Approximation algorithms for hard variants of the stable marriage and hospitals/residents problems

When ties and incomplete preference lists are permitted in the Stable Marriage and Hospitals/Residents problems, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is known to be NP-hard, even under very severe restrictions on the number, size and position of ties. In this paper, we describe polynomial-time 5/3-approximation algorithms for variants of these problems in which ties are on one side only and at the end of the preference lists. The particular variant is motivated by important applications in large scale centralised matching schemes

The Stable Roommates problem with short lists

We consider two variants of the classical Stable Roommates problem with Incomplete (but strictly ordered) preference lists SRI that are degree constrained, i.e., preference lists are of bounded length. The first variant, EGAL d-SRI, involves finding an egalitarian stable matching in solvable instances of SRI with preference lists of length at most d. We show that this problem is NP-hard even if d=3. On the positive side we give a (2d+3)/7-approximation algorithm for d={3,4,5} which improves on the known bound of 2 for the unbounded preference list case. In the second variant of SRI, called d-SRTI, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of d-SRTI admits a stable matching is NP-complete even if d=3. We also consider the "most stable" version of this problem and prove a strong inapproximability bound for the d=3 case. However for d=2 we show that the latter problem can be solved in polynomial time.Comment: short version appeared at SAGT 201

Rank Maximal Matchings -- Structure and Algorithms

Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and ranks on the edges denote preferences of the agents over posts. A matching M in G is rank-maximal if it matches the maximum number of applicants to their top-rank post, subject to this, the maximum number of applicants to their second rank post and so on. In this paper, we develop a switching graph characterization of rank-maximal matchings, which is a useful tool that encodes all rank-maximal matchings in an instance. The characterization leads to simple and efficient algorithms for several interesting problems. In particular, we give an efficient algorithm to compute the set of rank-maximal pairs in an instance. We show that the problem of counting the number of rank-maximal matchings is #P-Complete and also give an FPRAS for the problem. Finally, we consider the problem of deciding whether a rank-maximal matching is popular among all the rank-maximal matchings in a given instance, and give an efficient algorithm for the problem

Stable Noncrossing Matchings

Given a set of $n$ men represented by $n$ points lying on a line, and $n$ women represented by $n$ points lying on another parallel line, with each person having a list that ranks some people of opposite gender as his/her acceptable partners in strict order of preference. In this problem, we want to match people of opposite genders to satisfy people's preferences as well as making the edges not crossing one another geometrically. A noncrossing blocking pair w.r.t. a matching $M$ is a pair $(m,w)$ of a man and a woman such that they are not matched with each other but prefer each other to their own partners in $M$, and the segment $(m,w)$ does not cross any edge in $M$. A weakly stable noncrossing matching (WSNM) is a noncrossing matching that does not admit any noncrossing blocking pair. In this paper, we prove the existence of a WSNM in any instance by developing an $O(n^2)$ algorithm to find one in a given instance.Comment: This paper has appeared at IWOCA 201

Manipulation Strategies for the Rank Maximal Matching Problem

We consider manipulation strategies for the rank-maximal matching problem. In the rank-maximal matching problem we are given a bipartite graph $G = (A \cup P, E)$ such that $A$ denotes a set of applicants and $P$ a set of posts. Each applicant $a \in A$ has a preference list over the set of his neighbours in $G$, possibly involving ties. Preference lists are represented by ranks on the edges - an edge $(a,p)$ has rank $i$, denoted as $rank(a,p)=i$, if post $p$ belongs to one of $a$'s $i$-th choices. A rank-maximal matching is one in which the maximum number of applicants is matched to their rank one posts and subject to this condition, the maximum number of applicants is matched to their rank two posts, and so on. A rank-maximal matching can be computed in $O(\min(c \sqrt{n},n) m)$ time, where $n$ denotes the number of applicants, $m$ the number of edges and $c$ the maximum rank of an edge in an optimal solution. A central authority matches applicants to posts. It does so using one of the rank-maximal matchings. Since there may be more than one rank- maximal matching of $G$, we assume that the central authority chooses any one of them randomly. Let $a_1$ be a manipulative applicant, who knows the preference lists of all the other applicants and wants to falsify his preference list so that he has a chance of getting better posts than if he were truthful. In the first problem addressed in this paper the manipulative applicant $a_1$ wants to ensure that he is never matched to any post worse than the most preferred among those of rank greater than one and obtainable when he is truthful. In the second problem the manipulator wants to construct such a preference list that the worst post he can become matched to by the central authority is best possible or in other words, $a_1$ wants to minimize the maximal rank of a post he can become matched to

Super-stability in the Student-Project Allocation Problem with Ties

The Student-Project Allocation problem with lecturer preferences over Students ( Open image in new window ) involves assigning students to projects based on student preferences over projects, lecturer preferences over students, and the maximum number of students that each project and lecturer can accommodate. This classical model assumes that preference lists are strictly ordered. Here, we study a generalisation of Open image in new window where ties are allowed in the preference lists of students and lecturers, which we refer to as the Student-Project Allocation problem with lecturer preferences over Students with Ties ( Open image in new window ). We investigate stable matchings under the most robust definition of stability in this context, namely super-stability. We describe the first polynomial-time algorithm to find a super-stable matching or to report that no such matching exists, given an instance of Open image in new window . Our algorithm runs in O(L) time, where L is the total length of all the preference lists. Finally, we present results obtained from an empirical evaluation of the linear-time algorithm based on randomly-generated Open image in new window instances. Our main finding is that, whilst super-stable matchings can be elusive, the probability of such a matching existing is significantly higher if ties are restricted to the lecturers’ preference lists

Integer programming methods for special college admissions problems

We develop Integer Programming (IP) solutions for some special college admission problems arising from the Hungarian higher education admission scheme. We focus on four special features, namely the solution concept of stable score-limits, the presence of lower and common quotas, and paired applications. We note that each of the latter three special feature makes the college admissions problem NP-hard to solve. Currently, a heuristic based on the Gale-Shapley algorithm is being used in the application. The IP methods that we propose are not only interesting theoretically, but may also serve as an alternative solution concept for this practical application, and also for other ones

Profile-Based Optimal Matchings in the Student-Project Allocation Problem

In the Student/Project Allocation problem (spa) we seek to assign students to individual or group projects offered by lecturers. Students provide a list of projects they find acceptable in order of preference. Each student can be assigned to at most one project and there are constraints on the maximum number of students that can be assigned to each project and lecturer. We seek matchings of students to projects that are optimal with respect to profile, which is a vector whose rth component indicates how many students have their rth-choice project. We present an efficient algorithm for finding agreedy maximum matching in the spa context – this is a maximum matching whose profile is lexicographically maximum. We then show how to adapt this algorithm to find a generous maximum matching – this is a matching whose reverse profile is lexicographically minimum. Our algorithms involve finding optimal flows in networks. We demonstrate how this approach can allow for additional constraints, such as lecturer lower quotas, to be handled flexibly

An Algorithm for Strong Stability in the Student-Project Allocation Problem With Ties

We study a variant of the Student-Project Allocation problem with lecturer preferences over Students where ties are allowed in the preference lists of students and lecturers (spa-st). We investigate the concept of strong stability in this context. Informally, a matching is strongly stable if there is no student and lecturer l such that if they decide to form a private arrangement outside of the matching via one of l’s proposed projects, then neither party would be worse off and at least one of them would strictly improve. We describe the first polynomial-time algorithm to find a strongly stable matching or report that no such matching exists, given an instance of spa-st. Our algorithm runs in O(m2) time, where m is the total length of the students’ preference lists

b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic