Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis

TL;DR

This paper explores the complexity of NP-hard problems by extending algebraic techniques to relate the time complexity of MaxOnes and VCSP problems to the Exponential-Time Hypothesis, providing new lower bounds.

Contribution

It introduces new algebraic languages for MaxOnes and VCSP problems that connect their complexity to the Exponential-Time Hypothesis, extending previous methods.

Findings

Existence of special languages for MaxOnes and VCSP problems.

Relations established between problem complexity and ETH.

Provides new lower bounds for NP-hard problems.

Abstract

Obtaining lower bounds for NP-hard problems has for a long time been an active area of research. Recent algebraic techniques introduced by Jonsson et al. (SODA 2013) show that the time complexity of the parameterized SAT($\cdot$) problem correlates to the lattice of strong partial clones. With this ordering they isolated a relation $R$ such that SAT($R$) can be solved at least as fast as any other NP-hard SAT($\cdot$) problem. In this paper we extend this method and show that such languages also exist for the max ones problem (MaxOnes($\Gamma$)) and the Boolean valued constraint satisfaction problem over finite-valued constraint languages (VCSP($\Delta$)). With the help of these languages we relate MaxOnes and VCSP to the exponential time hypothesis in several different ways.