1,905 research outputs found
A domination algorithm for -instances of the travelling salesman problem
We present an approximation algorithm for -instances of the
travelling salesman problem which performs well with respect to combinatorial
dominance. More precisely, we give a polynomial-time algorithm which has
domination ratio . In other words, given a
-edge-weighting of the complete graph on vertices, our
algorithm outputs a Hamilton cycle of with the following property:
the proportion of Hamilton cycles of whose weight is smaller than that of
is at most . Our analysis is based on a martingale approach.
Previously, the best result in this direction was a polynomial-time algorithm
with domination ratio for arbitrary edge-weights. We also prove a
hardness result showing that, if the Exponential Time Hypothesis holds, there
exists a constant such that cannot be replaced by in the result above.Comment: 29 pages (final version to appear in Random Structures and
Algorithms
Algorithmic Analysis of Qualitative and Quantitative Termination Problems for Affine Probabilistic Programs
In this paper, we consider termination of probabilistic programs with
real-valued variables. The questions concerned are:
1. qualitative ones that ask (i) whether the program terminates with
probability 1 (almost-sure termination) and (ii) whether the expected
termination time is finite (finite termination); 2. quantitative ones that ask
(i) to approximate the expected termination time (expectation problem) and (ii)
to compute a bound B such that the probability to terminate after B steps
decreases exponentially (concentration problem).
To solve these questions, we utilize the notion of ranking supermartingales
which is a powerful approach for proving termination of probabilistic programs.
In detail, we focus on algorithmic synthesis of linear ranking-supermartingales
over affine probabilistic programs (APP's) with both angelic and demonic
non-determinism. An important subclass of APP's is LRAPP which is defined as
the class of all APP's over which a linear ranking-supermartingale exists.
Our main contributions are as follows. Firstly, we show that the membership
problem of LRAPP (i) can be decided in polynomial time for APP's with at most
demonic non-determinism, and (ii) is NP-hard and in PSPACE for APP's with
angelic non-determinism; moreover, the NP-hardness result holds already for
APP's without probability and demonic non-determinism. Secondly, we show that
the concentration problem over LRAPP can be solved in the same complexity as
for the membership problem of LRAPP. Finally, we show that the expectation
problem over LRAPP can be solved in 2EXPTIME and is PSPACE-hard even for APP's
without probability and non-determinism (i.e., deterministic programs). Our
experimental results demonstrate the effectiveness of our approach to answer
the qualitative and quantitative questions over APP's with at most demonic
non-determinism.Comment: 24 pages, full version to the conference paper on POPL 201
Markov selections for the 3D stochastic Navier-Stokes equations
We investigate the Markov property and the continuity with respect to the
initial conditions (strong Feller property) for the solutions to the
Navier-Stokes equations forced by an additive noise.
First, we prove, by means of an abstract selection principle, that there are
Markov solutions to the Navier-Stokes equations. Due to the lack of continuity
of solutions in the space of finite energy, the Markov property holds almost
everywhere in time. Then, depending on the regularity of the noise, we prove
that any Markov solution has the strong Feller property for regular initial
conditions.
We give also a few consequences of these facts, together with a new
sufficient condition for well-posedness.Comment: 59 pages; corrected several errors and typos, added reference
Holderian weak invariance principle under a Hannan type condition
We investigate the invariance principle in H{\"o}lder spaces for strictly
stationary martingale difference sequences. In particular, we show that the
sufficient condition on the tail in the i.i.d. case does not extend to
stationary ergodic martingale differences. We provide a sufficient condition on
the conditional variance which guarantee the invariance principle in H{\"o}lder
spaces. We then deduce a condition in the spirit of Hannan one.Comment: in Stochastic Processes and their Applications, Elsevier, 2016, 12
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