6,970 research outputs found
Focused Local Search for Random 3-Satisfiability
A local search algorithm solving an NP-complete optimisation problem can be
viewed as a stochastic process moving in an 'energy landscape' towards
eventually finding an optimal solution. For the random 3-satisfiability
problem, the heuristic of focusing the local moves on the presently
unsatisfiedclauses is known to be very effective: the time to solution has been
observed to grow only linearly in the number of variables, for a given
clauses-to-variables ratio sufficiently far below the critical
satisfiability threshold . We present numerical results
on the behaviour of three focused local search algorithms for this problem,
considering in particular the characteristics of a focused variant of the
simple Metropolis dynamics. We estimate the optimal value for the
``temperature'' parameter for this algorithm, such that its linear-time
regime extends as close to as possible. Similar parameter
optimisation is performed also for the well-known WalkSAT algorithm and for the
less studied, but very well performing Focused Record-to-Record Travel method.
We observe that with an appropriate choice of parameters, the linear time
regime for each of these algorithms seems to extend well into ratios -- much further than has so far been generally assumed. We discuss the
statistics of solution times for the algorithms, relate their performance to
the process of ``whitening'', and present some conjectures on the shape of
their computational phase diagrams.Comment: 20 pages, lots of figure
Behavior of heuristics and state space structure near SAT/UNSAT transition
We study the behavior of ASAT, a heuristic for solving satisfiability
problems by stochastic local search near the SAT/UNSAT transition. The
heuristic is focused, i.e. only variables in unsatisfied clauses are updated in
each step, and is significantly simpler, while similar to, walksat or Focused
Metropolis Search. We show that ASAT solves instances as large as one million
variables in linear time, on average, up to 4.21 clauses per variable for
random 3SAT. For K higher than 3, ASAT appears to solve instances at the ``FRSB
threshold'' in linear time, up to K=7.Comment: 12 pages, 6 figures, longer version available as MSc thesis of first
author at http://biophys.physics.kth.se/docs/ardelius_thesis.pd
Efficient Benchmarking of Algorithm Configuration Procedures via Model-Based Surrogates
The optimization of algorithm (hyper-)parameters is crucial for achieving
peak performance across a wide range of domains, ranging from deep neural
networks to solvers for hard combinatorial problems. The resulting algorithm
configuration (AC) problem has attracted much attention from the machine
learning community. However, the proper evaluation of new AC procedures is
hindered by two key hurdles. First, AC benchmarks are hard to set up. Second
and even more significantly, they are computationally expensive: a single run
of an AC procedure involves many costly runs of the target algorithm whose
performance is to be optimized in a given AC benchmark scenario. One common
workaround is to optimize cheap-to-evaluate artificial benchmark functions
(e.g., Branin) instead of actual algorithms; however, these have different
properties than realistic AC problems. Here, we propose an alternative
benchmarking approach that is similarly cheap to evaluate but much closer to
the original AC problem: replacing expensive benchmarks by surrogate benchmarks
constructed from AC benchmarks. These surrogate benchmarks approximate the
response surface corresponding to true target algorithm performance using a
regression model, and the original and surrogate benchmark share the same
(hyper-)parameter space. In our experiments, we construct and evaluate
surrogate benchmarks for hyperparameter optimization as well as for AC problems
that involve performance optimization of solvers for hard combinatorial
problems, drawing training data from the runs of existing AC procedures. We
show that our surrogate benchmarks capture overall important characteristics of
the AC scenarios, such as high- and low-performing regions, from which they
were derived, while being much easier to use and orders of magnitude cheaper to
evaluate
On Improving Local Search for Unsatisfiability
Stochastic local search (SLS) has been an active field of research in the
last few years, with new techniques and procedures being developed at an
astonishing rate. SLS has been traditionally associated with satisfiability
solving, that is, finding a solution for a given problem instance, as its
intrinsic nature does not address unsatisfiable problems. Unsatisfiable
instances were therefore commonly solved using backtrack search solvers. For
this reason, in the late 90s Selman, Kautz and McAllester proposed a challenge
to use local search instead to prove unsatisfiability. More recently, two SLS
solvers - Ranger and Gunsat - have been developed, which are able to prove
unsatisfiability albeit being SLS solvers. In this paper, we first compare
Ranger with Gunsat and then propose to improve Ranger performance using some of
Gunsat's techniques, namely unit propagation look-ahead and extended
resolution
Relaxation and Metastability in the RandomWalkSAT search procedure
An analysis of the average properties of a local search resolution procedure
for the satisfaction of random Boolean constraints is presented. Depending on
the ratio alpha of constraints per variable, resolution takes a time T_res
growing linearly (T_res \sim tau(alpha) N, alpha < alpha_d) or exponentially
(T_res \sim exp(N zeta(alpha)), alpha > alpha_d) with the size N of the
instance. The relaxation time tau(alpha) in the linear phase is calculated
through a systematic expansion scheme based on a quantum formulation of the
evolution operator. For alpha > alpha_d, the system is trapped in some
metastable state, and resolution occurs from escape from this state through
crossing of a large barrier. An annealed calculation of the height zeta(alpha)
of this barrier is proposed. The polynomial/exponentiel cross-over alpha_d is
not related to the onset of clustering among solutions.Comment: 23 pages, 11 figures. A mistake in sec. IV.B has been correcte
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