1,400 research outputs found
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
A Framework for Structured Quantum Search
A quantum algorithm for general combinatorial search that uses the underlying
structure of the search space to increase the probability of finding a solution
is presented. This algorithm shows how coherent quantum systems can be matched
to the underlying structure of abstract search spaces, and is analytically
simpler than previous structured search methods. The algorithm is evaluated
empirically with a variety of search problems, and shown to be particularly
effective for searches with many constraints. Furthermore, the algorithm
provides a simple framework for utilizing search heuristics. It also exhibits
the same phase transition in search difficulty as found for sophisticated
classical search methods, indicating it is effectively using the problem
structure.Comment: 18 pages, Latex, 7 figures, further information available at
ftp://parcftp.xerox.com/pub/dynamics/quantum.htm
Approximated Symbolic Computations over Hybrid Automata
Hybrid automata are a natural framework for modeling and analyzing systems
which exhibit a mixed discrete continuous behaviour. However, the standard
operational semantics defined over such models implicitly assume perfect
knowledge of the real systems and infinite precision measurements. Such
assumptions are not only unrealistic, but often lead to the construction of
misleading models. For these reasons we believe that it is necessary to
introduce more flexible semantics able to manage with noise, partial
information, and finite precision instruments. In particular, in this paper we
integrate in a single framework based on approximated semantics different over
and under-approximation techniques for hybrid automata. Our framework allows to
both compare, mix, and generalize such techniques obtaining different
approximated reachability algorithms.Comment: In Proceedings HAS 2013, arXiv:1308.490
Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem
In recent years, there has been much interest in phase transitions of
combinatorial problems. Phase transitions have been successfully used to
analyze combinatorial optimization problems, characterize their typical-case
features and locate the hardest problem instances. In this paper, we study
phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an
NP-hard combinatorial optimization problem that has many real-world
applications. Using random instances of up to 1,500 cities in which intercity
distances are uniformly distributed, we empirically show that many properties
of the problem, including the optimal tour cost and backbone size, experience
sharp transitions as the precision of intercity distances increases across a
critical value. Our experimental results on the costs of the ATSP tours and
assignment problem agree with the theoretical result that the asymptotic cost
of assignment problem is pi ^2 /6 the number of cities goes to infinity. In
addition, we show that the average computational cost of the well-known
branch-and-bound subtour elimination algorithm for the problem also exhibits a
thrashing behavior, transitioning from easy to difficult as the distance
precision increases. These results answer positively an open question regarding
the existence of phase transitions in the ATSP, and provide guidance on how
difficult ATSP problem instances should be generated
Phase coexistence and finite-size scaling in random combinatorial problems
We study an exactly solvable version of the famous random Boolean
satisfiability problem, the so called random XOR-SAT problem. Rare events are
shown to affect the combinatorial ``phase diagram'' leading to a coexistence of
solvable and unsolvable instances of the combinatorial problem in a certain
region of the parameters characterizing the model. Such instances differ by a
non-extensive quantity in the ground state energy of the associated diluted
spin-glass model. We also show that the critical exponent , controlling
the size of the critical window where the probability of having solutions
vanishes, depends on the model parameters, shedding light on the link between
random hyper-graph topology and universality classes. In the case of random
satisfiability, a similar behavior was conjectured to be connected to the onset
of computational intractability.Comment: 10 pages, 5 figures, to appear in J. Phys. A. v2: link to the XOR-SAT
probelm adde
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