31 research outputs found
A simple polynomial time algorithm to approximate the permanent within a simply exponential factor
We present a simple randomized polynomial time algorithm to approximate the
mixed discriminant of positive semidefinite matrices within a
factor . Consequently, the algorithm allows us to approximate in
randomized polynomial time the permanent of a given non-negative
matrix within a factor . When applied to approximating the permanent,
the algorithm turns out to be a simple modification of the well-known
Godsil-Gutman estimator
On the expressive power of planar perfect matching and permanents of bounded treewidth matrices
Valiant introduced some 25 years ago an algebraic model of computation along
with the complexity classes VP and VNP, which can be viewed as analogues of the
classical classes P and NP. They are defined using non-uniform sequences of
arithmetic circuits and provides a framework to study the complexity for
sequences of polynomials. Prominent examples of difficult (that is,
VNP-complete) problems in this model includes the permanent and hamiltonian
polynomials. While the permanent and hamiltonian polynomials in general are
difficult to evaluate, there have been research on which special cases of these
polynomials admits efficient evaluation. For instance, Barvinok has shown that
if the underlying matrix has bounded rank, both the permanent and the
hamiltonian polynomials can be evaluated in polynomial time, and thus are in
VP. Courcelle, Makowsky and Rotics have shown that for matrices of bounded
treewidth several difficult problems (including evaluating the permanent and
hamiltonian polynomials) can be solved efficiently. An earlier result of this
flavour is Kasteleyn's theorem which states that the sum of weights of perfect
matchings of a planar graph can be computed in polynomial time, and thus is in
VP also. For general graphs this problem is VNP-complete. In this paper we
investigate the expressive power of the above results. We show that the
permanent and hamiltonian polynomials for matrices of bounded treewidth both
are equivalent to arithmetic formulas. Also, arithmetic weakly skew circuits
are shown to be equivalent to the sum of weights of perfect matchings of planar
graphs.Comment: 14 page
On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries
We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and
prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We
prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution
to a convex optimization problem. We show that if Sigma(R, C) is sufficiently
large, then a random matrix D in Sigma(R, C) sampled from the uniform
probability measure in Sigma(R,C) with high probability is close to a
particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among
all matrices with row sums R, column sums C and entries between 0 and 1.
Similar results are obtained for 0-1 matrices with prescribed row and column
sums and assigned zeros in some positions.Comment: 26 pages, proofs simplified, results strengthene
Geometric versions of the 3-dimensional assignment problem under general norms
We discuss the computational complexity of special cases of the 3-dimensional
(axial) assignment problem where the elements are points in a Cartesian space
and where the cost coefficients are the perimeters of the corresponding
triangles measured according to a certain norm. (All our results also carry
over to the corresponding special cases of the 3-dimensional matching problem.)
The minimization version is NP-hard for every norm, even if the underlying
Cartesian space is 2-dimensional. The maximization version is polynomially
solvable, if the dimension of the Cartesian space is fixed and if the
considered norm has a polyhedral unit ball. If the dimension of the Cartesian
space is part of the input, the maximization version is NP-hard for every
norm; in particular the problem is NP-hard for the Manhattan norm and the
Maximum norm which both have polyhedral unit balls.Comment: 21 pages, 9 figure
Maximum Scatter TSP in Doubling Metrics
We study the problem of finding a tour of points in which every edge is
long. More precisely, we wish to find a tour that visits every point exactly
once, maximizing the length of the shortest edge in the tour. The problem is
known as Maximum Scatter TSP, and was introduced by Arkin et al. (SODA 1997),
motivated by applications in manufacturing and medical imaging. Arkin et al.
gave a -approximation for the metric version of the problem and showed
that this is the best possible ratio achievable in polynomial time (assuming ). Arkin et al. raised the question of whether a better approximation
ratio can be obtained in the Euclidean plane.
We answer this question in the affirmative in a more general setting, by
giving a -approximation algorithm for -dimensional doubling
metrics, with running time , where . As a corollary we obtain (i) an
efficient polynomial-time approximation scheme (EPTAS) for all constant
dimensions , (ii) a polynomial-time approximation scheme (PTAS) for
dimension , for a sufficiently large constant , and (iii)
a PTAS for constant and . Furthermore, we
show the dependence on in our approximation scheme to be essentially
optimal, unless Satisfiability can be solved in subexponential time