8,541 research outputs found
Utilitarian resource assignment
This paper studies a resource allocation problem introduced by Koutsoupias
and Papadimitriou. The scenario is modelled as a multiple-player game in which
each player selects one of a finite number of known resources. The cost to the
player is the total weight of all players who choose that resource, multiplied
by the ``delay'' of that resource. Recent papers have studied the Nash
equilibria and social optima of this game in terms of the cost
metric, in which the social cost is taken to be the maximum cost to any player.
We study the variant of this game, in which the social cost is taken to
be the sum of the costs to the individual players, rather than the maximum of
these costs. We give bounds on the size of the coordination ratio, which is the
ratio between the social cost incurred by selfish behavior and the optimal
social cost; we also study the algorithmic problem of finding optimal
(lowest-cost) assignments and Nash Equilibria. Additionally, we obtain bounds
on the ratio between alternative Nash equilibria for some special cases of the
problem.Comment: 19 page
Markov chain comparison
This is an expository paper, focussing on the following scenario. We have two
Markov chains, and . By some means, we have
obtained a bound on the mixing time of . We wish to compare
with in order to derive a corresponding bound on
the mixing time of . We investigate the application of the
comparison method of Diaconis and Saloff-Coste to this scenario, giving a
number of theorems which characterize the applicability of the method. We focus
particularly on the case in which the chains are not reversible. The purpose of
the paper is to provide a catalogue of theorems which can be easily applied to
bound mixing times.Comment: Published at http://dx.doi.org/10.1214/154957806000000041 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The Complexity of Approximately Counting Stable Matchings
We investigate the complexity of approximately counting stable matchings in
the -attribute model, where the preference lists are determined by dot
products of "preference vectors" with "attribute vectors", or by Euclidean
distances between "preference points" and "attribute points". Irving and
Leather proved that counting the number of stable matchings in the general case
is #P-complete. Counting the number of stable matchings is reducible to
counting the number of downsets in a (related) partial order and is
interreducible, in an approximation-preserving sense, to a class of problems
that includes counting the number of independent sets in a bipartite graph
(#BIS). It is conjectured that no FPRAS exists for this class of problems. We
show this approximation-preserving interreducibilty remains even in the
restricted -attribute setting when (dot products) or
(Euclidean distances). Finally, we show it is easy to count the number of
stable matchings in the 1-attribute dot-product setting.Comment: Fixed typos, small revisions for clarification, et
Matrix norms and rapid mixing for spin systems
We give a systematic development of the application of matrix norms to rapid
mixing in spin systems. We show that rapid mixing of both random update Glauber
dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the
associated dependency matrix is less than 1. We give improved analysis for the
case in which the diagonal of the dependency matrix is (as in heat
bath dynamics). We apply the matrix norm methods to random update and
systematic scan Glauber dynamics for coloring various classes of graphs. We
give a general method for estimating a norm of a symmetric nonregular matrix.
This leads to improved mixing times for any class of graphs which is hereditary
and sufficiently sparse including several classes of degree-bounded graphs such
as nonregular graphs, trees, planar graphs and graphs with given tree-width and
genus.Comment: Published in at http://dx.doi.org/10.1214/08-AAP532 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The legacy of nineteenth-century replicas for object cultural biographies : lessons in duplication from 1830s Fife
Peer reviewedPostprin
An approximation trichotomy for Boolean #CSP
We give a trichotomy theorem for the complexity of approximately counting the
number of satisfying assignments of a Boolean CSP instance. Such problems are
parameterised by a constraint language specifying the relations that may be
used in constraints. If every relation in the constraint language is affine
then the number of satisfying assignments can be exactly counted in polynomial
time. Otherwise, if every relation in the constraint language is in the
co-clone IM_2 from Post's lattice, then the problem of counting satisfying
assignments is complete with respect to approximation-preserving reductions in
the complexity class #RH\Pi_1. This means that the problem of approximately
counting satisfying assignments of such a CSP instance is equivalent in
complexity to several other known counting problems, including the problem of
approximately counting the number of independent sets in a bipartite graph. For
every other fixed constraint language, the problem is complete for #P with
respect to approximation-preserving reductions, meaning that there is no fully
polynomial randomised approximation scheme for counting satisfying assignments
unless NP=RP
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