206 research outputs found
Learned navigation in unknown terrains: A retraction method
The problem of learned navigation of a circular robot R, of radius delta (is greater than or equal to 0), through a terrain whose model is not a-priori known is considered. Two-dimensional finite-sized terrains populated by an unknown (but, finite) number of simple polygonal obstacles are also considered. The number and locations of the vertices of each obstacle are unknown to R. R is equipped with a sensor system that detects all vertices and edges that are visible from its present location. In this context two problems are covered. In the visit problem, the robot is required to visit a sequence of destination points, and in the terrain model acquisition problem, the robot is required to acquire the complete model of the terrain. An algorithmic framework is presented for solving these two problems using a retraction of the freespace onto the Voronoi diagram of the terrain. Algorithms are then presented to solve the visit problem and the terrain model acquisition problem
Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function
We review various aspects of cluster algebras and the ways in which they
appear in the study of loop-level amplitudes in planar
supersymmetric Yang-Mills theory. In particular, we highlight the different
forms of cluster-algebraic structure that appear in this theory's two-loop MHV
amplitudes---considered as functions, symbols, and at the level of their Lie
cobracket---and recount how the `nonclassical' part of these amplitudes can be
decomposed into specific functions evaluated on the or subalgebras
of Gr. We then extend this line of inquiry by searching for other
subalgebras over which these amplitudes can be decomposed. We focus on the case
of seven-particle kinematics, where we show that the nonclassical part of the
two-loop MHV amplitude is also constructible out of functions evaluated on the
and subalgebras of Gr, and that these decompositions are
themselves decomposable in terms of the same function. These nested
decompositions take an especially canonical form, which is dictated in each
case by constraints arising from the automorphism group of the parent algebra.Comment: 68 pages, 7 figures, 6 table
Block-and-hole graphs: Constructibility and -sparsity
We show that minimally 3-rigid block-and-hole graphs, with one block or one
hole, are characterised as those which are constructible from by vertex
splitting, and also, as those having associated looped face graphs which are
-tight. This latter property can be verified in polynomial time by a
form of pebble game algorithm. We also indicate connections to the rigidity
properties of polyhedral surfaces known as origami and to graph rigidity in
for .Comment: 17 page
Amplitudes at Infinity
We investigate the asymptotically large loop-momentum behavior of multi-loop
amplitudes in maximally supersymmetric quantum field theories in four
dimensions. We check residue-theorem identities among color-dressed leading
singularities in supersymmetric Yang-Mills theory to
demonstrate the absence of poles at infinity of all MHV amplitudes through
three loops. Considering the same test for supergravity leads
us to discover that this theory does support non-vanishing residues at infinity
starting at two loops, and the degree of these poles grow arbitrarily with
multiplicity. This causes a tension between simultaneously manifesting
ultraviolet finiteness---which would be automatic in a representation obtained
by color-kinematic duality---and gauge invariance---which would follow from
unitarity-based methods.Comment: 4+1+1 pages; 15 figures; details provided in ancillary Mathematica
file
Explicit linear kernels via dynamic programming
Several algorithmic meta-theorems on kernelization have appeared in the last
years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of
bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding
a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed
topological minor. Typically, these results guarantee the existence of linear
or polynomial kernels on sparse graph classes for problems satisfying some
generic conditions but, mainly due to their generality, it is not clear how to
derive from them constructive kernels with explicit constants. In this paper we
make a step toward a fully constructive meta-kernelization theory on sparse
graphs. Our approach is based on a more explicit protrusion replacement
machinery that, instead of expressibility in CMSO logic, uses dynamic
programming, which allows us to find an explicit upper bound on the size of the
derived kernels. We demonstrate the usefulness of our techniques by providing
the first explicit linear kernels for -Dominating Set and -Scattered Set
on apex-minor-free graphs, and for Planar-\mathcal{F}-Deletion on graphs
excluding a fixed (topological) minor in the case where all the graphs in
\mathcal{F} are connected.Comment: 32 page
A Dichotomy Theorem for the Approximate Counting of Complex-Weighted Bounded-Degree Boolean CSPs
We determine the computational complexity of approximately counting the total
weight of variable assignments for every complex-weighted Boolean constraint
satisfaction problem (or CSP) with any number of additional unary (i.e., arity
1) constraints, particularly, when degrees of input instances are bounded from
above by a fixed constant. All degree-1 counting CSPs are obviously solvable in
polynomial time. When the instance's degree is more than two, we present a
dichotomy theorem that classifies all counting CSPs admitting free unary
constraints into exactly two categories. This classification theorem extends,
to complex-weighted problems, an earlier result on the approximation complexity
of unweighted counting Boolean CSPs of bounded degree. The framework of the
proof of our theorem is based on a theory of signature developed from Valiant's
holographic algorithms that can efficiently solve seemingly intractable
counting CSPs. Despite the use of arbitrary complex weight, our proof of the
classification theorem is rather elementary and intuitive due to an extensive
use of a novel notion of limited T-constructibility. For the remaining degree-2
problems, in contrast, they are as hard to approximate as Holant problems,
which are a generalization of counting CSPs.Comment: A4, 10pt, 20 pages. This revised version improves its preliminary
version published under a slightly different title in the Proceedings of the
4th International Conference on Combinatorial Optimization and Applications
(COCOA 2010), Lecture Notes in Computer Science, Springer, Vol.6508 (Part I),
pp.285--299, Kailua-Kona, Hawaii, USA, December 18--20, 201
Lagrangian Insertion in the Light-Like Limit and the Super-Correlators/Super-Amplitudes Duality
In these notes we describe how to formulate the Lagrangian insertion
technique in a way that mimics generalized unitarity. We introduce a notion of
cuts in real space and show that the cuts of the correlators in the
super-correlators/super-amplitudes duality correspond to generalized unitarity
cuts of the equivalent amplitudes. The cuts consist of correlation functions of
operators in the chiral part of the stress-tensor multiplet as well as other
half-BPS operators. We will also discuss the application of the method to other
correlators as well as non-planar contributions.Comment: 25 pages, 9 figures, added references; some sections have been
expanded, a substantial part have been rearranged; v4 examples added to make
it easier to rea
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