MohammadTaghi Hajiaghayi, Gregory B. Sorkin
We prove that a random 3-SAT instance with clause-to-variable density less than 3.52 is satisfiable with high probability. The proof comes through an algorithm which selects (and sets) a variable depending on its degree and that of its complement.
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We consider "unconstrained" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$ variables, each equation containing $k \geq 3$ variables, and also consider a "constrained" model where every variable appears in at least two equations. Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiability of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT. We show that $m/n=1$ remains a sharp threshold for satisfiabilit…
The satisfiability threshold of random linear equations over finite commutative rings
本文研究有限交换环上随机线性方程的可满足性阈值，精确刻画亚线性及线性情形，发现主环下阈值与环无关，非主环则依赖环及其他参数。
The satisfiability threshold and solution space of random uniquely extendable constraint satisfaction problems
本文研究了随机唯一可扩展约束满足问题的可满足性阈值，证明了在特定条件下该阈值与经典k-XORSAT阈值一致。
The Satisfiability Threshold for K-XOR Games
本文研究 $K$-XORGAME 系统的可满足性阈值，证明其 $m/n$ 阈值存在且等于 $K$-XORSAT 系统的阈值。
Let $A$ be a random $m\times n$ matrix over the finite field $F_q$ with precisely $k$ non-zero entries per row and let $y\in F_q^m$ be a random vector chosen independently of $A$. We identify the threshold $m/n$ up to which the linear system $A x=y$ has a solution with high probability and analyse the geometry of the set of solutions. In the special case $q=2$, known as the random $k$-XORSAT problem, the threshold was determined by [Dubois and Mandler 2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof technique was subsequently extended to the cases $q=3,4$ [Falke and Goe…
We consider "unconstrained" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$ variables, each equation containing $k \ge 3$ variables, and also consider a "constrained" model where every variable appears in at least two equations. Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiability of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT. We show that $m/n=1$ remains a sharp threshold for satisfiability…
Partly on the basis of heuristic arguments from physics it has been suggested that the performance of certain types of algorithms on random $k$-SAT formulas is linked to phase transitions that affect the geometry of the set of satisfying assignments. But beyond intuition there has been scant rigorous evidence that "practical" algorithms are affected by these phase transitions. In this paper we prove that \walksat, a popular randomised satisfiability algorithm, fails on random $k$-SAT formulas not very far above clause/variable density where the set of satisfying assignments shatters into tiny,…
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.
We compute the probability of satisfiability of a class of random Horn-SAT formulae, motivated by a connection with the nonemptiness problem of finite tree automata. In particular, when the maximum clause length is 3, this model displays a curve in its parameter space along which the probability of satisfiability is discontinuous, ending in a second-order phase transition where it becomes continuous. This is the first case in which a phase transition of this type has been rigorously established for a random constraint satisfaction problem.
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 …
Satisfiability Thresholds for Regular Occupation Problems
本文基于密度演化方程，研究正则约束满足问题中的占据问题，并严格确定了所有k取值下2-in-k占据问题的可满足性阈值。
Using Maple, we implement a SAT solver based on the principle of inclusion-exclusion and the Bonferroni inequalities. Using randomly generated input, we investigate the performance of our solver as a function of the number of variables and number of clauses. We also test it against Maple's built-in tautology procedure. Finally, we implement the Lovász local lemma with Maple and discuss its applicability to SAT.
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