Stable Marriage in Euclidean Space

We study stable marriage problems in the d-Euclidean space. Under this setting, each agent is represented as a point in the d-dimensional space, and for each agent a, the preference of a is based on the sorting according to the Euclidean distances between a and agents from the opposite gender. Let δ(a,b) being the Euclidean distance between two points a and b. A man u prefers a woman w1 to another woman w2 if and only if δ(u,w1) < δ(u,w2). If δ(u,w1) = δ(u,w2), then u ranks w1 and w2 indifferently, and we say there is a tie between w1 and w2 in u's preference list. A lot of variants of Stable Marriage with Ties (SMT) have been shown to be NP-complete when ties occur in preference lists. In this paper, we study the most famous hard variants of SMT in d-Euclidean space, namely, Regret-SMT, Forced-SMT, and Egalitarian-SMT. We prove that with d=1, Forced-SMT and Regert-SMT can be solved in polynomial-time, while with d=2, all of the three problems are NP-hard. Then we show that if the preference list can be incomplete (agents are allowed to not give a full rank of the opposite gender), the three problems and another variant Max-SMTI are NP-hard even with d=1. Finally, we provide an algorithm to recognize whether a given preference profile can be embedded into 1-Euclidean space.