7 research outputs found
Computational Complexity and Phase Transitions
Phase transitions in combinatorial problems have recently been shown to be
useful in locating "hard" instances of combinatorial problems. The connection
between computational complexity and the existence of phase transitions has
been addressed in Statistical Mechanics and Artificial Intelligence, but not
studied rigorously.
We take a step in this direction by investigating the existence of sharp
thresholds for the class of generalized satisfiability problems defined by
Schaefer. In the case when all constraints are clauses we give a complete
characterization of such problems that have a sharp threshold.
While NP-completeness does not imply (even in this restricted case) the
existence of a sharp threshold, it "almost implies" this, since clausal
generalized satisfiability problems that lack a sharp threshold are either
1. polynomial time solvable, or
2. predicted, with success probability lower bounded by some positive
constant by across all the probability range, by a single, trivial procedure.Comment: A (slightly) revised version of the paper submitted to the 15th IEEE
Conference on Computational Complexit
Measuring the fluctuation-dissipation ratio in glassy systems with no perturbing field
A method is presented for measuring the integrated response in Ising spin
system without applying any perturbing field. Large-scale simulations are
performed in order to show how the method works. Very precise measurements of
the fluctuation-dissipation ratio are presented for 3 different Ising models:
the 2-dimensional ferromagnetic model, the mean-field diluted 3-spin model, and
the 3-dimensional Edwards-Anderson model.Comment: 4 pages, 4 figure
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
q-Overlaps in the Random Exact Cover Problem
We prove upper and lower bounds for the threshold of the q-overlap-k-Exact
cover problem.
These results are motivated by the one-step replica symmetry breaking
approach of Statistical Physics, and the hope of using an approach based on
that of Mezard et al. (2005) to rigorously prove that for some values of the
order parameter the overlap distribution of k-Exact Cover has discontinuous
support.Comment: In Proceedings FROM 2023, arXiv:2309.1295