11 research outputs found
Random k-SAT and the Power of Two Choices
We study an Achlioptas-process version of the random k-SAT process: a bounded
number of k-clauses are drawn uniformly at random at each step, and exactly one
added to the growing formula according to a particular rule. We prove the
existence of a rule that shifts the satisfiability threshold. This extends a
well-studied area of probabilistic combinatorics (Achlioptas processes) to
random CSP's. In particular, while a rule to delay the 2-SAT threshold was
known previously, this is the first proof of a rule to shift the threshold of
k-SAT for k >= 3.
We then propose a gap decision problem based upon this semi-random model. The
aim of the problem is to investigate the hardness of the random k-SAT decision
problem, as opposed to the problem of finding an assignment or certificate of
unsatisfiability. Finally, we discuss connections to the study of Achlioptas
random graph processes.Comment: 13 page
Convergence of Achlioptas processes via differential equations with unique solutions
In Achlioptas processes, starting from an empty graph, in each step two
potential edges are chosen uniformly at random, and using some rule one of them
is selected and added to the evolving graph. The evolution of the rescaled size
of the largest component in such variations of the Erd\H{o}s--R\'enyi random
graph process has recently received considerable attention, in particular for
for Bollob\'as's `product rule'. In this paper we establish the following
result for rules such as the product rule: the limit of the rescaled size of
the `giant' component exists and is continuous provided that a certain system
of differential equations has a unique solution. In fact, our result applies to
a very large class of Achlioptas-like processes.
Our proof relies on a general idea which relates the evolution of stochastic
processes to an associated system of differential equations. Provided that the
latter has a unique solution, our approach shows that certain discrete
quantities converge (after appropriate rescaling) to this solution.Comment: 18 pages. Revised and expanded versio
The augmented multiplicative coalescent and critical dynamic random graph models
Random graph models with limited choice have been studied extensively with
the goal of understanding the mechanism of the emergence of the giant
component. One of the standard models are the Achlioptas random graph processes
on a fixed set of vertices. Here at each step, one chooses two edges
uniformly at random and then decides which one to add to the existing
configuration according to some criterion. An important class of such rules are
the bounded-size rules where for a fixed , all components of size
greater than are treated equally. While a great deal of work has gone into
analyzing the subcritical and supercritical regimes, the nature of the critical
scaling window, the size and complexity (deviation from trees) of the
components in the critical regime and nature of the merging dynamics has not
been well understood. In this work we study such questions for general
bounded-size rules. Our first main contribution is the construction of an
extension of Aldous's standard multiplicative coalescent process which
describes the asymptotic evolution of the vector of sizes and surplus of all
components. We show that this process, referred to as the standard augmented
multiplicative coalescent (AMC) is `nearly' Feller with a suitable topology on
the state space. Our second main result proves the convergence of suitably
scaled component size and surplus vector, for any bounded-size rule, to the
standard AMC. The key ingredients here are a precise analysis of the asymptotic
behavior of various susceptibility functions near criticality and certain
bounds from [8], on the size of the largest component in the barely subcritical
regime.Comment: 49 page