372 research outputs found
An Empirical Analysis of Search in GSAT
We describe an extensive study of search in GSAT, an approximation procedure
for propositional satisfiability. GSAT performs greedy hill-climbing on the
number of satisfied clauses in a truth assignment. Our experiments provide a
more complete picture of GSAT's search than previous accounts. We describe in
detail the two phases of search: rapid hill-climbing followed by a long plateau
search. We demonstrate that when applied to randomly generated 3SAT problems,
there is a very simple scaling with problem size for both the mean number of
satisfied clauses and the mean branching rate. Our results allow us to make
detailed numerical conjectures about the length of the hill-climbing phase, the
average gradient of this phase, and to conjecture that both the average score
and average branching rate decay exponentially during plateau search. We end by
showing how these results can be used to direct future theoretical analysis.
This work provides a case study of how computer experiments can be used to
improve understanding of the theoretical properties of algorithms.Comment: See http://www.jair.org/ for any accompanying file
When Gravity Fails: Local Search Topology
Local search algorithms for combinatorial search problems frequently
encounter a sequence of states in which it is impossible to improve the value
of the objective function; moves through these regions, called plateau moves,
dominate the time spent in local search. We analyze and characterize plateaus
for three different classes of randomly generated Boolean Satisfiability
problems. We identify several interesting features of plateaus that impact the
performance of local search algorithms. We show that local minima tend to be
small but occasionally may be very large. We also show that local minima can be
escaped without unsatisfying a large number of clauses, but that systematically
searching for an escape route may be computationally expensive if the local
minimum is large. We show that plateaus with exits, called benches, tend to be
much larger than minima, and that some benches have very few exit states which
local search can use to escape. We show that the solutions (i.e., global
minima) of randomly generated problem instances form clusters, which behave
similarly to local minima. We revisit several enhancements of local search
algorithms and explain their performance in light of our results. Finally we
discuss strategies for creating the next generation of local search algorithms.Comment: See http://www.jair.org/ for any accompanying file
Adiabatic Quantum Computing for Random Satisfiability Problems
The discrete formulation of adiabatic quantum computing is compared with
other search methods, classical and quantum, for random satisfiability (SAT)
problems. With the number of steps growing only as the cube of the number of
variables, the adiabatic method gives solution probabilities close to 1 for
problem sizes feasible to evaluate via simulation on current computers.
However, for these sizes the minimum energy gaps of most instances are fairly
large, so the good performance scaling seen for small problems may not reflect
asymptotic behavior where costs are dominated by tiny gaps. Moreover, the
resulting search costs are much higher than for other methods. Variants of the
quantum algorithm that do not match the adiabatic limit give lower costs, on
average, and slower growth than the conventional GSAT heuristic method.Comment: added discussion of discrete adiabatic method, and simulations with
30 bits 8 pages, 8 figure
Single-Step Quantum Search Using Problem Structure
The structure of satisfiability problems is used to improve search algorithms
for quantum computers and reduce their required coherence times by using only a
single coherent evaluation of problem properties. The structure of random k-SAT
allows determining the asymptotic average behavior of these algorithms, showing
they improve on quantum algorithms, such as amplitude amplification, that
ignore detailed problem structure but remain exponential for hard problem
instances. Compared to good classical methods, the algorithm performs better,
on average, for weakly and highly constrained problems but worse for hard
cases. The analytic techniques introduced here also apply to other quantum
algorithms, supplementing the limited evaluation possible with classical
simulations and showing how quantum computing can use ensemble properties of NP
search problems.Comment: 39 pages, 12 figures. Revision describes further improvement with
multiple steps (section 7). See also
http://www.parc.xerox.com/dynamics/www/quantum.htm
A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
A quantum system will stay near its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough. This quantum
adiabatic behavior is the basis of a new class of algorithms for quantum
computing. We test one such algorithm by applying it to randomly generated,
hard, instances of an NP-complete problem. For the small examples that we can
simulate, the quantum adiabatic algorithm works well, and provides evidence
that quantum computers (if large ones can be built) may be able to outperform
ordinary computers on hard sets of instances of NP-complete problems.Comment: 15 pages, 6 figures, email correspondence to [email protected] ; a
shorter version of this article appeared in the April 20, 2001 issue of
Science; see http://www.sciencemag.org/cgi/content/full/292/5516/47
The Potential of Restarts for ProbSAT
This work analyses the potential of restarts for probSAT, a quite successful
algorithm for k-SAT, by estimating its runtime distributions on random 3-SAT
instances that are close to the phase transition. We estimate an optimal
restart time from empirical data, reaching a potential speedup factor of 1.39.
Calculating restart times from fitted probability distributions reduces this
factor to a maximum of 1.30. A spin-off result is that the Weibull distribution
approximates the runtime distribution for over 93% of the used instances well.
A machine learning pipeline is presented to compute a restart time for a
fixed-cutoff strategy to exploit this potential. The main components of the
pipeline are a random forest for determining the distribution type and a neural
network for the distribution's parameters. ProbSAT performs statistically
significantly better than Luby's restart strategy and the policy without
restarts when using the presented approach. The structure is particularly
advantageous on hard problems.Comment: Eurocast 201
Portfolios in Stochastic Local Search: Efficiently Computing Most Probable Explanations in Bayesian Networks
Portfolio methods support the combination of different algorithms and heuristics, including stochastic local search (SLS) heuristics, and have been identified as a promising approach to solve computationally hard problems. While successful in experiments, theoretical foundations and analytical results for portfolio-based SLS heuristics are less developed. This article aims to improve the understanding of the role of portfolios of heuristics in SLS. We emphasize the problem of computing most probable explanations (MPEs) in Bayesian networks (BNs). Algorithmically, we discuss a portfolio-based SLS algorithm for MPE computation, Stochastic Greedy Search (SGS). SGS supports the integration of different initialization operators (or initialization heuristics) and different search operators (greedy and noisy heuristics), thereby enabling new analytical and experimental results. Analytically, we introduce a novel Markov chain model tailored to portfolio-based SLS algorithms including SGS, thereby enabling us to analytically form expected hitting time results that explain empirical run time results. For a specific BN, we show the benefit of using a homogenous initialization portfolio. To further illustrate the portfolio approach, we consider novel additive search heuristics for handling determinism in the form of zero entries in conditional probability tables in BNs. Our additive approach adds rather than multiplies probabilities when computing the utility of an explanation. We motivate the additive measure by studying the dramatic impact of zero entries in conditional probability tables on the number of zero-probability explanations, which again complicates the search process. We consider the relationship between MAXSAT and MPE, and show that additive utility (or gain) is a generalization, to the probabilistic setting, of MAXSAT utility (or gain) used in the celebrated GSAT and WalkSAT algorithms and their descendants. Utilizing our Markov chain framework, we show that expected hitting time is a rational function - i.e. a ratio of two polynomials - of the probability of applying an additive search operator. Experimentally, we report on synthetically generated BNs as well as BNs from applications, and compare SGSs performance to that of Hugin, which performs BN inference by compilation to and propagation in clique trees. On synthetic networks, SGS speeds up computation by approximately two orders of magnitude compared to Hugin. In application networks, our approach is highly competitive in Bayesian networks with a high degree of determinism. In addition to showing that stochastic local search can be competitive with clique tree clustering, our empirical results provide an improved understanding of the circumstances under which portfolio-based SLS outperforms clique tree clustering and vice versa
Improving WalkSAT for Random 3-SAT Problems
Stochastic local search (SLS) algorithms are well known for their ability to efficiently find models of random instances of the Boolean satisfiability (SAT) problems. One of the most famous SLS algorithms for SAT is called WalkSAT, which has wide influence and performs well on most of random 3-SAT instances. However, the performance of WalkSAT lags far behind on random 3-SAT instances equal to or greater than the phase transition ratio. Motivated by this limitation, in the present work, firstly an allocation strategy is introduced and utilized in WalkSAT to determine the initial assignment, leading to a new algorithm called WalkSATvav. The experimental results show that WalkSATvav significantly outperforms the state-of-the-art SLS solvers on random 3-SAT instances at the phase transition for SAT Competition 2017. However, WalkSATvav cannot rival its competitors on random 3-SAT instances greater than the phase transition ratio. Accordingly, WalkSATvav is further improved for such instances by utilizing a combination of an improved genetic algorithm and an improved ant colony algorithm, which complement each other in guiding the search direction. The resulting algorithm, called WalkSATga, is far better than WalkSAT and significantly outperforms some previous known SLS solvers on random 3-SAT instances greater than the phase transition ratio from SAT Competition 2017. Finally, a new SAT solver called WalkSATlg, which combines WalkSATvav and WalkSATga, is proposed, which is competitive with the winner of random satisfiable category of SAT competition 2017 on random 3-SAT problem
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