144,469 research outputs found
A Stable Marriage Requires Communication
The Gale-Shapley algorithm for the Stable Marriage Problem is known to take
steps to find a stable marriage in the worst case, but only
steps in the average case (with women and men). In
1976, Knuth asked whether the worst-case running time can be improved in a
model of computation that does not require sequential access to the whole
input. A partial negative answer was given by Ng and Hirschberg, who showed
that queries are required in a model that allows certain natural
random-access queries to the participants' preferences. A significantly more
general - albeit slightly weaker - lower bound follows from Segal's general
analysis of communication complexity, namely that Boolean queries
are required in order to find a stable marriage, regardless of the set of
allowed Boolean queries.
Using a reduction to the communication complexity of the disjointness
problem, we give a far simpler, yet significantly more powerful argument
showing that Boolean queries of any type are indeed required for
finding a stable - or even an approximately stable - marriage. Notably, unlike
Segal's lower bound, our lower bound generalizes also to (A) randomized
algorithms, (B) allowing arbitrary separate preprocessing of the women's
preferences profile and of the men's preferences profile, (C) several variants
of the basic problem, such as whether a given pair is married in every/some
stable marriage, and (D) determining whether a proposed marriage is stable or
far from stable. In order to analyze "approximately stable" marriages, we
introduce the notion of "distance to stability" and provide an efficient
algorithm for its computation
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 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 communication rounds for arbitrary preferences. We
also present a faster randomized variant which requires rounds.
This run-time can be improved to 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
A Note on Distributed Stable Matching
We consider the distributed complexity of the stable mar-riage problem. In this problem, the communication graph is undirected and bipartite, and each node ranks its neigh-bors. Given a matching of the nodes, a pair of unmatched nodes is called blocking if they prefer each other to their assigned match. A matching is called stable if it does not induce any blocking pair. In the distributed model, nodes exchange messages in each round over the communication links, until they find a stable matching. We show that if mes-sages may contain at most B bits each, then any distributed algorithm that solves the stable marriage problem requires Ω( n/B log n) communication rounds in the worst case, even for graphs of diameter O(log n), where n is the num-ber of nodes in the graph. Furthermore, the lower bound holds even if we allow the output to contain O( n) block-ing pairs. We also consider ε-stability, where a pair is called ε-blocking if they can improve the quality of their match by more than an ε fraction, for some 0 ≤ ε ≤ 1. Our lower bound extends to ε-stability where ε is arbitrarily close to 1/2. We also present a simple distributed algorithm for ε-stability whose time complexity is O(n/ε).
Matching Theory for Future Wireless Networks: Fundamentals and Applications
The emergence of novel wireless networking paradigms such as small cell and
cognitive radio networks has forever transformed the way in which wireless
systems are operated. In particular, the need for self-organizing solutions to
manage the scarce spectral resources has become a prevalent theme in many
emerging wireless systems. In this paper, the first comprehensive tutorial on
the use of matching theory, a Nobelprize winning framework, for resource
management in wireless networks is developed. To cater for the unique features
of emerging wireless networks, a novel, wireless-oriented classification of
matching theory is proposed. Then, the key solution concepts and algorithmic
implementations of this framework are exposed. Then, the developed concepts are
applied in three important wireless networking areas in order to demonstrate
the usefulness of this analytical tool. Results show how matching theory can
effectively improve the performance of resource allocation in all three
applications discussed
Stable Matchings for the Room-mates Problem
We show that, given two matchings for a room-mates problem of which say the second is stable, and given a non-empty subset of agents S if (a) no agent in S prefers the first matching to the second, and (b) no agent in S and his room-mate in S under the second matching prefer each other to their respective room-mates in the first matching, then no room-mate of an agent in S prefers the second matching to the first. This result is a strengthening of a result originally due to Knuth (1976). In a paper by Sasaki and Toda (1992) it is shown that if a marriage problem has more than one stable matchings, then given any one stable matching, it is possible to add agents and thereby obtain exactly one stable matching, whose restriction over the original set of agents, coincides with the given stable matching. We are able to extend this result here to the domain of room-mates problems. We also extend a result due to Roth and Sotomayor (1990) originally established for two-sided matching problems in the following manner: If in a room-mates problem, the number of agents increases, then given any stable matching for the old problem and any stable matching for the new one, there is at least one agent who is acceptable to this new agent who prefers the new matching to the old one and his room-mate under the new matching prefers the old matching to the new one. Sasaki and Toda (1992) shows that the solution correspondence which selects the set of all stable matchings, satisfies Pareto Optimality, Anonymity, Consistency and Converse Consistency on the domain of marriage problems. We show here that if a solution correspondence satisfying Consistency and Converse Consistency agrees with the solution correspondence comprising stable matchings for all room-mates problems involving four or fewer agents, then it must agree with the solution correspondence comprising stable matchings for all room-mates problems.Stable matchings, Room-mate problem
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