关于随机可满足过程

The random 2-SAT partition function

We show that throughout the satisfiable phase the normalised number of satisfying assignments of a random $2$-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. The proof is based on showing that the Belief Propagation algorithm renders the correct marginal probability that a variable is set to `true' under a uniformly random satisfying assignment.

随机MAX SAT、随机MAX CUT及其相变

Random MAX SAT, Random MAX CUT, and Their Phase Transitions

给定一个由$n$变量和$\cn$随机子句组成的2-SAT公式$F$，一次变量赋值可满足的最大子句数是多少？我们将答案从$(3/4)cn$和$cn$的平凡界限中绑定出来。我们证明了对于$c<1$，可满足的子句的期望数目是$\cn-Θ(1/n)$ ;对于大的$c$，它是$((3/4)c + Θ(\sqrt{c}))n$ ;对于$c = 1+\eps$，它至少是$(1+\eps-O(\eps^3))n$，至多是$(1+\eps-Ω(\eps^3/\ln \eps))n$ ;并且在"缩放窗口“$c= 1+Θ(n^{-1/3})$中，它是$cn-Θ(1)$。特别地，正如决策问题经历相变一样，我们的优化问题也在相同的临界值$c=1$处经历相变。几乎所有的结果都是在没有参考决策2-SAT的类似命题的情况下建立的，作为副产品，我们重现了许多结果，包括许多关于2-SAT缩放窗口的已知内容。我们认为“在线”版本的MAX-2-SAT，并表明，对于一个版本，明显的贪婪算法是最优的。我们只能将最简单的MAX-2-SAT结果扩展到MAX-k-SAT，但我们猜想一个类似于民间传说可满足性阈值猜想的“MAX-k-SAT限制函数”，但即使对于$k=2$也是开放的。这两个猜想都没有直接暗示另一个，但很自然地进一步推测它们之间的联系。最后，对于随机MAXCUT（稀疏随机图中最大割的大小），我们证明了类似的结果。

The Power of Choice for Random Satisfiability

We consider Achlioptas processes for k-SAT formulas. We create a semi-random formula with n variables and m clauses, where each clause is a choice, made on-line, between two or more uniformly random clauses. Our goal is to delay the satisfiability/unsatisfiability transition, keeping the formula satisfiable up to densities m/n beyond the satisfiability threshold alpha_k for random k-SAT. We show that three choices suffice to raise the threshold for any k >= 3, and that two choices suffice for all 3 <= k <= 25. We also show that two choices suffice to lower the threshold for all k >= 3, making …

Biased random k-SAT

The basic random $k$-SAT problem is: Given a set of $n$ Boolean variables, and $m$ clauses of size $k$ picked uniformly at random from the set of all such clauses on our variables, is the conjunction of these clauses satisfiable? Here we consider a variation of this problem where there is a bias towards variables occurring positive -- i.e. variables occur negated w.p. $0<p< \frac{1}{2}$ and positive otherwise -- and study how the satisfiability threshold depends on $p$. For $p<\frac{1}{2}$ this model breaks many of the symmetries of the original random $k$-SAT problem, e.g. the distribution of…

关于可满足k-CSP的近似性：VI

On Approximability of Satisfiable $k$-CSPs: VI

本文针对成对连通分布上的三元相关性证明了局部与全局逆定理，并利用密度增量论证将其应用于性质测试与加法组合学，首次为有限域上的受限3-AP问题给出了合理界限。

On the satisfiability of random regular signed SAT formulas

Regular signed SAT is a variant of the well-known satisfiability problem in which the variables can take values in a fixed set V \subset [0,1], and the `literals' have the form "x \le a" or "x \ge a". We answer some open question regarding random regular signed k-SAT formulas: the probability that a random formula is satisfiable increases with |V|; there is a constant upper bound on the ratio m/n of clauses m over variables n, beyond which a random formula is asypmtotically almost never satisfied; for k=2 and V=[0,1], there is a phase transition at m/n=2.

关于可满足k-CSP的近似性：IV

On Approximability of Satisfiable k-CSPs: IV

本文证明了满足温和连通性条件下一般三元相关性的稳定性定理，刻画了高相关函数必来源于常数大小阿贝尔群的结构，并给出改进的直积定理和限制逆定理，应用于PCP与组合数学。

Structure of random r-SAT below the pure literal threshold

It is well known that there is a sharp density threshold for a random $r$-SAT formula to be satisfiable, and a similar, smaller, threshold for it to be satisfied by the pure literal rule. Also, above the satisfiability threshold, where a random formula is with high probability (whp) unsatisfiable, the unsatisfiability is whp due to a large "minimal unsatisfiable subformula" (MUF). By contrast, we show that for the (rare) unsatisfiable formulae below the pure literal threshold, the unsatisfiability is whp due to a unique MUF with smallest possible "excess", failing this whp due to a unique MUF …

Analysing Survey Propagation Guided Decimation on Random Formulas

Let $\varPhi$ be a uniformly distributed random $k$-SAT formula with $n$ variables and $m$ clauses. For clauses/variables ratio $m/n \leq r_{k\text{-SAT}} \sim 2^k\ln2$ the formula $\varPhi$ is satisfiable with high probability. However, no efficient algorithm is known to provably find a satisfying assignment beyond $m/n \sim 2k \ln(k)/k$ with a non-vanishing probability. Non-rigorous statistical mechanics work on $k$-CNF led to the development of a new efficient "message passing algorithm" called \emph{Survey Propagation Guided Decimation} [Mézard et al., Science 2002]. Experiments conducted …

On belief propagation guided decimation for random k-SAT

Let F be a uniformly distributed random k-SAT formula with n variables and m clauses. Non-constructive arguments show that F is satisfiable for clause/variable ratios m/n< r(k)~2^k ln 2 with high probability. Yet no efficient algorithm is know to find a satisfying assignment for densities as low as m/n r(k).ln(k)/k with a non-vanishing probability. In fact, the density m/n r(k).ln(k)/k seems to form a barrier for a broad class of local search algorithms. One of the very few algorithms that plausibly seemed capable of breaking this barrier is a message passing algorithm called Belief Propagatio…

非齐次随机2-SAT

Inhomogeneous random 2-SAT

本文引入非齐次随机2-SAT变体，通过变量类型和核函数$W$定义子句出现概率，确定参数$ρ^*(W)$为积分算子谱半径，证明其小于1时实例渐近可满足、大于1时不可满足，扩展了齐次随机2-SAT的经典阈值结果。

A countdown process, with application to the rank of random matrices over $\mathbb F_q(n)$

Motivated by the work of Fulman and Goldstein, comparing the distribution of the corank of random matrices in $\mathbb F_q[n]$ with the limit distribution as $n \to \infty$, we define a countdown process, driven by independent geometric random variables related to random integer partitions. Analysis of this process leads to sharper bounds on the total variation distance.