3,706 research outputs found
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference
We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF
inference problems. The core of our method is a very efficient bounding
procedure, which combines scalable semidefinite programming (SDP) and a
cutting-plane method for seeking violated constraints. In order to further
speed up the computation, several strategies have been exploited, including
model reduction, warm start and removal of inactive constraints.
We analyze the performance of the proposed method under different settings,
and demonstrate that our method either outperforms or performs on par with
state-of-the-art approaches. Especially when the connectivities are dense or
when the relative magnitudes of the unary costs are low, we achieve the best
reported results. Experiments show that the proposed algorithm achieves better
approximation than the state-of-the-art methods within a variety of time
budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page
Discrete Convex Functions on Graphs and Their Algorithmic Applications
The present article is an exposition of a theory of discrete convex functions
on certain graph structures, developed by the author in recent years. This
theory is a spin-off of discrete convex analysis by Murota, and is motivated by
combinatorial dualities in multiflow problems and the complexity classification
of facility location problems on graphs. We outline the theory and algorithmic
applications in combinatorial optimization problems
Precise Coulomb wave functions for a wide range of complex l, eta and z
A new algorithm to calculate Coulomb wave functions with all of its arguments
complex is proposed. For that purpose, standard methods such as continued
fractions and power/asymptotic series are combined with direct integrations of
the Schrodinger equation in order to provide very stable calculations, even for
large values of |eta| or |Im(l)|. Moreover, a simple analytic continuation for
Re(z) < 0 is introduced, so that this zone of the complex z-plane does not pose
any problem. This code is particularly well suited for low-energy calculations
and the calculation of resonances with extremely small widths. Numerical
instabilities appear, however, when both |eta| and |Im(l)| are large and
|Re(l)| comparable or smaller than |Im(l)|
Symbolic integration of a product of two spherical bessel functions with an additional exponential and polynomial factor
We present a mathematica package that performs the symbolic calculation of
integrals of the form \int^{\infty}_0 e^{-x/u} x^n j_{\nu} (x) j_{\mu} (x) dx
where and denote spherical Bessel functions of
integer orders, with and . With the real parameter
and the integer , convergence of the integral requires that . The package provides analytical result for the integral in its most
simplified form. The novel symbolic method employed enables the calculation of
a large number of integrals of the above form in a fraction of the time
required for conventional numerical and Mathematica based brute-force methods.
We test the accuracy of such analytical expressions by comparing the results
with their numerical counterparts.Comment: 17 pages; updated references for the introductio
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