390 research outputs found
Below All Subsets for Some Permutational Counting Problems
We show that the two problems of computing the permanent of an
matrix of -bit integers and counting the number of
Hamiltonian cycles in a directed -vertex multigraph with
edges can be reduced to relatively
few smaller instances of themselves. In effect we derive the first
deterministic algorithms for these two problems that run in time in
the worst case. Classic time algorithms for the two
problems have been known since the early 1960's. Our algorithms run in
time.Comment: Corrected several technical errors, added comment on how to use the
algorithm for ATSP, and changed title slightly to a more adequate on
Below All Subsets for Some Permutational Counting Problems
We show that the two problems of computing the permanent of an n*n matrix of poly(n)-bit integers and counting the number of Hamiltonian cycles in a directed n-vertex multigraph with exp(poly(n)) edges can be reduced to relatively few smaller instances of themselves. In effect we derive the first deterministic algorithms for these two problems that run in o(2^n) time in the worst case. Classic poly(n)2^n time algorithms for the two problems have been known since the early 1960\u27s.
Our algorithms run in 2^{n-Omega(sqrt{n/log(n)})} time
Potts q-color field theory and scaling random cluster model
We study structural properties of the q-color Potts field theory which, for
real values of q, describes the scaling limit of the random cluster model. We
show that the number of independent n-point Potts spin correlators coincides
with that of independent n-point cluster connectivities and is given by
generalized Bell numbers. Only a subset of these spin correlators enters the
determination of the Potts magnetic properties for q integer. The structure of
the operator product expansion of the spin fields for generic q is also
identified. For the two-dimensional case, we analyze the duality relation
between spin and kink field correlators, both for the bulk and boundary cases,
obtaining in particular a sum rule for the kink-kink elastic scattering
amplitudes.Comment: 27 pages; 6 figures. Published version, some comments and references
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Patterson-Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for Outer space
We quantitatively relate the Patterson-Sullivant currents and generic
stretching factors for free group automorphisms to the asymmetric Lipschitz
metric on Outer space and to Guirardel's intersection number.Comment: some minor updates and revisions; 18 pages, no figures; to appear in
the Pacific Journal of Mathematic
The relativistic two-body potentials of constraint theory from summation of Feynman diagrams
The relativistic two-body potentials of constraint theory for systems
composed of two spin-0 or two spin-1/2 particles are calculated, in
perturbation theory, by means of the Lippmann-Schwinger type equation that
relates them to the scattering amplitude. The cases of scalar and vector
interactions with massless photons are considered. The two-photon exchange
contributions, calculated with covariant propagators,are globally free of
spurious infra-red singularities and produce at leading order O(\alpha^4)
effects that can be represented in three-dimensional x-space by local
potentials proportional to (\alpha/r)^2. Leading contributions of n-photon
exchange diagrams produce terms proportional to (\alpha/r)^n. The series of
leading contributions are summed. The resulting potentials are functions, in
the c.m. frame, of r and of the total energy. Their forms are compatible with
Todorov's minimal substitution rules proposed in the quasipotential approach.Comment: 60 pages, Latex, with four pages of figures included at the end of
the article in a Latex file calling the FEYNMAN macropackag
Directed Hamiltonicity and Out-Branchings via Generalized Laplacians
We are motivated by a tantalizing open question in exact algorithms: can we
detect whether an -vertex directed graph has a Hamiltonian cycle in time
significantly less than ? We present new randomized algorithms that
improve upon several previous works:
1. We show that for any constant and prime we can count the
Hamiltonian cycles modulo in
expected time less than for a constant that depends only on and
. Such an algorithm was previously known only for the case of counting
modulo two [Bj\"orklund and Husfeldt, FOCS 2013].
2. We show that we can detect a Hamiltonian cycle in
time and polynomial space, where is the size of the maximum
independent set in . In particular, this yields an time
algorithm for bipartite directed graphs, which is faster than the
exponential-space algorithm in [Cygan et al., STOC 2013].
Our algorithms are based on the algebraic combinatorics of "incidence
assignments" that we can capture through evaluation of determinants of
Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed
graphs. In addition to the novel algorithms for directed Hamiltonicity, we use
the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting
out-branchings. Specifically, we give an -time randomized algorithm
for detecting out-branchings with at least internal vertices, improving
upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015].
We also present an algebraic algorithm for the directed -Leaf problem, based
on a non-standard monomial detection problem
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