The Stable Roommates Problem with Globally Ranked Pairs

Abraham, David; Levavi, Ariel; Manlove, David F.; O'Malley, Gregg

doi:10.1080/15427951.2008.10129167

Cited by 28 publications

(20 citation statements)

References 29 publications

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“…The Stable Roommates problem with Globally Ranked Pairs (SR-GRP) [1,3] is a variant of the Stable Roommates problem involving ties and incomplete lists in which each pair of compatible agents {p, q} has a weight w({p, q}) assigned to their potential pairing, and the preference lists of each agent can be derived from these weights in the obvious manner: given two compatible pairs {p, q} and {p, r}, p prefers q to r if and only if w({p, q}) > w({p, r}). This problem can be restricted to give the Stable Marriage problem with Ties, Incomplete lists, and Globally Ranked Pairs (SMTI-GRP) by splitting the agents into two sets as per the Stable Marriage problem.…”

Section: Introductionmentioning

confidence: 99%

Mathematical models for stable matching problems with ties and incomplete lists

Delorme

García

Gondzio

et al. 2019

European Journal of Operational Research

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We present new integer linear programming (ILP) models for N P-hard optimisation problems in instances of the Stable Marriage problem with Ties and Incomplete lists (SMTI) and its many-to-one generalisation, the Hospitals / Residents problem with Ties (HRT). These models can be used to efficiently solve these optimisation problems when applied to (i) instances derived from real-world applications, and (ii) larger instances that are randomlygenerated. In the case of SMTI, we consider instances arising from the pairing of children with adoptive families, where preferences are obtained from a quality measure of each possible pairing of child to family. In this case we seek a maximum weight stable matching. We present new algorithms for preprocessing instances of SMTI with ties on both sides, as well as new ILP models. Algorithms based on existing state-of-the-art models only solve 6 of our 22 real-world instances within an hour per instance, and our new models incorporating dummy variables and constraint merging, together with preprocessing and a warm start, solve all 22 instances within a mean runtime of a minute. For HRT, we consider instances derived from the problem of assigning junior doctors to foundation posts in Scottish hospitals. Here we seek a maximum size stable matching. We show how to extend our models for SMTI to HRT and reduce the average running time for real-world HRT instances by two orders of magnitude. We also show that our models outperform by a wide margin all known state-of-the-art models on larger randomly-generated instances of SMTI and HRT.

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Section: Introductionmentioning

confidence: 99%

Mathematical models for stable matching problems with ties and incomplete lists

Delorme

García

Gondzio

et al. 2019

European Journal of Operational Research

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“…Note that there exist no edges in either G F or G C between the nodes belonging to the same set. From Equation (8), in S2H games a node u has an opportunity to obtain two-hop benefit only if it is a vertex of a triangle uvw such that (uv), (vw) ∈ G C and (uw) ∈ G F but the two-hop benefit it actually obtains depends on the contributions made by the nodes u, v, w. Thus in Figure 4, the only nodes that have an opportunity to obtain two-hop benefit are the nodes from set A and D 1 .…”

Section: Theorem 6 the Bounds On The Price Of Anarchy In Theorem 5 Armentioning

confidence: 99%

“…uv u over all the nodes, we get (n − 3k)q + 3k · k = nq + 3k(k − q). We already mentioned that the only nodes that have an opportunity to obtain two-hop benefit are the nodes in A and D 1 . Now recall that in Figure 4, for every node u ∈ A and w ∈ D 1 , we have (uw) ∈ G F .…”

Section: Theorem 6 the Bounds On The Price Of Anarchy In Theorem 5 Armentioning

confidence: 99%

“…More precisely, it becomes the "correlated" version of stable matching [1] for which a stable matching is known to exist for arbitrary graphs, and the 2-PoA (quality of stable matching compared to the optimum one) is bounded by 2 [4]. For k > 1, this becomes a many-to-many version of stable matching, where each node is allowed to match with k partners.…”

Section: Related Work and Impact Of Two-hop Benefitmentioning

confidence: 99%

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Friend of My Friend: Network Formation with Two-Hop Benefit

2014

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Abstract. How and why people form ties is a critical issue for understanding the fabric of social networks. In contrast to most existing work, we are interested in settings where agents are neither so myopic as to consider only the benefit they derive from their immediate neighbors, nor do they consider the effects on the entire network when forming connections. Instead, we consider games on networks where a node tries to maximize its utility taking into account the benefit it gets from the nodes it is directly connected to (called direct benefit), as well as the benefit it gets from the nodes it is acquainted with via a two-hop connection (called two-hop benefit). We call such games Two-Hop Games. The decision to consider only two hops stems from the observation that human agents rarely consider "contacts of a contact of a contact" (3-hop contacts) or further while forming their relationships. We consider several versions of Two-Hop games which are extensions of well-studied games. While the addition of two-hop benefit changes the properties of these games significantly, we prove that in many important cases good equilibrium solutions still exist, and bound the change in the price of anarchy due to two-hop benefit both theoretically and in simulation.

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“…Although in the roommates case (for general graphs) stable matchings might not exist, Diamantoudi et al [15] showed that there are always sequences of resolutions leading to a stable matching if it exists. Furthermore, the problem has been studied in constrained stable matching problems [19][20][21] and for preferences with special structure [4,22,27].…”

Section: Related Workmentioning

confidence: 99%

Maintaining Near-Popular Matchings

Bhattacharya

Hoefer

Huang

et al. 2015

Lecture Notes in Computer Science

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We study dynamic matching problems in graphs among agents with preferences. Agents and/or edges of the graph arrive and depart iteratively over time. The goal is to maintain matchings that are favorable to the agent population and stable over time. More formally, we strive to keep nearpopular matchings with a small unpopularity factor by making only a small amortized number of changes to the matching per round. Our main result is an algorithm to maintain matchings with unpopularity factor (∆ + k) by making an amortized number of O(∆ + ∆ 2 /k) changes per round, for any k > 0. Here ∆ denotes the maximum degree of any agent in any round. We complement this result by a variety of lower bounds indicating that matchings with smaller factor do not exist or cannot be maintained using our algorithm.As a byproduct, we obtain several additional results of independent interest. First, our algorithm implies existence of matchings with small unpopularity factors in graphs with bounded degree. Second, given any matching M and any value α ≥ 1, we provide an efficient algorithm to compute a matching M with unpopularity factor α over M if it exists. Finally, our results show the absence of voting paths in two-sided instances, even if we restrict to sequences of matchings with larger unpopularity factors (below ∆).

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The Stable Roommates Problem with Globally Ranked Pairs

Cited by 28 publications

References 29 publications

Mathematical models for stable matching problems with ties and incomplete lists

Mathematical models for stable matching problems with ties and incomplete lists

Friend of My Friend: Network Formation with Two-Hop Benefit

Maintaining Near-Popular Matchings

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