298 research outputs found
Clifford Algebras Meet Tree Decompositions
We introduce the Non-commutative Subset Convolution - a convolution of functions useful when working with determinant-based algorithms. In order to compute it efficiently, we take advantage of Clifford algebras, a generalization of quaternions used mainly in the quantum field theory.
We apply this tool to speed up algorithms counting subgraphs parameterized by the treewidth of a graph. We present an O^*((2^omega + 1)^{tw})-time algorithm for counting Steiner trees and an O^*((2^omega + 2)^{tw})-time algorithm for counting Hamiltonian cycles, both of which improve the previously known upper bounds. The result for Steiner Tree also translates into a deterministic algorithm for Feedback Vertex Set. All of these constitute the best known running times of deterministic algorithms for decision versions of these problems and they match the best obtained running times for pathwidth parameterization under assumption omega = 2
Homology of Distributive Lattices
We outline the theory of sets with distributive operations: multishelves and
multispindles, with examples provided by semi-lattices, lattices and skew
lattices. For every such a structure we define multi-term distributive homology
and show some of its properties. The main result is a complete formula for the
homology of a finite distributive lattice. We also indicate the answer for
unital spindles and conjecture the general formula for semi-lattices and some
skew lattices. Then we propose a generalization of a lattice as a set with a
number of idempotent operations satisfying the absorption law.Comment: 30 pages, 3 tables, 3 figure
Fast Algorithms for Join Operations on Tree Decompositions
Treewidth is a measure of how tree-like a graph is. It has many important
algorithmic applications because many NP-hard problems on general graphs become
tractable when restricted to graphs of bounded treewidth. Algorithms for
problems on graphs of bounded treewidth mostly are dynamic programming
algorithms using the structure of a tree decomposition of the graph. The
bottleneck in the worst-case run time of these algorithms often is the
computations for the so called join nodes in the associated nice tree
decomposition.
In this paper, we review two different approaches that have appeared in the
literature about computations for the join nodes: one using fast zeta and
M\"obius transforms and one using fast Fourier transforms. We combine these
approaches to obtain new, faster algorithms for a broad class of vertex subset
problems known as the [\sigma,\rho]-domination problems. Our main result is
that we show how to solve [\sigma,\rho]-domination problems in arithmetic operations. Here, t is the treewidth, s is the
(fixed) number of states required to represent partial solutions of the
specific [\sigma,\rho]-domination problem, and n is the number of vertices in
the graph. This reduces the polynomial factors involved compared to the
previously best time bound (van Rooij, Bodlaender, Rossmanith, ESA 2009) of arithmetic operations. In particular, this removes
the dependence of the degree of the polynomial on the fixed number of
states~.Comment: An earlier version appeared in "Treewidth, Kernels, and Algorithms.
Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday"
LNCS 1216
An overview of knot Floer homology
Knot Floer homology is an invariant for knots discovered by the authors and,
independently, Jacob Rasmussen. The discovery of this invariant grew naturally
out of studying how a certain three-manifold invariant, Heegaard Floer
homology, changes as the three-manifold undergoes Dehn surgery along a knot.
Since its original definition, thanks to the contributions of many researchers,
knot Floer homology has emerged as a useful tool for studying knots in its own
right. We give here a few selected highlights of this theory, and then move on
to some new algebraic developments in the computation of knot Floer homology
Extensor-coding
We devise an algorithm that approximately computes the number of paths of
length in a given directed graph with vertices up to a multiplicative
error of . Our algorithm runs in time . The algorithm is based on associating with
each vertex an element in the exterior (or, Grassmann) algebra, called an
extensor, and then performing computations in this algebra. This connection to
exterior algebra generalizes a number of previous approaches for the longest
path problem and is of independent conceptual interest. Using this approach, we
also obtain a deterministic time algorithm
to find a -path in a given directed graph that is promised to have few of
them. Our results and techniques generalize to the subgraph isomorphism problem
when the subgraphs we are looking for have bounded pathwidth. Finally, we also
obtain a randomized algorithm to detect -multilinear terms in a multivariate
polynomial given as a general algebraic circuit. To the best of our knowledge,
this was previously only known for algebraic circuits not involving negative
constants.Comment: To appear at STOC 2018: Symposium on Theory of Computing, June 23-27,
2018, Los Angeles, CA, US
Singular foliations for M-theory compactification
We use the theory of singular foliations to study
compactifications of eleven-dimensional supergravity on eight-manifolds
down to spaces, allowing for the possibility that the internal
part of the supersymmetry generator is chiral on some locus
which does not coincide with . We show that the complement must be a dense open subset of and that admits a singular foliation
endowed with a longitudinal structure and defined by a
closed one-form , whose geometry is determined by the
supersymmetry conditions. The singular leaves are those leaves which meet
. When is a Morse form, the chiral locus is a
finite set of points, consisting of isolated zero-dimensional leaves and of
conical singularities of seven-dimensional leaves. In that case, we describe
the topology of using results from Novikov theory. We also
show how this description fits in with previous formulas which were extracted
by exploiting the structures which exist on the
complement of .Comment: 66 pages, 6 tables, 4 figures; v2: added discussion of limit
$kappa=0
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