185 research outputs found
Fine-grained dichotomies for the Tutte plane and Boolean #CSP
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of
evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard
almost everywhere, and the remaining points admit polynomial-time algorithms.
Dell, Husfeldt, and Wahl\'en [9] and Husfeldt and Taslaman [12], in combination
with Curticapean [7], extended the #P-hardness results to tight lower bounds
under the counting exponential time hypothesis #ETH, with the exception of the
line , which was left open. We complete the dichotomy theorem for the
Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs
of a given -vertex graph cannot be determined in time unless
#ETH fails.
Another dichotomy theorem we strengthen is the one of Creignou and Hermann
[6] for counting the number of satisfying assignments to a constraint
satisfaction problem instance over the Boolean domain. We prove that all
#P-hard cases are also hard under #ETH. The main ingredient is to prove that
the number of independent sets in bipartite graphs with vertices cannot be
computed in time unless #ETH fails. In order to prove our results,
we use the block interpolation idea by Curticapean [7] and transfer it to
systems of linear equations that might not directly correspond to
interpolation.Comment: 16 pages, 1 figur
Optimal Data Collection For Informative Rankings Expose Well-Connected Graphs
Given a graph where vertices represent alternatives and arcs represent
pairwise comparison data, the statistical ranking problem is to find a
potential function, defined on the vertices, such that the gradient of the
potential function agrees with the pairwise comparisons. Our goal in this paper
is to develop a method for collecting data for which the least squares
estimator for the ranking problem has maximal Fisher information. Our approach,
based on experimental design, is to view data collection as a bi-level
optimization problem where the inner problem is the ranking problem and the
outer problem is to identify data which maximizes the informativeness of the
ranking. Under certain assumptions, the data collection problem decouples,
reducing to a problem of finding multigraphs with large algebraic connectivity.
This reduction of the data collection problem to graph-theoretic questions is
one of the primary contributions of this work. As an application, we study the
Yahoo! Movie user rating dataset and demonstrate that the addition of a small
number of well-chosen pairwise comparisons can significantly increase the
Fisher informativeness of the ranking. As another application, we study the
2011-12 NCAA football schedule and propose schedules with the same number of
games which are significantly more informative. Using spectral clustering
methods to identify highly-connected communities within the division, we argue
that the NCAA could improve its notoriously poor rankings by simply scheduling
more out-of-conference games.Comment: 31 pages, 10 figures, 3 table
Compressed Genotyping
Significant volumes of knowledge have been accumulated in recent years
linking subtle genetic variations to a wide variety of medical disorders from
Cystic Fibrosis to mental retardation. Nevertheless, there are still great
challenges in applying this knowledge routinely in the clinic, largely due to
the relatively tedious and expensive process of DNA sequencing. Since the
genetic polymorphisms that underlie these disorders are relatively rare in the
human population, the presence or absence of a disease-linked polymorphism can
be thought of as a sparse signal. Using methods and ideas from compressed
sensing and group testing, we have developed a cost-effective genotyping
protocol. In particular, we have adapted our scheme to a recently developed
class of high throughput DNA sequencing technologies, and assembled a
mathematical framework that has some important distinctions from 'traditional'
compressed sensing ideas in order to address different biological and technical
constraints.Comment: Submitted to IEEE Transaction on Information Theory - Special Issue
on Molecular Biology and Neuroscienc
Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees
An edge coloring of a graph G is called interval edge coloring if for each v ? V(G) the set of colors on edges incident to v forms an interval of integers. A graph G is interval colorable if there is an interval coloring of G. For an interval colorable graph G, by the interval chromatic index of G, denoted by ?\u27_i(G), we mean the smallest number k such that G is interval colorable with k colors. A bipartite graph G is called (?,?)-biregular if each vertex in one part has degree ? and each vertex in the other part has degree ?. A graph G is called (?*,?*)-bipartite if G is a subgraph of an (?,?)-biregular graph and the maximum degree in one part is ? and the maximum degree in the other part is ?.
In the paper we study the problem of interval edge colorings of (k*,2*)-bipartite graphs, for k ? {3,4,5}, and of (5*,3*)-bipartite graphs. We prove that every (5*,2*)-bipartite graph admits an interval edge coloring using at most 6 colors, which can be found in O(n^{3/2}) time, and we prove that an interval edge 5-coloring of a (5*,2*)-bipartite graph can be found in O(n^{3/2}) time, if it exists. We show that every (4^*,2^*)-bipartite graph admits an interval edge 4-coloring, which can be found in O(n) time. The two following problems of interval edge coloring are known to be NP-complete: 6-coloring of (6,3)-biregular graphs (Asratian and Casselgren (2006)) and 5-coloring of (5*,5*)-bipartite graphs (Giaro (1997)). In the paper we prove NP-completeness of 5-coloring of (5*,3*)-bipartite graphs
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