The asymptotic k-SAT threshold

Abstract

MSC: 60C05, 05C80.

On the max‐cut of sparse random graphs

“…In this paper, however, we are concerned with the average case analysis of the Max-Cut problem. Last decade we have seen a dramatic progress improving our understanding of various randomly generated constraint satisfaction models such as the random K-SAT problem, the random XOR-SAT problem, proper coloring of a random graph, independence ratio of a random graph, and many related problems [8,9,20]. These problems broadly fall into the class of so-called anti-ferromagnetic spin glass models, borrowing a terminology from statistical physics.…”

“…First we claim that W (x, 1/4) = 2w(x) for any x satisfying (5). For β = 1/4, it is easy to see from (7) and (8) that θ 1 = θ 2 and then t = x/2 follows from (9). Hence, it is the same computation scenario as the special setting in the beginning of Section 4.3, then W (x, 1/4) =2 log 2 − 4x 2 + log P(2θ(x), 1, 4x)…”

“…Lemma 1.2 For every x satisfying (5) and β ∈ (0, 1/2), the system (7), (8), and (9) has a unique solution.…”

“…Finally, we numerically compute x l in (13) based on the bisection method, in which for a given x satisfying (5) we use the command "FindRoot" in Mathematica to search for a solution to the equation system (7)(8)(9) and L(x, β, t, θ 1 , θ 2 ) = 2w(x) inside the region β ∈ [10 −10 , 1/4 − 10 −10 ]. If the search succeeds, we set x as an upper bound of x l , otherwise we set x as a lower bound of x l .…”

Inside the clustering window for random linear equations

“…They explain many phenomena, most notably why some random CSP's are algorithmically very challenging (eg, [1,18]). Intuition gained from these hypotheses has led to some very impressive heuristics (eg, Survey Propogation [4,30,32], and the best of the random r-SAT algorithms whose performance has been rigorously proven [10]), and some remarkably tight rigorous bounds on various satisfiability thresholds [11][12][13][14]. Ding, Sly and Sun recently used an approach outlined by these hypotheses to prove the k-SAT conjecture [19], with a determination of the k-SAT satisfiability threshold for all large k. It is clear that, in order to approach many of the outstanding challenges around random CSP's, we need to understand clustering.…”