Abstract

Obtaining lower bounds for NP-hard problems has for a long time been an active area of research. Algebraic techniques introduced by Jonsson et al. (2017) [4] show that the fine-grained time complexity of the parameterized ▪ problem correlates to the lattice of strong partial clones. With this ordering they isolated a relation R such that ▪ can be solved at least as fast as any other NP-hard ▪ problem. In this paper we extend this method and show that such languages also exist for the surjective SAT problem, the max ones problem, the propositional abduction problem, and the Boolean valued constraint satisfaction problem over finite-valued constraint languages. These languages may be interesting when investigating the borderline between polynomial time, subexponential time and exponential-time algorithms since they in a precise sense can be regarded as NP-hard problems with minimum time complexity. Indeed, with the help of these languages we relate all of the above problems to the exponential time hypothesis (ETH) in several different ways.

Highlights

  • In this article we study the fine-grained complexity of NP-hard optimization problems and logical reasoning problems, with a particular focus on describing the relative complexity of the problems in each class

  • After successfully accomplishing this for the four problems under consideration we explore the likelihood of obtaining subexponential time algorithms, in light of the exponential-time hypothesis, where we obtain several strong equivalent characterizations

  • The particular conjecture that the 3-SAT problem is not solvable in subexponential time is known as the exponential-time hypothesis (ETH) [9]

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Summary

Introduction

In this article we study the fine-grained complexity of NP-hard optimization problems and logical reasoning problems, with a particular focus on describing the relative complexity of the problems in each class. For each problem class under consideration we are interested in determining an intractable problem which is ‘maximally easy’, in the sense that there cannot exist any other intractable problem in the class with a strictly lower (exponential) running time. After successfully accomplishing this for the four problems under consideration we explore the likelihood of obtaining subexponential time algorithms, in light of the exponential-time hypothesis, where we obtain several strong equivalent characterizations

Background
Aims and methods
Preliminaries
Problem definitions
Objective
Size-preserving reductions and subexponential time
Operations and relations
Clone theory
The surjective SAT problem
The propositional abduction problem
The Max-Ones problem
The VCSP problem
The broader picture
Subexponential time and the exponential-time hypothesis
Lower bounds for surjective satisfiability
Lower bounds for propositional abduction
Lower bounds for max-ones
Lower bounds for VCSP
Wrapping up
Future research

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