Preference Elicitation in Matching Markets Via Interviews: A Study of Offline Benchmarks

In this paper we study two-sided matching markets in which the participants do not fully know their preferences and need to go through some costly deliberation process in order to learn their preferences. We assume that such deliberations are carried out via interviews, thus the problem is to find a good strategy for interviews to be carried out in order to minimize their use, whilst leading to a stable matching. One way to evaluate the performance of an interview strategy is to compare it against a nave ïalgorithm that conducts all interviews. We argue however that a more meaningful comparison would be against an optimal offline algorithm that has access to agents' preference orderings under complete information. We show that, unless P=NP, no offline algorithm can compute the optimal interview strategy in polynomial time. If we are additionally aiming for a particular stable matching, we provide restricted settings under which efficient optimal offline algorithms exist

Parameterized Complexity of Stable Roommates with Ties and Incomplete Lists Through the Lens of Graph Parameters

We continue and extend previous work on the parameterized complexity analysis of the NP-hard Stable Roommates with Ties and Incomplete Lists problem, thereby strengthening earlier results both on the side of parameterized hardness as well as on the side of fixed-parameter tractability. Other than for its famous sister problem Stable Marriage which focuses on a bipartite scenario, Stable Roommates with Incomplete Lists allows for arbitrary acceptability graphs whose edges specify the possible matchings of each two agents (agents are represented by graph vertices). Herein, incomplete lists and ties reflect the fact that in realistic application scenarios the agents cannot bring all other agents into a linear order. Among our main contributions is to show that it is W[1]-hard to compute a maximum-cardinality stable matching for acceptability graphs of bounded treedepth, bounded tree-cut width, and bounded feedback vertex number (these are each time the respective parameters). However, if we "only" ask for perfect stable matchings or the mere existence of a stable matching, then we obtain fixed-parameter tractability with respect to tree-cut width but not with respect to treedepth. On the positive side, we also provide fixed-parameter tractability results for the parameter feedback edge set number

An Integer Programming Approach to the Student-Project Allocation Problem with Preferences over Projects

The Student-Project Allocation problem with preferences over Projects (SPA-P) involves sets of students, projects and lecturers, where the students and lecturers each have preferences over the projects. In this context, we typically seek a stable matching of students to projects (and lecturers). However, these stable matchings can have different sizes, and the problem of finding a maximum stable matching (MAX-SPA-P) is NP-hard. There are two known approximation algorithms for MAX-SPA-P, with performance guarantees of 2 and 32 . In this paper, we describe an Integer Programming (IP) model to enable MAX-SPA-P to be solved optimally. Following this, we present results arising from an empirical analysis that investigates how the solution produced by the approximation algorithms compares to the optimal solution obtained from the IP model, with respect to the size of the stable matchings constructed, on instances that are both randomly-generated and derived from real datasets. Our main finding is that the 32 -approximation algorithm finds stable matchings that are very close to having maximum cardinality

Fast distributed almost stable marriages

In their seminal work on the Stable Marriage Problem, Gale and Shapley describe an algorithm which finds a stable matching in $O(n^2)$ communication rounds. Their algorithm has a natural interpretation as a distributed algorithm where each player is represented by a single processor. In this distributed model, Floreen, Kaski, Polishchuk, and Suomela recently showed that for bounded preference lists, terminating the Gale-Shapley algorithm after a constant number of rounds results in an almost stable matching. In this paper, we describe a new deterministic distributed algorithm which finds an almost stable matching in $O(\log^5 n)$ communication rounds for arbitrary preferences. We also present a faster randomized variant which requires $O(\log^2 n)$ rounds. This run-time can be improved to $O(1)$ rounds for "almost regular" (and in particular complete) preferences. To our knowledge, these are the first sub-polynomial round distributed algorithms for any variant of the stable marriage problem with unbounded preferences.Comment: Various improvements in version 2: algorithms for general (not just "almost regular") preferences; deterministic variant of the algorithm; streamlined proof of approximation guarante