34,852 research outputs found
Efficient algorithms for generalized Stable Marriage and Roommates problems
We consider a generalization of the Stable Roommates problem (SR), in which preference lists may be partially ordered and forbidden pairs may be present, denoted by SRPF. This includes, as a special case, a corresponding generalization of the classical Stable Marriage problem (SM), denoted by SMPF. By extending previous work of Feder, we give a two-step reduction from SRPF to 2-SAT. This has many consequences, including fast algorithms for a range of problems associated with finding "optimal" stable matchings and listing all solutions, given variants of SR and SM. For example, given an SMPF instance I, we show that there exists an O(m) "succinct" certificate for the unsolvability of I, an O(m) algorithm for finding all the super-stable pairs in I, an O(m+kn) algorithm for listing all the super-stable matchings in I, an O(m<sup>1.5</sup>) algorithm for finding an egalitarian super-stable matching in I, and an O(m) algorithm for finding a minimum regret super-stable matching in I, where n is the number of men, m is the total length of the preference lists, and k is the number of super-stable matchings in I. Analogous results apply in the case of SRPF
The Complexity of Approximately Counting Stable Roommate Assignments
We investigate the complexity of approximately counting stable roommate
assignments in two models: (i) the -attribute model, in which the preference
lists are determined by dot products of "preference vectors" with "attribute
vectors" and (ii) the -Euclidean model, in which the preference lists are
determined by the closeness of the "positions" of the people to their
"preferred positions". Exactly counting the number of assignments is
#P-complete, since Irving and Leather demonstrated #P-completeness for the
special case of the stable marriage problem. We show that counting the number
of stable roommate assignments in the -attribute model () and the
3-Euclidean model() is interreducible, in an approximation-preserving
sense, with counting independent sets (of all sizes) (#IS) in a graph, or
counting the number of satisfying assignments of a Boolean formula (#SAT). This
means that there can be no FPRAS for any of these problems unless NP=RP. As a
consequence, we infer that there is no FPRAS for counting stable roommate
assignments (#SR) unless NP=RP. Utilizing previous results by the authors, we
give an approximation-preserving reduction from counting the number of
independent sets in a bipartite graph (#BIS) to counting the number of stable
roommate assignments both in the 3-attribute model and in the 2-Euclidean
model. #BIS is complete with respect to approximation-preserving reductions in
the logically-defined complexity class #RH\Pi_1. Hence, our result shows that
an FPRAS for counting stable roommate assignments in the 3-attribute model
would give an FPRAS for all of #RH\Pi_1. We also show that the 1-attribute
stable roommate problem always has either one or two stable roommate
assignments, so the number of assignments can be determined exactly in
polynomial time
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An O(n3 [square root of] log n) algorithm for the optimal stable marriage problem
We give an O(n^3 √logn) time algorithm for the optimal stable marriage problem. This algorithm finds a stable marriage that minimizes an objective function defined over all stable marriages in a given problem instance.Irving, Leather, and Gusfield have previously provided a solution to this problem that runs in O(n^4) time [ILG87]. In addition, Feder has claimed that an O(n^3 log n) time algorithm exists [F89]. Our result is an asymptotic improvement over both cases.As part of our solution, we solve a special blue-red matching problem, and illustrate a technique for simulating Hopcroft and Karp's maximum-matching algorithm [HK73] on the transitive closure of a graph
On covering expander graphs by Hamilton cycles
The problem of packing Hamilton cycles in random and pseudorandom graphs has
been studied extensively. In this paper, we look at the dual question of
covering all edges of a graph by Hamilton cycles and prove that if a graph with
maximum degree satisfies some basic expansion properties and contains
a family of edge disjoint Hamilton cycles, then there also
exists a covering of its edges by Hamilton cycles. This
implies that for every and every there exists
a covering of all edges of by Hamilton cycles
asymptotically almost surely, which is nearly optimal.Comment: 19 pages. arXiv admin note: some text overlap with arXiv:some
math/061275
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