1,828 research outputs found
Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs
We present a deterministic way of assigning small (log bit) weights to the
edges of a bipartite planar graph so that the minimum weight perfect matching
becomes unique. The isolation lemma as described in (Mulmuley et al. 1987)
achieves the same for general graphs using a randomized weighting scheme,
whereas we can do it deterministically when restricted to bipartite planar
graphs. As a consequence, we reduce both decision and construction versions of
the matching problem to testing whether a matrix is singular, under the promise
that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm
for bipartite planar graphs. This improves the earlier known bounds of
non-uniform SPL by (Allender et al. 1999) and by (Miller and Naor 1995,
Mahajan and Varadarajan 2000). It also rekindles the hope of obtaining a
deterministic parallel algorithm for constructing a perfect matching in
non-bipartite planar graphs, which has been open for a long time. Our
techniques are elementary and simple
Fast Parallel Fixed-Parameter Algorithms via Color Coding
Fixed-parameter algorithms have been successfully applied to solve numerous
difficult problems within acceptable time bounds on large inputs. However, most
fixed-parameter algorithms are inherently \emph{sequential} and, thus, make no
use of the parallel hardware present in modern computers. We show that parallel
fixed-parameter algorithms do not only exist for numerous parameterized
problems from the literature -- including vertex cover, packing problems,
cluster editing, cutting vertices, finding embeddings, or finding matchings --
but that there are parallel algorithms working in \emph{constant} time or at
least in time \emph{depending only on the parameter} (and not on the size of
the input) for these problems. Phrased in terms of complexity classes, we place
numerous natural parameterized problems in parameterized versions of AC. On
a more technical level, we show how the \emph{color coding} method can be
implemented in constant time and apply it to embedding problems for graphs of
bounded tree-width or tree-depth and to model checking first-order formulas in
graphs of bounded degree
The Value of Help Bits in Randomized and Average-Case Complexity
"Help bits" are some limited trusted information about an instance or
instances of a computational problem that may reduce the computational
complexity of solving that instance or instances. In this paper, we study the
value of help bits in the settings of randomized and average-case complexity.
Amir, Beigel, and Gasarch (1990) show that for constant , if instances
of a decision problem can be efficiently solved using less than bits of
help, then the problem is in P/poly. We extend this result to the setting of
randomized computation: We show that the decision problem is in P/poly if using
help bits, instances of the problem can be efficiently solved with
probability greater than . The same result holds if using less than
help bits (where is the binary entropy function),
we can efficiently solve fraction of the instances correctly with
non-vanishing probability. We also extend these two results to non-constant but
logarithmic . In this case however, instead of showing that the problem is
in P/poly we show that it satisfies "-membership comparability," a notion
known to be related to solving instances using less than bits of help.
Next we consider the setting of average-case complexity: Assume that we can
solve instances of a decision problem using some help bits whose entropy is
less than when the instances are drawn independently from a particular
distribution. Then we can efficiently solve an instance drawn from that
distribution with probability better than .
Finally, we show that in the case where is super-logarithmic, assuming
-membership comparability of a decision problem, one cannot prove that the
problem is in P/poly by a "black-box proof.
Unfoldings and Deformations of Rational and Logarithmic Foliations
We study codimension one foliations in projective space \PP^n over \CC by
looking at its first order perturbations: unfoldings and deformations. We give
special attention to foliations of rational and logarithmic type.
For a differential form \omega defining a codimension one foliation, we
present a graded module \UU(\omega), related to the first order unfoldings of
\omega. If \omega is a generic form of rational or logarithmic type, as a first
application of the construction of \UU(\omega), we classify the first order
deformations that arise from first order unfoldings. Then, we count the number
of isolated points in the singular set of \omega, in terms of a Hilbert
polynomial associated to \UU(\omega).
We review the notion of regularity of \omega in terms of a long complex of
graded modules that we also introduce in this work. We use this complex to
prove that, for generic rational and logarithmic foliations, \omega is regular
if and only if every unfolding is trivial up to isomorphism.Comment: Final version. 25 page
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