用于求解正则(3,4)-SAT实例集的修正警示传播算法

利用极小不可满足公式的临界特性,可以将任意的一个3-CNF公式多项式时间归约转换为一个正则(3,4)-CNF公式,从而得到一个保留NP完全性的正则(3,4)-SAT问题。警示传播算法(Warning Propagation,WP)在归约转换后的正则(3,4)-SAT实例集上高概率收敛,但在任意一个实例上都无法判断公式的可满足性,因此算法求解失效。对于一个归约转换后的正则(3,4)-CNF公式,每一变元出现的正负次数之差具有趋于稳定的结构特征,基于该特征,提出基于变元正负出现次数规则的WP算法来求解归约转换后的正则(3,4)-SAT实例。实验结果表明,修正的WP算法对正则公式的可满足性判定有效,从而可以利用公式的正则性特征进一步研究WP算法的收敛性特征条件。     

一个正则NP-完全问题及其不可近似性

通过一个适当的归约变换,可以将一个CNF (conjunctive normal form)公式变换为另一个具有某种特殊结构或性质的公式,使两者具有相同的可满足性.带有正则结构的CNF公式的因子图在图论中具有某些良好的性质和结果,可以用于研究公式的可满足性和计算复杂性.极小不可满足公式具有一个临界特征,公式本身不可满足,从原始公式中删去任意一个子句后得到的公式可满足.借助此临界特性,给出了一个从3-CNF公式到正则(3,4)-CNF公式的多项式归约转换.这里,正则(3,4)-CNF公式是指公式中每个子句的长度恰为3,每个变元出现的次数恰为4.因此,正则(3,4)-SAT问题是一个NP-完全问题,并且MAX(3,4)-SAT是不可近似问题.     

正则3-SAT问题的相变现象

通过对3-CNF公式加以限制,要求其中每个变元出现的次数相同,引出正则3-SAT问题。进一步,通过对两种子句产生机制形成的(3,s)-CNF公式进行可满足性观察,发现在规模较小的情况下,正则3-CNF公式比非正则3-CNF公式更容易满足。从而推测与非正则3-SAT问题相比,正则3-SAT问题的相变点有偏移现象。最后,从变元自由度的角度对这一现象给出了定性解释。     

极小不可满足公式在多项式归约中的应用

合取范式(CNF)公式F是极小不可满足的,如果F不可满足,并且从F中删去任意一个子句后得到的公式可满足,(r,s)-CNF是限制CNF公式中每个子句恰有r个不同的文字,且每个变元出现的次数不超过s次的公式类,对应的满足性问题(r,s)-SAT指实例公式限制于(r,s)-CNF.对于正整数r≥3,有一个临界函数f(r),使得(r,f(r))-CNF中的公式都是可满足的,而(r,f(r)+1)-SAT却是NP-完全的.函数f是否可计算是一个开问题,除了知道f(3)=3,f(4)=4外,只能估计f(r)的界.描述了极小不可满足公式在CNF公式类之间转换中的作用.为使转换过程中引入较少的新变元,给出了CNF公式到3-CNF公式的一种新的转换方法,对于长度为l(＞3)的子句,仅需引入|l/2|个新变元.并且,给出了CNF到(r,s)-CNF公式转换以及(r,s)-CNF中不可满足公式构造的原理和方法.     

SLS算法求解平衡正则(k,2r)-CNF公式

可满足性问题的求解算法和结构性质研究是计算机科学中重要问题之一，为寻求某些CNF公式子类问题有效算法或算法改进途径，对公式的结构加以某些限制，其中限定子句长度为恒定常数和变元出现次数是常见的处理方式。研究具有正则结构且每个变元正负出现均衡的结构化公式的可满足性问题求解，其随机生成模型的构建及随机实验测试有助于观察解分布状况。并且，随机局部搜索算法在求解具有一定规则结构CNF公式实例中具有良好效率。本文集中研究平衡正则(k, 2r)-CNF公式的求解问题，即限制每个子句的长度为k，每个变元出现的次数为偶数2r,并且每个变元正负出现的次数在相等情况下的可满足性问题求解。给出BR(n,  k, 2r)模型，以此模型来生成具有特殊结构的平衡正则(k, 2r)-CNF公式实例，利用随机局部搜索算法求解问题。通过限制初始指派的0文字和1文字各占一半且均匀生成，以WalkSAT算法和NSAT算法做实验对比，发现对于平衡正则(k, 2r)-CNF公式，实例具有明显效率。     

MAX-k-SAT的PTAS归约等价性

通过构造适当的极小不可满足公式,利用子句拼接技术,引入了一个一般化的从k-CNF公式(k≥3)到3-CNF公式之间的归约转换。基于该转换,给出了一个真值指派的转换算法,并证明了MAX-k-SAT与MAX-3-SAT是PTAS归约等价的。因此,对于k,t≥3,MAX-k-SAT与MAX-t-SAT是PTAS归约等价的。     

正则(3,4)-CNF公式的社区结构

通过构造适当的极小不可满足公式以实现在多项式时间内将3-CNF公式归约转换为一个正则(3,4)-CNF公式,转换后的公式与原公式具有相同的可满足性,同时公式的结构也发生相应的变化。图的社区结构反映了图的模块特性,文中将CNF公式转化为相应的图,研究公式图的模块特性与公式某些性质之间的关系。将归约前后的两类公式转换为相应的图并研究其模块特性,发现转换后得到的正则(3,4)-CNF公式具有较高的模块度。此外,在使用DPLL(Davis Putnam Logemann Loveland)算法求解CNF公式的过程中,发生冲突时利用冲突驱动子句学习策略,得到一个学习子句并将其添加到原公式中,使得原公式的模块度降低。研究发现:将DPLL算法与冲突驱动子句学习策略结合应用到正则(3,4)-CNF公式时,其学习子句所包含的绝大部分变元位于不同的社区中。     

多文字可满足SAT问题的相变点上界

可满足(SAT)问题是指：是否存在一组布尔变元赋值,使得随机合取范式(CNF)公式中每个子句至少有1个文字为真。多文字可满足SAT问题是指：是否存在一组布尔变元赋值,使得随机CNF公式中每个子句至少有2个文字为真。此问题仍然是一个NP难问题。定义约束密度α为CNF公式子句数与变元数之比,对该问题的相变点上界α^*进行了研究。如果α>α^*,则多文字可满足SAT问题高概率不可满足。通过一阶矩一个简单的推断,可以证明α^*=-ln 2/ln(1-(k+1)/2^k),当k=3时,α^*=1。利用Kirousis等人的局部最大值技术,提升了多文字可满足3-SAT问题的相变点上界α^*=0.7193。最后,选择了大量数据进行实验验证,结果表明,理论结果与实验结果相吻合。     

取定s的严格d-正则随机(3,2s)-SAT问题的可满足临界

3-CNF公式的随机难解实例生成对于揭示3-SAT问题的难解实质和设计满足性测试的有效算法有着重要意义.对于整数k>2和s>0,如果在一个k-CNF公式中每个变量正负出现次数均为s,则称该公式是严格正则（k,2s）-CNF公式.受严格正则（k,2s）-CNF公式的结构特征启发,提出每个变量正负出现次数之差的绝对值均为d的严格d-正则（k,2s）-CNF公式,并使用新提出的SDRRK2S模型生成严格d-正则随机（k,2s）-CNF公式.取定整数5<s<11,模拟实验显示,严格d-正则随机（3,2s）-SAT问题存在SAT-UNSAT相变现象和HARD-EASY相变现象.因此,立足于3-CNF公式的随机难解实例生成,研究了严格d-正则随机（3,2s）-SAT问题在s取定时的可满足临界.通过构造一个特殊随机实验和使用一阶矩方法,得到了严格d-正则随机（3,2s）-SAT问题在s取定时可满足临界值的一个下界.模拟实验结果验证了理论证明所得下界的正确性.     

基于寻找可满足2SAT子问题的SAT算法*

可满足问题（SAT）是一个NP-Hard问题。提出了一种求解SAT的新算法（FFSAT）。该算法将SAT问题转换为寻找一个可满足的2-SAT子问题。SAT问题虽然是NP完全问题,但是当所有子句长度不大于2时,SAT问题可以在线性时间求解。使用2-SAT算法-BinSat求解2-SAT子问题,当它不满足时,根据赋值选择新的2-SAT子问题。实验结果表明,采用本算法的结果优于UnitWalk。     

A Probabilistic Algorithm for k -SAT Based on Limited Local Search and Restart

Abstract. A simple probabilistic algorithm for solving the NP-complete problem k -SAT is reconsidered. This algorithm follows a well-known local-search paradigm: randomly guess an initial assignment and then, guided by those clauses that are not satisfied, by successively choosing a random literal from such a clause and changing the corresponding truth value, try to find a satisfying assignment. Papadimitriou [11] introduced this random approach and applied it to the case of 2-SAT, obtaining an expected O(n ² ) time bound. The novelty here is to restart the algorithm after 3n unsuccessful steps of local search. The analysis shows that for any satisfiable k -CNF formula with n variables the expected number of repetitions until a satisfying assignment is found this way is (2⋅ (k-1)/ k) ⁿ . Thus, for 3-SAT the algorithm presented here has a complexity which is within a polynomial factor of (\frac 4 3 ) ⁿ . This is the fastest and also the simplest among those algorithms known up to date for 3-SAT achieving an o(2 ⁿ ) time bound. Also, the analysis is quite simple compared with other such algorithms considered before.     

随机正则(k,r)-SAT问题的可满足临界

研究k-SAT问题实例中每个变元恰好出现r=2s次,且每个变元对应的正、负文字都出现s次的严格随机正则（k,r）-SAT问题.通过构造一个特殊的独立随机实验,结合一阶矩方法,给出了严格随机正则（k,r）-SAT问题可满足临界值的上界.由于严格正则情形与正则情形的可满足临界值近似相等,因此得到了随机正则（k,r）-SAT问题可满足临界值的新上界.该上界不仅小于当前已有的随机正则（k,r）-SAT问题的可满足临界值上界,而且还小于一般的随机k-SAT问题的可满足临界值.因此,这也从理论上解释了在相变点处的随机正则（k,r）-SAT问题实例通常比在相应相变点处同规模的随机k-SAT问题实例更难满足的原因.最后,数值分析结果验证了所给上界的正确性.     

Generalizations of matched CNF formulas

A CNF formula is called matched if its associated bipartite graph (whose vertices are clauses and variables) has a matching that covers all clauses. Matched CNF formulas are satisfiable and can be recognized efficiently by matching algorithms. We generalize this concept and cover clauses by collections of bicliques (complete bipartite graphs). It turns out that such generalization indeed gives rise to larger classes of satisfiable CNF formulas which we term biclique satisfiable. We show, however, that the recognition of biclique satisfiable CNF formulas is NP-complete, and remains NP-hard if the size of bicliques is bounded. A satisfiable CNF formula is called var-satisfiable if it remains satisfiable under arbitrary replacement of literals by their complements. Var-satisfiable CNF formulas can be viewed as the best possible generalization of matched CNF formulas as every matched CNF formula and every biclique satisfiable CNF formula is var-satisfiable. We show that recognition of var-satisfiable CNF formulas is ₂ ^P-complete, answering a question posed by Kleine Büning and Zhao.     

A Probabilistic Algorithm for k -SAT Based on Limited Local Search and Restart

A simple probabilistic algorithm for solving the NP-complete problem k -SAT is reconsidered. This algorithm follows a well-known local-search paradigm: randomly guess an initial assignment and then, guided by those clauses that are not satisfied, by successively choosing a random literal from such a clause and changing the corresponding truth value, try to find a satisfying assignment. Papadimitriou [11] introduced this random approach and applied it to the case of 2-SAT, obtaining an expected O(n ² ) time bound. The novelty here is to restart the algorithm after 3n unsuccessful steps of local search. The analysis shows that for any satisfiable k -CNF formula with n variables the expected number of repetitions until a satisfying assignment is found this way is (2? (k-1)/ k) ⁿ . Thus, for 3-SAT the algorithm presented here has a complexity which is within a polynomial factor of (\frac 4 3 ) ⁿ . This is the fastest and also the simplest among those algorithms known up to date for 3-SAT achieving an o(2 ⁿ ) time bound. Also, the analysis is quite simple compared with other such algorithms considered before.     

Generalizations of matched CNF formulas

A CNF formula is called matched if its associated bipartite graph (whose vertices are clauses and variables) has a matching that covers all clauses. Matched CNF formulas are satisfiable and can be recognized efficiently by matching algorithms. We generalize this concept and cover clauses by collections of bicliques (complete bipartite graphs). It turns out that such generalization indeed gives rise to larger classes of satisfiable CNF formulas which we term biclique satisfiable. We show, however, that the recognition of biclique satisfiable CNF formulas is NP-complete, and remains NP-hard if the size of bicliques is bounded. A satisfiable CNF formula is called var-satisfiable if it remains satisfiable under arbitrary replacement of literals by their complements. Var-satisfiable CNF formulas can be viewed as the best possible generalization of matched CNF formulas as every matched CNF formula and every biclique satisfiable CNF formula is var-satisfiable. We show that recognition of var-satisfiable CNF formulas is _P ² P-complete, answering a question posed by Kleine Büning and Zhao.     

Exact Algorithms for MAX-SAT

The maximum satisfiability problem (MAX-SAT) is stated as follows: Given a boolean formula in CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-SAT is MAX-SNP-complete and received much attention recently. One of the challenges posed by Alber, Gramm and Niedermeier in a recent survey paper asks: Can MAX-SAT be solved in less than 2ⁿ “steps”? Here, n is the number of distinct variables in the formula and a step may take polynomial time of the input. We answered this challenge positively by showing that a popular algorithm based on branch-and-bound is bounded by O(2ⁿ) in time, where n is the maximum number of occurrences of any variable in the input.When the input formula is in 2-CNF, that is, each clause has at most two literals, MAX-SAT becomes MAX-2-SAT and the decision version of MAX-2-SAT is still NP-complete. The best bound of the known algorithms for MAX-2-SAT is O(m2^m/5), where m is the number of clauses. We propose an efficient decision algorithm for MAX-2-SAT whose time complexity is bound by O(n2ⁿ). This result is substantially better than the previously known results. Experimental results also show that our algorithm outperforms any algorithm we know on MAX-2-SAT.     

Almost 2-SAT is fixed-parameter tractable

We consider the following problem. Given a 2-cnf formula, is it possible to remove at most k clauses so that the resulting 2-cnf formula is satisfiable? This problem is known to different research communities in theoretical computer science under the names Almost 2-SAT, All-but-k 2-SAT, 2-cnf deletion, and 2-SAT deletion. The status of the fixed-parameter tractability of this problem is a long-standing open question in the area of parameterized complexity. We resolve this open question by proposing an algorithm that solves this problem in O(k¹⁵×k×m³) time showing that this problem is fixed-parameter tractable.     

An interpolation problem associated with H-optimal design in systems with distributed input lags

A variety of H^∞ optimal design problems reduce to interpolation of compressed multiplication operators, f(s) → π_k(w(s)f(s)), where w(s) is a given rational function and the subspace K is of the form K=H² φ(s)H². Here we consider φ(s) = (1-e^α-5)/(s - α), which stands for a distributed delay in a system's input. The interpolation scheme we develop, adapts to a broader class of distributed lags, namely, those determined by transfer functions of the form B(e^−s)/b(s), where B(z) and b(s) are polynomials and b(s) = 0 implies B(e^−s) = 0.