Conditional lower bounds for sparse parameterized 2-CSP: A streamlined proof

TL;DR

This paper provides a simplified proof of a known conditional lower bound for solving sparse 2-CSP problems parameterized by the number of constraints, assuming the ETH, which impacts the complexity analysis of related problems.

Contribution

It offers a streamlined proof of an existing lower bound result for 2-CSPs under ETH, simplifying the understanding of the problem's computational hardness.

Findings

Reaffirms the ETH-based lower bound for 2-CSPs with k constraints.

Simplifies the proof technique for the existing lower bound.

Supports the complexity assumptions used in parameterized complexity theory.

Abstract

Assuming the Exponential Time Hypothesis (ETH), a result of Marx (ToC'10) implies that there is no $f(k)\cdot n^{o(k/\log k)}$ time algorithm that can solve 2-CSPs with $k$ constraints (over a domain of arbitrary large size $n$) for any computable function $f$. This lower bound is widely used to show that certain parameterized problems cannot be solved in time $f(k)\cdot n^{o(k/\log k)}$ time (assuming the ETH). The purpose of this note is to give a streamlined proof of this result.