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Biased random k-SAT
Random Structures and Algorithms ( IF 0.9 ) Pub Date : 2021-02-16 , DOI: 10.1002/rsa.20996
Joel Larsson 1 , Klas Markström 2
Affiliation  

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—that is, variables occur negated w.p. urn:x-wiley:rsa:media:rsa20996:rsa20996-math-0001 and positive otherwise—and study how the satisfiability threshold depends on p. For urn:x-wiley:rsa:media:rsa20996:rsa20996-math-0002 this model breaks many of the symmetries of the original random k-SAT problem, for example, the distribution of satisfying assignments in the Boolean cube is no longer uniform. For any fixed k, we find the asymptotics of the threshold as p approaches 0 or urn:x-wiley:rsa:media:rsa20996:rsa20996-math-0003. The former confirms earlier predictions based on numerical studies and heuristic methods from statistical physics.

中文翻译:

有偏随机 k-SAT

基本的随机k- SAT 问题是:给定一组n 个布尔变量,从我们变量的所有此类子句集合中随机均匀挑选m个大小为k 的子句,这些子句的合取是否可满足?在这里,我们考虑这个问题的一个变体,其中对出现正值的变量有偏见——也就是说,变量出现负 wpurn:x-wiley:rsa:media:rsa20996:rsa20996-math-0001和正反例——并研究可满足性阈值如何取决于p。因为urn:x-wiley:rsa:media:rsa20996:rsa20996-math-0002这个模型打破了原始随机k- SAT 问题的许多对称性,例如,布尔立方体中满足分配的分布不再是均匀的。对于任何固定k,当p接近 0 或 时,我们找到阈值的渐近线urn:x-wiley:rsa:media:rsa20996:rsa20996-math-0003。前者证实了早期基于数值研究和统计物理学启发式方法的预测。
更新日期:2021-02-16
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