Stable Marriage with Covering Constraints-A Complete Computational Trichotomy

We consider stable marriage problems equipped with covering constraints: here the input distinguishes a subset of women as well as a subset of men, and we seek a matching with fewest number of blocking pairs that matches all of the distinguished people. This concept strictly generalizes the notion of arranged marriages introduced by Knuth in 1976, in which the partner of each distinguished person is fixed a priori. Our main result is a complete computational complexity trichotomy of the stable marriage problem with covering constraints, into polynomial-time solvable cases, fixed-parameter tractable cases, and cases that are W[1]-hard, for every choice among a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed. Thereby, we fully answer an open problem of Hamada et al. (ESA 2011).