Superlinear Lower Bounds Based on ETH

Authors András Z. Salamon , Michael Wehar

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Author Details

András Z. Salamon
  • School of Computer Science, University of St Andrews, UK
Michael Wehar
  • Computer Science Department, Swarthmore College, PA, USA


We greatly appreciate the help and suggestions that we received. We are especially grateful to Kenneth Regan and Jonathan Buss who shared a manuscript [Buss and Regan, 2014] on speed-up results relating time and space. We also thank Michael Fischer, Mike Paterson, and Nick Pippenger, who tracked down two manuscripts related to circuit simulations. In addition, we thank Karl Bringmann whose advice helped us to better align this work with recent advances in fine-grained complexity. Likewise, we recognize helpful discussions with Henning Fernau and all of the participants at the workshop on Modern Aspects of Complexity Within Formal Languages (sponsored by DFG). Finally, we very much appreciate all of the feedback from Joseph Swernofsky, suggestions from Ryan Williams, and comments from anonymous referees.

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András Z. Salamon and Michael Wehar. Superlinear Lower Bounds Based on ETH. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 55:1-55:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Complexity classes
  • Circuit Satisfiability
  • Conditional Lower Bounds
  • Exponential Time Hypothesis
  • Limited Nondeterminism


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