4,262 research outputs found
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
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
Reweighted belief propagation and quiet planting for random K-SAT
We study the random K-satisfiability problem using a partition function where
each solution is reweighted according to the number of variables that satisfy
every clause. We apply belief propagation and the related cavity method to the
reweighted partition function. This allows us to obtain several new results on
the properties of random K-satisfiability problem. In particular the
reweighting allows to introduce a planted ensemble that generates instances
that are, in some region of parameters, equivalent to random instances. We are
hence able to generate at the same time a typical random SAT instance and one
of its solutions. We study the relation between clustering and belief
propagation fixed points and we give a direct evidence for the existence of
purely entropic (rather than energetic) barriers between clusters in some
region of parameters in the random K-satisfiability problem. We exhibit, in
some large planted instances, solutions with a non-trivial whitening core; such
solutions were known to exist but were so far never found on very large
instances. Finally, we discuss algorithmic hardness of such planted instances
and we determine a region of parameters in which planting leads to satisfiable
benchmarks that, up to our knowledge, are the hardest known.Comment: 23 pages, 4 figures, revised for readability, stability expression
correcte
Quiet Planting in the Locked Constraint Satisfaction Problems
We study the planted ensemble of locked constraint satisfaction problems. We
describe the connection between the random and planted ensembles. The use of
the cavity method is combined with arguments from reconstruction on trees and
first and second moment considerations; in particular the connection with the
reconstruction on trees appears to be crucial. Our main result is the location
of the hard region in the planted ensemble. In a part of that hard region
instances have with high probability a single satisfying assignment.Comment: 21 pages, revised versio
Hiding Satisfying Assignments: Two are Better than One
The evaluation of incomplete satisfiability solvers depends critically on the
availability of hard satisfiable instances. A plausible source of such
instances consists of random k-SAT formulas whose clauses are chosen uniformly
from among all clauses satisfying some randomly chosen truth assignment A.
Unfortunately, instances generated in this manner tend to be relatively easy
and can be solved efficiently by practical heuristics. Roughly speaking, as the
formula's density increases, for a number of different algorithms, A acts as a
stronger and stronger attractor. Motivated by recent results on the geometry of
the space of satisfying truth assignments of random k-SAT and NAE-k-SAT
formulas, we introduce a simple twist on this basic model, which appears to
dramatically increase its hardness. Namely, in addition to forbidding the
clauses violated by the hidden assignment A, we also forbid the clauses
violated by its complement, so that both A and complement of A are satisfying.
It appears that under this "symmetrization'' the effects of the two attractors
largely cancel out, making it much harder for algorithms to find any truth
assignment. We give theoretical and experimental evidence supporting this
assertion.Comment: Preliminary version appeared in AAAI 200
Simplest random K-satisfiability problem
We study a simple and exactly solvable model for the generation of random
satisfiability problems. These consist of random boolean constraints
which are to be satisfied simultaneously by logical variables. In
statistical-mechanics language, the considered model can be seen as a diluted
p-spin model at zero temperature. While such problems become extraordinarily
hard to solve by local search methods in a large region of the parameter space,
still at least one solution may be superimposed by construction. The
statistical properties of the model can be studied exactly by the replica
method and each single instance can be analyzed in polynomial time by a simple
global solution method. The geometrical/topological structures responsible for
dynamic and static phase transitions as well as for the onset of computational
complexity in local search method are thoroughly analyzed. Numerical analysis
on very large samples allows for a precise characterization of the critical
scaling behaviour.Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor
errors and references correcte
Algorithms for the workflow satisfiability problem engineered for counting constraints
The workflow satisfiability problem (WSP) asks whether there exists an
assignment of authorized users to the steps in a workflow specification that
satisfies the constraints in the specification. The problem is NP-hard in
general, but several subclasses of the problem are known to be fixed-parameter
tractable (FPT) when parameterized by the number of steps in the specification.
In this paper, we consider the WSP with user-independent counting constraints,
a large class of constraints for which the WSP is known to be FPT. We describe
an efficient implementation of an FPT algorithm for solving this subclass of
the WSP and an experimental evaluation of this algorithm. The algorithm
iteratively generates all equivalence classes of possible partial solutions
until, whenever possible, it finds a complete solution to the problem. We also
provide a reduction from a WSP instance to a pseudo-Boolean SAT instance. We
apply this reduction to the instances used in our experiments and solve the
resulting PB SAT problems using SAT4J, a PB SAT solver. We compare the
performance of our algorithm with that of SAT4J and discuss which of the two
approaches would be more effective in practice
Hiding solutions in random satisfiability problems: A statistical mechanics approach
A major problem in evaluating stochastic local search algorithms for
NP-complete problems is the need for a systematic generation of hard test
instances having previously known properties of the optimal solutions. On the
basis of statistical mechanics results, we propose random generators of hard
and satisfiable instances for the 3-satisfiability problem (3SAT). The design
of the hardest problem instances is based on the existence of a first order
ferromagnetic phase transition and the glassy nature of excited states. The
analytical predictions are corroborated by numerical results obtained from
complete as well as stochastic local algorithms.Comment: 5 pages, 4 figures, revised version to app. in PR
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