37,653 research outputs found
Connect Four and Graph Decomposition
We introduce the standard decomposition, a way of decomposing a labeled graph
into a sum of certain labeled subgraphs. We motivate this graph-theoretic
concept by relating it to Connect Four decompositions of standard sets. We
prove that all standard decompositions can be generated in polynomial time,
which implies that all Connect Four decompositions can be generated in
polynomial time
On the Enumeration of all Minimal Triangulations
We present an algorithm that enumerates all the minimal triangulations of a
graph in incremental polynomial time. Consequently, we get an algorithm for
enumerating all the proper tree decompositions, in incremental polynomial time,
where "proper" means that the tree decomposition cannot be improved by removing
or splitting a bag
Hypergraph Acyclicity and Propositional Model Counting
We show that the propositional model counting problem #SAT for CNF- formulas
with hypergraphs that allow a disjoint branches decomposition can be solved in
polynomial time. We show that this class of hypergraphs is incomparable to
hypergraphs of bounded incidence cliquewidth which were the biggest class of
hypergraphs for which #SAT was known to be solvable in polynomial time so far.
Furthermore, we present a polynomial time algorithm that computes a disjoint
branches decomposition of a given hypergraph if it exists and rejects
otherwise. Finally, we show that some slight extensions of the class of
hypergraphs with disjoint branches decompositions lead to intractable #SAT,
leaving open how to generalize the counting result of this paper
On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials
In this paper, we are interested in developing polynomial decomposition
techniques to reformulate real valued multivariate polynomials into
difference-of-sums-of-squares (namely, D-SOS) and
difference-of-convex-sums-of-squares (namely, DC-SOS). Firstly, we prove that
the set of D-SOS and DC-SOS polynomials are vector spaces and equivalent to the
set of real valued polynomials. Moreover, the problem of finding D-SOS and
DC-SOS decompositions are equivalent to semidefinite programs (SDP) which can
be solved to any desired precision in polynomial time. Some important algebraic
properties and the relationships among the set of sums-of-squares (SOS)
polynomials, positive semidefinite (PSD) polynomials, convex-sums-of-squares
(CSOS) polynomials, SOS-convex polynomials, D-SOS and DC-SOS polynomials are
discussed. Secondly, we focus on establishing several practical algorithms for
constructing D-SOS and DC-SOS decompositions for any polynomial without solving
SDP. Using DC-SOS decomposition, we can reformulate polynomial optimization
problems in the realm of difference-of-convex (DC) programming, which can be
handled by efficient DC programming approaches. Some examples illustrate how to
use our methods for constructing D-SOS and DC-SOS decompositions. Numerical
performance of D-SOS and DC-SOS decomposition algorithms and their parallelized
methods are tested on a synthetic dataset with 1750 randomly generated large
and small sized sparse and dense polynomials. Some real-world applications in
higher order moment portfolio optimization problems, eigenvalue complementarity
problems, Euclidean distance matrix completion problems, and Boolean polynomial
programs are also presented.Comment: 47 pages, 19 figure
Polynomial subspace decomposition for broadband angle of arrival estimation
In this paper we study the impact of polynomial or broadband subspace decompositions on any subsequent processing, which here uses the example of a broadband angle of arrival estimation technique using a recently proposed polynomial MUSIC (P-MUSIC) algorithm. The subspace decompositions are performed by iterative polynomial EVDs, which differ in their approximations to diagonalise and spectrally majorise s apce-time covariance matrix.We here show that a better diagonalisation has a significant impact on the accuracy of defining broadband signal and noise subspaces, demonstrated by a much higher accuracy of the P-MUSIC spectrum
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