112,815 research outputs found
Finding Simple Shortest Paths and Cycles
The problem of finding multiple simple shortest paths in a weighted directed
graph has many applications, and is considerably more difficult than
the corresponding problem when cycles are allowed in the paths. Even for a
single source-sink pair, it is known that two simple shortest paths cannot be
found in time polynomially smaller than (where ) unless the
All-Pairs Shortest Paths problem can be solved in a similar time bound. The
latter is a well-known open problem in algorithm design. We consider the
all-pairs version of the problem, and we give a new algorithm to find
simple shortest paths for all pairs of vertices. For , our algorithm runs
in time (where ), which is almost the same bound as
for the single pair case, and for we improve earlier bounds. Our approach
is based on forming suitable path extensions to find simple shortest paths;
this method is different from the `detour finding' technique used in most of
the prior work on simple shortest paths, replacement paths, and distance
sensitivity oracles.
Enumerating simple cycles is a well-studied classical problem. We present new
algorithms for generating simple cycles and simple paths in in
non-decreasing order of their weights; the algorithm for generating simple
paths is much faster, and uses another variant of path extensions. We also give
hardness results for sparse graphs, relative to the complexity of computing a
minimum weight cycle in a graph, for several variants of problems related to
finding simple paths and cycles.Comment: The current version includes new results for undirected graphs. In
Section 4, the notion of an (m,n) reduction is generalized to an f(m,n)
reductio
Using TPA to count linear extensions
A linear extension of a poset is a permutation of the elements of the set
that respects the partial order. Let denote the number of linear
extensions. It is a #P complete problem to determine exactly for an
arbitrary poset, and so randomized approximation algorithms that draw randomly
from the set of linear extensions are used. In this work, the set of linear
extensions is embedded in a larger state space with a continuous parameter ?.
The introduction of a continuous parameter allows for the use of a more
efficient method for approximating called TPA. Our primary result is
that it is possible to sample from this continuous embedding in time that as
fast or faster than the best known methods for sampling uniformly from linear
extensions. For a poset containing elements, this means we can approximate
to within a factor of with probability at least using an expected number of random bits and comparisons in the poset
which is at most Comment: 12 pages, 4 algorithm
Isoperimetric inequalities, shapes of F{\o}lner sets and groups with Shalom's property
We prove an isoperimetric inequality for groups. As an application, we obtain
lower bound on F{\o}lner functions in various nilpotent-by-cyclic groups. Under
a regularity assumption, we obtain a characterization of F{\o}lner functions of
these groups. As another application, we evaluate the asymptotics of the
F{\o}lner function of . We construct new
examples of groups with Shalom's property , in particular
among nilpotent-by-cyclic and lacunary hyperbolic groups. Among these examples
we find groups with property , which are direct products of
lacunary hyperbolic groups and have arbitrarily large F{\o}lner functions
Algorithms for group isomorphism via group extensions and cohomology
The isomorphism problem for finite groups of order n (GpI) has long been
known to be solvable in time, but only recently were
polynomial-time algorithms designed for several interesting group classes.
Inspired by recent progress, we revisit the strategy for GpI via the extension
theory of groups.
The extension theory describes how a normal subgroup N is related to G/N via
G, and this naturally leads to a divide-and-conquer strategy that splits GpI
into two subproblems: one regarding group actions on other groups, and one
regarding group cohomology. When the normal subgroup N is abelian, this
strategy is well-known. Our first contribution is to extend this strategy to
handle the case when N is not necessarily abelian. This allows us to provide a
unified explanation of all recent polynomial-time algorithms for special group
classes.
Guided by this strategy, to make further progress on GpI, we consider
central-radical groups, proposed in Babai et al. (SODA 2011): the class of
groups such that G mod its center has no abelian normal subgroups. This class
is a natural extension of the group class considered by Babai et al. (ICALP
2012), namely those groups with no abelian normal subgroups. Following the
above strategy, we solve GpI in time for central-radical
groups, and in polynomial time for several prominent subclasses of
central-radical groups. We also solve GpI in time for
groups whose solvable normal subgroups are elementary abelian but not
necessarily central. As far as we are aware, this is the first time there have
been worst-case guarantees on a -time algorithm that tackles
both aspects of GpI---actions and cohomology---simultaneously.Comment: 54 pages + 14-page appendix. Significantly improved presentation,
with some new result
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