# Coupon Collectors, q-Binomial Coefficients and the Unsatisfiability Threshold

@inproceedings{Kaporis2001CouponCQ, title={Coupon Collectors, q-Binomial Coefficients and the Unsatisfiability Threshold}, author={Alexis C. Kaporis and Lefteris M. Kirousis and Yannis C. Stamatiou and Malvina Vamvakari and Michele A. A. Zito}, booktitle={ICTCS}, year={2001} }

The problem of determining the unsatisfiability threshold for random 3-SAT formulas consists in determining the clause to variable ratio that marks the (experimentally observed) abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. An upper bound of 4.506 was announced by Dubois et al. in 1999 but, to the best of our knowledge, no complete…

## 16 Citations

On the satisfiability threshold and clustering of solutions of random 3-SAT formulas

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2008

On the Diameter of the Set of Satisfying Assignments in Random Satisfiable k-CNF Formulas

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 2011

It is shown that for all densities above a density that is slightly above the satisfiability threshold, the diameter is almost surely zero (a very dense satisfiable formula is expected to have only one satisfying assignment).

Planted Random 3 SAT with a Small Fraction of 1-Clauses

- Computer Science
- 2012

This work focuses on the case where the number of clauses satisfied by exactly one literal is small, both in absolute and in relative terms, and presents polynomial time algorithms for the two distribution families.

It's All about the Support: A New Perspective on the Satisfiability Problem

- Computer ScienceJ. Satisf. Boolean Model. Comput.
- 2007

A new algorithm is presented that is effective for random 3CNF formulas in the “hard” near-threshold regime and rigorously proves its effectiveness for a certain distribution over satisfiable k-C NF formulas known as the planted distribution.

Speed and concentration of the covering time for structured coupon collectors

- MathematicsAdvances in Applied Probability
- 2020

The covering time for much more general random variables X is studied; general criteria for T being sharply concentrated around its mean, precise tools to estimate that mean are given, as well as examples where T fails to be concentrated and when structural properties in the distribution of X allow for a very different behaviour of T relative to the symmetric/uniform case.

Belief propagation algorithms for constraint satisfaction problems

- Computer Science
- 2006

This thesis shows that survey propagation, which is the most effective heuristic for random 3-SAT problems with density of clauses close to the conjectured satisfiability threshold, is in fact a belief propagation algorithm, and defines a parameterized distribution on partial assignments, and shows that applying belief propagation to this distribution recovers a known family of algorithms.

A spectral technique for random satisfiable 3CNF formulas

- Computer Science, PhysicsSODA '03
- 2003

It is shown that for any constants 0 ≤ η2,η3 ≤ 1 there is a constant dmin so that for all d ≥ dmin a spectral algorithm similar to the graph coloring algorithm of Alon and Kahale will find a satisfying assignment with high probability for p1 = d/n2, p2 =η2d/n 2, and p3 = η3d/ n2.

Fixed-point logics, descriptive complexity, and random satisfiability

- Computer Science, Mathematics
- 2002

The goal is to identify the boundary between the fragments of UP that collapse to FO and those whose collapse is unlikely or will be hard to settle and establish precise connections between the areas of finite model theory and propositional proof complexity.

Average-case analysis for combinatorial problems

- Mathematics
- 2006

This thesis considers the average case analysis of algorithms, focusing primarily on NP-hard combinatorial optimization problems. It includes a catalog of distributions frequently used in…

Modèles d'urnes et phénomènes de seuils en combinatoire analytique.

- Philosophy
- 2005

Cette these traite de phenomenes de seuils et de modeles d'urnes, en adoptant le point de vue de la combinatoire analytique. On traite ici trois problemes qui illustrent cette approche: la transition…

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