Hervé Daudé

An Average-Case Analysis of the Gaussian Algorithm for Lattice Reduction

The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, its probability distribution decays geometrically, and the dynamics are characterized by a conditional invariant measure. The proofs make use of connections between lattice reduction, continued fractions, continuants, and functional operators. Analysis in the discrete model and detailed numerical data are also presented.

Satisfiability threshold for random XOR-CNF formulas

Abstract Various experimental investigations have shown a sharp transition between satisfiability and unsatisfiability of CNF formulas with L clauses over n variables as c=L/n is varied. For 2-SAT it has been shown theoretically that a threshold phenomenon occurs at the critical value c=1. For 3-SAT experimental results show a sharp transition near c=4 but no such threshold phenomenon has already been proved. Noticing that the XOR-SAT problem (in which one uses the ‘exclusive or’ instead of the usual ‘or’) is a special case of satisfiability which is solvable in polynomial time as decision problem as well as counting problem leads to the natural question: is there a satisfiability threshold for XOR-SAT? In this paper, we answer this question in establishing a threshold phenomenon for XOR-SAT, with associated critical value c=1. So, consider randomly generated XOR-CNF formulas F. We prove that F is satisfiable with probability 1−o(1) whenever c 1 as n tends to infinity. Indeed, in following the nice terminology classification given by Erdos and Renyi for random graphs, we obtain much better: we exhibit a probability distribution function that gives a complete understanding of the transition from satisfiability to unsatisfiability for random XOR-SAT formulas.

Generalized satisfiability problems: minimal elements and phase transitions

We develop a probabilistic model on the generalized satisfiability problems defined by Schaefer (in: Proceedings of the 10th STOC, San Diego, CA, USA, Association for Computing Machinery, New York, 1978, pp. 216-226) for which the arity of the constraints is fixed in order to study the associated phase transition. We establish new results on minimal elements associated with such generalized satisfiability problems. These results are the keys of the exploration we conduct on the location and on the nature of the phase transition for generalized satisfiability. We first prove that the phase transition occurs at the same scale for every reasonable problem and we provide lower and upper bounds for the associated critical ratio. Our framework allows one to get these bounds in a uniform way, in particular, we obtain a lower bound proportional to the number of variables for k-SAT without analyzing any algorithm. Finally, we reveal the seed of coarseness for the phase transition of generalized satisfiability: 2-XOR-SAT.

Combinatorial sharpness criterion and phase transition classification for random CSPs

We investigate the nature of the phase transition (sharp or coarse) for random constraint satisfaction problems. We first give a sharp threshold criterion specified for CSPs, which is derived from Friedgut-Bourgains one. Thus, we get a complete and precise classification of the nature of the threshold for symmetric Boolean CSPs. In particular we show that it is governed by two local properties strongly related to the problems 1-SAT and 2-XOR-SAT.

Approximating the Satisfiability Threshold for Random k -XOR-formulas

In this paper we study random linear systems with

Pairs of SAT-assignments in random Boolean formulæ

k > 3

Smooth and sharp thresholds for random {k}-XOR-CNF satisfiability

variables per equation over the finite field GF(2), or equivalently k-XOR-CNF formulas. In a previous paper Creignou and Daude proved that there exists a phase transition exhibiting a sharp threshold, for the consistency (satisfiability) of such systems (formulas). The control parameter for this transition is the ratio of the number of equations to the number of variables, and the scale for which the transition occurs remains somewhat elusive. In this paper we establish, for any

An analysis of the Gaussian algorithm for lattice reduction

k > 3

The SAT–UNSAT transition for random constraint satisfaction problems

, non-trivial lower and upper estimates of the value of the control ratio for which the phase transition occurs. For

Phase transition for random quantified XOR-formulas

k=3