X3SAT is Decidable in Time (original) (raw)

A new algorithm for Solving 3-CNF-SAT problem

2017

NP-Complete problems have an important attribute that if one NP-Complete problem can be solved in polynomial time, all NP-Complete problems will have a polynomial solution. The 3-CNF-SAT problem is a NP-Complete problem and the primary method to solve it checks all values of the truth table. This task is of the {\Omega}(2^n) time order. This paper shows that by changing the viewpoint towards the problem, it is possible to know if a 3-CNF-SAT problem is satisfiable in time O(n^10) or not? In this paper, the value of all clauses are considered as false. With this presumption, any of the values inside the truth table can be shown in string form in order to define the set of compatible clauses for each of the strings. So, rather than processing strings, their clauses will be processed implicating that instead of 2^n strings, (O(n^3)) clauses are to be processed; therefore, the time and space complexity of the algorithm would be polynomial.

On the complexity of SAT

… of Computer Science, 1999. 40th Annual …, 1999

We show that non-deterministic time N T IM E(n) is not contained in deterministic time n 2−ǫ and poly-logarithmic space, for any ǫ > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n 2−ǫ ) and polylogarithmic space. A similar result is presented for uniform circuits.

A new bound for 3-satisfiable MaxSat and its algorithmic application

Information and Computation, 2013

Let F be a CNF formula with n variables and m clauses. F is 3-satisfiable if for any 3 clauses in F , there is a truth assignment which satisfies all of them. Lieberherr and Specker (1982) and, later, Yannakakis (1994) proved that in each 3-satisfiable CNF formula at least 2 3 of its clauses can be satisfied by a truth assignment. We improve this result by showing that every 3-satisfiable CNF formula F contains a subset of variables U , such that some truth assignment τ will satisfy at least 2 3 m + 1 3 m U + ρn ′ clauses, where m is the number of clauses of F , m U is the number of clauses of F containing a variable from U , n ′ is the total number of variables in clauses not containing a variable in U , and ρ is a positive absolute constant. Both U and τ can be found in polynomial time. We use our result to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. In 3-S-MAXSAT-AE, we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2 3 m + k clauses, where k is the parameter.

A simplified NP-complete satisfiability problem

Discrete Applied Mathematics, 1984

3-SAT is NP-complete when restricted to instances where each variable appears in at most four clauses. When no variable appears in more than three clauses, 3-SAT is trivial and SAT is NPcomplete. When no variable appears in more than two clauses, SAT may be solved in linear time.

A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and its Algorithmic Application

For a formula FFF in conjunctive normal form (CNF), let rmsat(F){\rm sat}(F)rmsat(F) be the maximum number of clauses of FFF that can be satisfied by a truth assignment, and let mmm be the number of clauses in FFF. It is well-known that for every CNF formula FFF, rmsat(F)gem/2{\rm sat}(F)\ge m/2rmsat(F)gem/2 and the bound is tight when FFF consists of conflicting unit clauses (x)(x)(x) and (barx)(\bar{x})(barx). Since each truth assignment satisfies exactly one clause in each pair of conflicting unit clauses, it is natural to reduce FFF to the unit-conflict free (UCF) form. If FFF is UCF, then Lieberherr and Specker (J. ACM 28(2):411-421, 1981) proved that rmsat(F)gepm{\rm sat}(F)\ge pmrmsat(F)gepm, where p=(sqrt5−1)/2p=(\sqrt{5}-1)/2p=(sqrt51)/2. We introduce another reduction that transforms a UCF CNF formula FFF into a UCF CNF formula F′F'F, which has a complete matching, i.e., there is an injective map from the variables to the clauses, such that each variable maps to a clause containing that variable or its negation. The formula F′F'F is obtained from FFF by deleting some clauses and the variables contained only in the deleted clauses. We prove that rmsat(F)gepm′+(m−m′)+n′(2−3p)/2{\rm sat}(F)\ge pm' + (m-m') + n'(2-3p)/2rmsat(F)gepm+(mm)+n(23p)/2, where n′n'n and m′m'm are the number of variables and clauses in F′F'F, respectively. This improves the Lieberherr-Specker lower bound on rmsat(F){\rm sat}(F)rmsat(F). We show that our new bound has an algorithmic application by considering the following parameterized problem: given a UCF CNF formula FFF decide whether rmsat(F)gepm+k,{\rm sat}(F)\ge p m + k,rmsat(F)gepm+k, where kkk is the parameter. This problem was introduced by Mahajan and Raman (J. Algorithms 31(2):335--354, 1999) who asked whether the problem is fixed-parameter tractable. We use the new bound to show that the problem is fixed-parameter tractable by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most lfloor(7+3sqrt5)krfloor\lfloor (7+3\sqrt{5})k\rfloorlfloor(7+3sqrt5)krfloor variables

On the Complexity of SAT (Revised)

2008

We show 1 that non-deterministic time N T IM E(n) is not contained in deterministic time n √ 2− and polylogarithmic space, for any > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n √ 2− ) and poly-logarithmic space. A similar result is presented for uniform circuits.

The Problem " 3-completeness at One True Literal "

The problem " 3-completeness at One True Literal " is a classical example for an NP problem. Stated shortly it is defined as follows: Are given the set of Boolean variables А=and the set C of 3-element disjunctions. Does such a set of logical values for the variables exist, so that all the disjunctions to be " true " , and in each disjunction one and only one variable is " true ". In the first part of this paper is shown how the problem can be reduced for polynomial time to two problems from the Graph Theory, in particular: vertices coverage with independent cliques or search for maximum independent set of vertices. In the second part the authors propose an algorithm for sequential solving of the disjunctions, based on work with lists which effectiveness strongly depends on the ordering of disjunctions and in many cases works for polynomial time.

The complexity of Unique k-SAT: An Isolation Lemma for k-CNFs

Journal of Computer and System Sciences, 2008

We provide some evidence that Unique k-SAT is as hard to solve as general k-SAT, where k-SAT denotes the satisfiability problem for k-CNFs with at most k literals in each clause and Unique k-SAT is the promise version where the given formula has 0 or 1 solutions. Namely, defining for each k 1, s k = inf{δ 0 | ∃ a O(2 δn)-time randomized algorithm for k-SAT} and, similarly, σ k = inf{δ 0 | ∃ a O(2 δn)-time randomized algorithm for Unique k-SAT}, we show that lim k→∞ s k = lim k→∞ σ k. As a corollary, we prove that, if Unique 3-SAT can be solved in time 2 n for every > 0, then so can k-SAT for all k 3. Our main technical result is an Isolation Lemma for k-CNFs, which shows that a given satisfiable k-CNF can be efficiently probabilistically reduced to a uniquely satisfiable k-CNF, with non-trivial, albeit exponentially small, success probability.

On Satisfiability

Among the NP-hard problems is 3SAT [2, 4], whichasks if there exists a satisfying assignment for theBoolean variables in a conjunctive normal form formula.This paper gives a mapping of 3SAT problemsinto multi-dimensional definite integrals. These integralstake on values greater than 1-# if the Booleanformula is satisfiable and less than 1/2 when it is not.They thus serve as e#ective indicators of the satisfiabilityof Boolean formulas. The number of dimensionsof the integral is the...