Uniquely Satisfiable d-Regular (k,s)-SAT Instances.

Zufeng Fu,Daoyun Xu

doi:10.3390/e22050569

Abstract

Unique k-SAT is the promised version of k-SAT where the given formula has 0 or 1 solution and is proved to be as difficult as the general k-SAT. For any , and , a parsimonious reduction from k-CNF to d-regular (k,s)-CNF is given. Here regular (k,s)-CNF is a subclass of CNF, where each clause of the formula has exactly k distinct variables, and each variable occurs in exactly s clauses. A d-regular (k,s)-CNF formula is a regular (k,s)-CNF formula, in which the absolute value of the difference between positive and negative occurrences of every variable is at most a nonnegative integer d. We prove that for all , and . The critical function is the maximal value of s, such that every d-regular (k,s)-CNF formula is satisfiable. In this study, denotes the minimal value of s such that there exists a uniquely satisfiable d-regular (k,s)-CNF formula. We further show that for and , there exists a uniquely satisfiable d-regular -CNF formula. Moreover, for , we have that .

Highlights

Satisfiability Problem (SAT) is a central problem in theoretical computer science of deciding whether a given Conjunction Normal Formula (CNF) is satisfiable
From Theorem 5, it demonstrates that there exists a polynomial time reduction from k-SAT to d-regular (k, s)-SAT for any constants k ≥ 3, s ≥ f (k) + 1 and (s + d)/2 > k − 1
D-regular (k, s)-SAT problem is NP-complete in this case

Summary

Introduction

Satisfiability Problem (SAT) is a central problem in theoretical computer science of deciding whether a given Conjunction Normal Formula (CNF) is satisfiable. The constrained density of the regular (3,4)-CNF is much smaller than the SAT-UNSAT phase transition point of the random 3-SAT problem α(3) ≈ 4.267 in [12]. This shows that a random regular (3,4)-CNF formula is satisfiable with high probability, but the regular (3,4)-SAT problem is NP-complete. In [15], Johannsen, Razgon and Wahlström presented an algorithm for solving a CNF formula in which the number of occurrences of each literal is at most d Their results demonstrated that the CNF formulas with some restrictions on the number of occurrences (positive or negative) of each variable have its own characteristics. We give a parsimonious reduction from k-CNF to d-regular (k, s)-CNF, and further explain the constrained density is not enough to describe the structural features of a CNF formula

Related Works

Notations

A Parsimonious Polynomial Time Reduction

Conclusions