.

A Illustration of Construction of Instance in the Proof of Theorem 2

Fig. 2.

An illustration of the construction of instance I from a formula g in the proof of Theorem 2. The matching edges corresponding to a satisfying assignment for g, say $x_1=1$, $x_2=0$, $x_3=0$, are shown in blue colour. (Color figure online)