We investigate the complexity of approximately counting stable matchings in the k-attribute model, where the preference lists are determined by dot products of "preference vectors" with "attribute vectors", or by Euclidean distances between "preference points" and "attribute points". Irving and Leather [16] proved that counting the number of stable matchings in the general case is #P -complete. Counting the number of stable matchings is reducible to counting the number of downsets in a (related) partial order [16] and is interreducible, in an approximation-preserving sense, to a class of problems that includes counting the number of independent sets in a bipartite graph (#BIS) [7]. It is conjectured that no FPRAS exists for this class of problems. We show this approximation-preserving interreducibilty remains even in the restricted k-attribute setting when k ≥ 3 (dot products) or k ≥ 2 (Euclidean distances). Finally, we show it is easy to count the number of stable matchings in the 1-attribute dot-product setting.considered unstable and the man-woman pair (M, w) is called a blocking pair. (M and w would prefer to drop their current partners and pair up with each other.) If a matching has no blocking pairs, then we call it a stable matching. In 1962, Gale and Shapley proved that every stable matching instance has a stable matching, and described an O(n 2 ) algorithm for finding one [8].The stable matching problem has many variants, where ties in the preference lists could be allowed, where people might have partial preference lists (i.e. someone might prefer to remain single rather than be paired with certain members of the opposite sex), generalizations to men/women/pets, universities and applicants, students and projects, etc. Some of these generalizations have also been well-studied and, indeed, algorithms for finding stable matchings are used for assigning residents to hospitals in Scotland, Canada, and the USA [4,20,22].In this paper, we concentrate solely on the classical problem, so the term "matching instance" will refer to one where the number of men is equal to the number of women, and each man or women has their own full totally-ordered (i.e. no ties allowed) preference list for the opposite sex.Irving and Leather [16] demonstrated that counting the number of stable matchings for a given instance is #P -complete. This completeness result relies on the connection between stable marriages and downsets in a related partial order (explained in more detail in Section 3), as counting the number of downsets in a partial order is another classical #P -complete problem [21].Knowing that exactly counting stable matchings is difficult (under standard complexitytheoretic assumptions), one might turn to methods for approximately counting this number. In particular, we would like to find a fully-polynomial randomized approximation scheme (an FPRAS) for this task, i.e. an algorithm that provides an arbitrarily close approximation in time polynomial in the input size and the desired error -see Section 2 for a formal de...

We present a linear expected time algorithm for finding maximum cardinality matchings in sparse random graphs. This is optimal and improves on previous results by a logarithmic factor.

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We give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random ET of the given generalized series-parallel graph. Note that the class of generalized seriesparallel graphs includes all outerplanar graphs. We can perform the counting in time O(m∆ 3 ), where ∆ is the maximum degree of the graph with m edges. We use O(m∆ 2 log ∆) bits to store intermediate values during our computations. To date, these are the first known polynomial-time algorithms to count or sample ETs of any class of graphs; there are no other known polynomial-time algorithms to even approximately count or sample ETs of any other class of graphs. The problem of counting ETs is known to be ♯P -complete for general graphs (Brightwell and Winkler, 2005 [3]) and also for planar graphs (Creed, 2009 [4]).

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