35,090 research outputs found
Exact Covers via Determinants
Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such
that every hyperedge includes one vertex from each part, the k-dimensional
matching problem asks whether there is a disjoint collection of the hyperedges
which covers all vertices. We show it can be solved by a randomized polynomial
space algorithm in time O*(2^(n(k-2)/k)). The O*() notation hides factors
polynomial in n and k.
When we drop the partition constraint and permit arbitrary hyperedges of
cardinality k, we obtain the exact cover by k-sets problem. We show it can be
solved by a randomized polynomial space algorithm in time O*(c_k^n), where
c_3=1.496, c_4=1.642, c_5=1.721, and provide a general bound for larger k.
Both results substantially improve on the previous best algorithms for these
problems, especially for small k, and follow from the new observation that
Lovasz' perfect matching detection via determinants (1979) admits an embedding
in the recently proposed inclusion-exclusion counting scheme for set covers,
despite its inability to count the perfect matchings
Monotone Projection Lower Bounds from Extended Formulation Lower Bounds
In this short note, we reduce lower bounds on monotone projections of
polynomials to lower bounds on extended formulations of polytopes. Applying our
reduction to the seminal extended formulation lower bounds of Fiorini, Massar,
Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014;
J. ACM, 2017), we obtain the following interesting consequences.
1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size
projection of the permanent; this both rules out a natural attempt at a
monotone lower bound on the Boolean permanent, and shows that the permanent is
not complete for non-negative polynomials in VNP under monotone
p-projections.
2. The cut polynomials and the perfect matching polynomial (or "unsigned
Pfaffian") are not monotone p-projections of the permanent. The latter, over
the Boolean and-or semi-ring, rules out monotone reductions in one of the
natural approaches to reducing perfect matchings in general graphs to perfect
matchings in bipartite graphs.
As the permanent is universal for monotone formulas, these results also imply
exponential lower bounds on the monotone formula size and monotone circuit size
of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18;
Received: November 10, 2015, Revised: July 27, 2016, Published: December 22,
201
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Projected Power Iteration for Network Alignment
The network alignment problem asks for the best correspondence between two
given graphs, so that the largest possible number of edges are matched. This
problem appears in many scientific problems (like the study of protein-protein
interactions) and it is very closely related to the quadratic assignment
problem which has graph isomorphism, traveling salesman and minimum bisection
problems as particular cases. The graph matching problem is NP-hard in general.
However, under some restrictive models for the graphs, algorithms can
approximate the alignment efficiently. In that spirit the recent work by Feizi
and collaborators introduce EigenAlign, a fast spectral method with convergence
guarantees for Erd\H{o}s-Reny\'i graphs. In this work we propose the algorithm
Projected Power Alignment, which is a projected power iteration version of
EigenAlign. We numerically show it improves the recovery rates of EigenAlign
and we describe the theory that may be used to provide performance guarantees
for Projected Power Alignment.Comment: 8 page
Towards a better approximation for sparsest cut?
We give a new -approximation for sparsest cut problem on graphs
where small sets expand significantly more than the sparsest cut (sets of size
expand by a factor bigger, for some small ; this
condition holds for many natural graph families). We give two different
algorithms. One involves Guruswami-Sinop rounding on the level- Lasserre
relaxation. The other is combinatorial and involves a new notion called {\em
Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which
we show exists in the input graph. Both algorithms run in time . We also show similar approximation algorithms in graphs with
genus with an analogous local expansion condition. This is the first
algorithm we know of that achieves -approximation on such general
family of graphs
Adaptive statistical recognition of hand-printed Telugu characters
A brief description of statistical and syntactic pattern matching techniques is presented with an emphasis on statistical techniques. The characteristics of the Telugu script are described. A subset of 16 characters, which are both easy and hard to recognize, is selected for the dictionary of standard characters. A weighted linear difference polynomial of features is used to recognize Telugu characters. The features were Fourier descriptors of projection profiles and cross sections taken in various directions. Algorithms for obtaining the projection profiles cross sections and adaptive learning method are presented. The system was trained and tested with a set of 8 nano-written samples of each of 16 different Telugu characters. More than 90% of the 123 samples were correctly recognized by the system. Results of numerous trials examining the different features and classification techniques are discussed
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