For fixed graphs $F$ and $H$, a graph $G\subseteq F$ is $H$-saturated if there no copy of $H$ in $G$, but for any edge $e\in E(F)\setminus E(G)$, $G+e$. The saturation number $F$, denoted $sat(F,H)$, the minimum edges an subgraph $F$. In this paper, we study numbers $tK_{l,l,l}$ complete tripartite $K_{n_1,n_2,n_3}$. $t\ge 1$, $l\ge 1$ $n_1,n_2$ $n_3$ sufficiently large, determine $sat(K_{n_1,n...

Of fundamental importance for SAT solving is the translation to CNF. One of the basic tasks is to find metrics for determining what are “good” translations, i.e., what makes the resulting SAT problem easy to solve. We introduce a new measure hd(F ), the “hardness”, for formulas F in conjunctive normal form (i.e., clause-sets). hd(F ) for unsatisfiable clause-sets has been studied in [10,12]. Ho...

An (encoded) decision problem over Σ is a pair (E,F ) where E=words that encode instances of the problem, F=words to be accepted. We use strings in a technical sense, borrowed from Computability. With any NP problem (E,F ) we associate a set of strings |LogE(F )| called the reduced logogram of F relative to E, which conveys structural information on E, F , and how F is embedded in E. We define ...

The (k, s)-SAT problem is the satisfiability problem restricted to instances where each clause has exactly k literals and every variable occurs at most s times. It is known that there exists a function f such that for s ≤ f(k) all (k, s)-SAT instances are satisfiable, but (k, f(k)+1)-SAT is already NP-complete (k ≥ 3). We prove that f(k) = O(2 · log k/k), improving upon the best know upper boun...

Of fundamental importance for SAT solving is the translation to CNF. One of the basic tasks is to find metrics for determining what are “good” translations, i.e., what makes the resulting SAT problem easy to solve. We introduce a new measure hd(F ), the “hardness”, for formulas F in conjunctive normal form (i.e., clause-sets). hd(F ) for unsatisfiable clause-sets has been studied in [6,8]. Howe...

We present two simple algorithms for SAT and prove upper bounds on their running time. Given a Boolean formula F in conjunctive normal form, the first algorithm finds a satisfying assignment for F (if any) by repeating the following: Choose an assignment A at random and search for a satisfying assignment inside a Hamming ball around A (the radius of the ball depends on F ). We show that this al...

Of fundamental importance for SAT solving is the translation to CNF. One of the basic tasks is to find metrics for determining what are “good” translations, i.e., what makes the resulting SAT problem easy to solve. We introduce a new measure hd(F ), the “hardness”, for formulas F in conjunctive normal form (i.e., clause-sets). hd(F ) for unsatisfiable clause-sets has been studied in [12,14]. Ho...

Background and objective: Magnetic resonance imaging (MRI) is the most sensitive technique to detect multiple sclerosis (MS) plaques in central nervous system. In some cases, the patients who were suspected to MS, Whereas MRI images are normal, but whether patients don’t have MS plaques or MRI images are not enough optimized enough in order to show MS plaques? The aim of the current study is ...

The research community on complex networks has developed techniques of analysis and algorithms that can be used by the SAT community to improve our knowledge about the structure of industrial SAT instances. It is often argued that modern SAT solvers are able to exploit this hidden structure, without a precise definition of this notion. In this paper, we show that most industrial SAT instances h...