Three fast algorithms for four problems in stable marriage
SIAM Journal on Computing
An efficient algorithm for the “optimal” stable marriage
Journal of the ACM (JACM)
Every finite distributive lattice is a set of stable matchings for a small stable marriage instance
Journal of Combinatorial Theory Series A
The stable marriage problem: structure and algorithms
The stable marriage problem: structure and algorithms
Approximate counting, uniform generation and rapidly mixing Markov chains
Information and Computation
Algorithms for random generation and counting: a Markov chain approach
Algorithms for random generation and counting: a Markov chain approach
The Geometry of Fractional Stable Matchings and its Applications
Mathematics of Operations Research
Finding a Level Ideal of a Poset
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
Longest common subsequence as private search
Proceedings of the 8th ACM workshop on Privacy in the electronic society
The complexity of approximately counting stable matchings
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
An experimental comparison of single-sided preference matching algorithms
Journal of Experimental Algorithmics (JEA)
Center stable matchings and centers of cover graphs of distributive lattices
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Procedural fairness in stable marriage problems
The 10th International Conference on Autonomous Agents and Multiagent Systems - Volume 3
The complexity of approximately counting stable matchings
Theoretical Computer Science
The complexity of approximately counting stable roommate assignments
Journal of Computer and System Sciences
Mathematics of Operations Research
Time hierarchies for sampling distributions
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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We study the behavior of random walks along the edges of the stable marriage lattice for various restricted families of allowable preference sets. In the "k-attribute model," each man is valued in each of k attributes, and each woman's ranking of the men is determined by a linear function, representing her relative ranking of those attributes; men's rankings of the women are determined similarly. We show that sampling with a random walk on the marriage lattice can take exponential time, even when k = 2. Moreover, we show that the marriage lattices arising in the k-attribute model are more restrictive than in the general setting; previously such a restriction had only been shown for the sets of preference lists. The second model we consider is the "k-range model," where each person lies in a position in [i, i + k - 1], for some i, on every preference list of the opposite sex. When k = 1 there is a unique stable marriage. When k = 2 there already can be an exponential number of stable marriages, but we show that a random walk on the stable marriage lattice always converges quickly to equilibrium. However, when k ≥ 5, there are preference sets such that the random walk on the lattice will require exponential time to converge. Lastly, we show that in the extreme case where each gender's rankings of the other are restricted to one of just a constant k possible preference lists, there are still instances for which the Markov chain mixes exponentially slowly, even when k = 4. This oversimplification of the general model helps elucidate why Markov chains based on spouse-swapping are not good approaches to sampling, even in specialized scenarios.