55 research outputs found
A permanent formula for the Jones polynomial
The permanent of a square matrix is defined in a way similar to the
determinant, but without using signs. The exact computation of the permanent is
hard, but there are Monte-Carlo algorithms that can estimate general
permanents. Given a planar diagram of a link L with crossings, we define a
7n by 7n matrix whose permanent equals to the Jones polynomial of L. This
result accompanied with recent work of Freedman, Kitaev, Larson and Wang
provides a Monte-Carlo algorithm to any decision problem belonging to the class
BQP, i.e. such that it can be computed with bounded error in polynomial time
using quantum resources.Comment: To appear in Advances in Applied Mathematic
Integration and Optimization of Multivariate Polynomials by Restriction onto a Random Subspace
We consider the problem of efficient integration of an n-variate polynomial
with respect to the Gaussian measure in R^n and related problems of complex
integration and optimization of a polynomial on the unit sphere. We identify a
class of n-variate polynomials f for which the integral of any positive integer
power f^p over the whole space is well-approximated by a properly scaled
integral over a random subspace of dimension O(log n). Consequently, the
maximum of f on the unit sphere is well-approximated by a properly scaled
maximum on the unit sphere in a random subspace of dimension O(log n). We
discuss connections with problems of combinatorial counting and applications to
efficient approximation of a hafnian of a positive matrix.Comment: 15 page
Counting Popular Matchings in House Allocation Problems
We study the problem of counting the number of popular matchings in a given
instance. A popular matching instance consists of agents A and houses H, where
each agent ranks a subset of houses according to their preferences. A matching
is an assignment of agents to houses. A matching M is more popular than
matching M' if the number of agents that prefer M to M' is more than the number
of people that prefer M' to M. A matching M is called popular if there exists
no matching more popular than M. McDermid and Irving gave a poly-time algorithm
for counting the number of popular matchings when the preference lists are
strictly ordered.
We first consider the case of ties in preference lists. Nasre proved that the
problem of counting the number of popular matching is #P-hard when there are
ties. We give an FPRAS for this problem.
We then consider the popular matching problem where preference lists are
strictly ordered but each house has a capacity associated with it. We give a
switching graph characterization of popular matchings in this case. Such
characterizations were studied earlier for the case of strictly ordered
preference lists (McDermid and Irving) and for preference lists with ties
(Nasre). We use our characterization to prove that counting popular matchings
in capacitated case is #P-hard
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