Superlinear Lower Bounds Based on ETH

TL;DR

This paper develops new techniques to establish superlinear lower bounds for polynomial time problems based on ETH, advancing the understanding of computational complexity and conditional lower bounds.

Contribution

It introduces methods for proving superlinear lower bounds for problems like CircuitSAT and k-Clique based on ETH, moving beyond SETH-based results.

Findings

CircuitSAT with m gates and log(m) inputs cannot be decided in linear time unless ETH is false

k-Clique cannot be decided in linear time in the size of the graph for fixed k unless ETH is false

Progress towards unconditional superlinear lower bounds for natural polynomial-time problems

Abstract

We introduce techniques for proving superlinear conditional lower bounds for polynomial time problems. In particular, we show that CircuitSAT for circuits with m gates and log(m) inputs (denoted by log-CircuitSAT) is not decidable in essentially-linear time unless the exponential time hypothesis (ETH) is false and k-Clique is decidable in essentially-linear time in terms of the graph's size for all fixed k. Such conditional lower bounds have previously only been demonstrated relative to the strong exponential time hypothesis (SETH). Our results therefore offer significant progress towards proving unconditional superlinear time complexity lower bounds for natural problems in polynomial time.