11 research outputs found
Exact counting of Euler Tours for Graphs of Bounded Treewidth
In this paper we give a simple polynomial-time algorithm to exactly count the
number of Euler Tours (ETs) of any Eulerian graph of bounded treewidth. The
problems of counting ETs are known to be #P-complete for general graphs
(Brightwell and Winkler, (Brightwell and Winkler, 2005). To date, no
polynomial-time algorithm for counting Euler tours of any class of graphs is
known except for the very special case of series-parallel graphs (which have
treewidth 2).Comment: 16 pages, draf
The Complexity of Approximately Counting Stable Matchings
We investigate the complexity of approximately counting stable matchings in
the -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 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 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). It is conjectured that no FPRAS exists for this class of problems. We
show this approximation-preserving interreducibilty remains even in the
restricted -attribute setting when (dot products) or
(Euclidean distances). Finally, we show it is easy to count the number of
stable matchings in the 1-attribute dot-product setting.Comment: Fixed typos, small revisions for clarification, et
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
Finding a Maximum Matching in a Sparse Random Graph in O(n) Expected Time
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.
Alan Frieze ∗
An n-lift of a digraph K, is a digraph with vertex set V (K) × [n] and for each directed edge (i,j) ∈ E(K) there is a perfect matching between fibers {i} × [n] and {j} × [n], with edges directed from fiber i to fiber j. If these matchings are chosen independently and uniformly at random then we say that we have a random n-lift. We show that if h is sufficiently large then a random n-lift of the complete digraph �Kh is hamiltonian whp.