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

TL;DR

The paper presents a streamlined, expander-based embedding proof to establish the conditional ETH lower bound for solving $2$-$\mathsf{CSP}$ with $k$ constraints, improving conceptual clarity over the original treewidth-centric approach. By leveraging Leighton–Rao routing on bounded-degree expanders, the authors embed an input graph into a small-expander host with depth $O\left(\frac{|V(G)|+|E(G)|}{k}\log k\right)$, enabling a CSP reduction that preserves satisfiability while expanding the alphabet to $\Sigma^d$. This yields near-tight, fixed-$k$ formulations: if there exists an algorithm running in time $f(k)\cdot|\Sigma|^{\alpha k/\log k}$ for some $\alpha>0$, ETH would fail. The results generalize to sparse and bounded-degree instances and provide a clear framework for deriving ETH-based lower bounds for a range of parameterized problems through CSP embeddings. Overall, the work tightens the toolkit for conditional lower bounds in CSP-complexity by offering a simpler, quantifiable embedding strategy with explicit depth/congestion control.

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.

Theorems & Definitions (23)

• Theorem 1.1: Chen et al. CHKX06aCHKX06b
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4: Marx marx-toc-treewidth
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7: Alon and Marx DBLP:journals/siamdm/AlonM11
• Theorem 2.2: Alon alon2021explicit
• Theorem 2.3: Alon alon
• proof