827 research outputs found
Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances
This paper first analyzes the resolution complexity of two random CSP models
(i.e. Model RB/RD) for which we can establish the existence of phase
transitions and identify the threshold points exactly. By encoding CSPs into
CNF formulas, it is proved that almost all instances of Model RB/RD have no
tree-like resolution proofs of less than exponential size. Thus, we not only
introduce new families of CNF formulas hard for resolution, which is a central
task of Proof-Complexity theory, but also propose models with both many hard
instances and exact phase transitions. Then, the implications of such models
are addressed. It is shown both theoretically and experimentally that an
application of Model RB/RD might be in the generation of hard satisfiable
instances, which is not only of practical importance but also related to some
open problems in cryptography such as generating one-way functions.
Subsequently, a further theoretical support for the generation method is shown
by establishing exponential lower bounds on the complexity of solving random
satisfiable and forced satisfiable instances of RB/RD near the threshold.
Finally, conclusions are presented, as well as a detailed comparison of Model
RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively,
exhibit three different kinds of phase transition behavior in NP-complete
problems.Comment: 19 pages, corrected mistakes in Theorems 5 and
A Simple Model to Generate Hard Satisfiable Instances
In this paper, we try to further demonstrate that the models of random CSP
instances proposed by [Xu and Li, 2000; 2003] are of theoretical and practical
interest. Indeed, these models, called RB and RD, present several nice
features. First, it is quite easy to generate random instances of any arity
since no particular structure has to be integrated, or property enforced, in
such instances. Then, the existence of an asymptotic phase transition can be
guaranteed while applying a limited restriction on domain size and on
constraint tightness. In that case, a threshold point can be precisely located
and all instances have the guarantee to be hard at the threshold, i.e., to have
an exponential tree-resolution complexity. Next, a formal analysis shows that
it is possible to generate forced satisfiable instances whose hardness is
similar to unforced satisfiable ones. This analysis is supported by some
representative results taken from an intensive experimentation that we have
carried out, using complete and incomplete search methods.Comment: Proc. of 19th IJCAI, pp.337-342, Edinburgh, Scotland, 2005. For more
information, please click
http://www.nlsde.buaa.edu.cn/~kexu/papers/ijcai05-abstract.ht
Heuristic average-case analysis of the backtrack resolution of random 3-Satisfiability instances
An analysis of the average-case complexity of solving random 3-Satisfiability
(SAT) instances with backtrack algorithms is presented. We first interpret
previous rigorous works in a unifying framework based on the statistical
physics notions of dynamical trajectories, phase diagram and growth process. It
is argued that, under the action of the Davis--Putnam--Loveland--Logemann
(DPLL) algorithm, 3-SAT instances are turned into 2+p-SAT instances whose
characteristic parameters (ratio alpha of clauses per variable, fraction p of
3-clauses) can be followed during the operation, and define resolution
trajectories. Depending on the location of trajectories in the phase diagram of
the 2+p-SAT model, easy (polynomial) or hard (exponential) resolutions are
generated. Three regimes are identified, depending on the ratio alpha of the
3-SAT instance to be solved. Lower sat phase: for small ratios, DPLL almost
surely finds a solution in a time growing linearly with the number N of
variables. Upper sat phase: for intermediate ratios, instances are almost
surely satisfiable but finding a solution requires exponential time (2 ^ (N
omega) with omega>0) with high probability. Unsat phase: for large ratios,
there is almost always no solution and proofs of refutation are exponential. An
analysis of the growth of the search tree in both upper sat and unsat regimes
is presented, and allows us to estimate omega as a function of alpha. This
analysis is based on an exact relationship between the average size of the
search tree and the powers of the evolution operator encoding the elementary
steps of the search heuristic.Comment: to appear in Theoretical Computer Scienc
Computational Complexity for Physicists
These lecture notes are an informal introduction to the theory of
computational complexity and its links to quantum computing and statistical
mechanics.Comment: references updated, reprint available from
http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm
The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
Among various algorithms designed to exploit the specific properties of
quantum computers with respect to classical ones, the quantum adiabatic
algorithm is a versatile proposition to find the minimal value of an arbitrary
cost function (ground state energy). Random optimization problems provide a
natural testbed to compare its efficiency with that of classical algorithms.
These problems correspond to mean field spin glasses that have been extensively
studied in the classical case. This paper reviews recent analytical works that
extended these studies to incorporate the effect of quantum fluctuations, and
presents also some original results in this direction.Comment: 151 pages, 21 figure
Unweighted Stochastic Local Search can be Effective for Random CSP Benchmarks
We present ULSA, a novel stochastic local search algorithm for random binary
constraint satisfaction problems (CSP). ULSA is many times faster than the
prior state of the art on a widely-studied suite of random CSP benchmarks.
Unlike the best previous methods for these benchmarks, ULSA is a simple
unweighted method that does not require dynamic adaptation of weights or
penalties. ULSA obtains new record best solutions satisfying 99 of 100
variables in the challenging frb100-40 benchmark instance
Conformant Planning as a Case Study of Incremental QBF Solving
We consider planning with uncertainty in the initial state as a case study of
incremental quantified Boolean formula (QBF) solving. We report on experiments
with a workflow to incrementally encode a planning instance into a sequence of
QBFs. To solve this sequence of incrementally constructed QBFs, we use our
general-purpose incremental QBF solver DepQBF. Since the generated QBFs have
many clauses and variables in common, our approach avoids redundancy both in
the encoding phase and in the solving phase. Experimental results show that
incremental QBF solving outperforms non-incremental QBF solving. Our results
are the first empirical study of incremental QBF solving in the context of
planning and motivate its use in other application domains.Comment: added reference to extended journal article; revision (camera-ready,
to appear in the proceedings of AISC 2014, volume 8884 of LNAI, Springer
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