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Strong ETH and Resolution via Games and the Multiplicity of Strategies

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Abstract

We consider a proof system intermediate between regular Resolution, in which no variable can be resolved more than once along any refutation path, and general Resolution. We call \(\delta \)-regular Resolution such system, in which at most a fraction \(\delta \) of the variables can be resolved more than once along each refutation path (however, the re-resolved variables along different paths do not need to be the same). We show that when for \(\delta \) not too large, \(\delta \)-regular Resolution is consistent with the Strong Exponential Time Hypothesis (SETH). More precisely, for large n and k, we show that there are unsatisfiable k-CNF formulas which require \(\delta \)-regular Resolution refutations of size \(2^{(1 - \epsilon _k)n}\), where n is the number of variables and \(\epsilon _k=\widetilde{O}(k^{-1/4})\) and \(\delta =\widetilde{O}(k^{-1/4})\) is the number of variables that can be resolved multiple times.

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Notes

  1. The weighted Arithmetic Mean - Geometric Mean inequality says that given non-negative numbers \(a_1,\ldots ,a_n\) and non-negative weights \(w_1,\ldots ,w_n\) then

    $$\begin{aligned} \prod _i a_i^{w_i}\le \left( \frac{\sum _i w_i a_i}{w}\right) ^w, \end{aligned}$$

    where \(w=\sum _i w_i\). We applied this inequality with \(a_i=\frac{e^2\ell }{|Z_i|+1}\) and \(w_i=|Z_i|+1\).

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Acknowledgments

We would like to thank Nicola Galesi for discussions on the topic. We would also like to thank Jakob Nordström and Massimo Lauria for discussions on Resolution size and strong width lower bounds.

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Authors and Affiliations

  1. School of Computer Science and Communication, KTH Royal Institute of Technology, Stockholm, Sweden

    Ilario Bonacina

  2. Saarland University and the Cluster of Excellence, MMCI, Saarbrücken, Germany

    Navid Talebanfard

Authors
  1. Ilario Bonacina
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  2. Navid Talebanfard
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Corresponding author

Correspondence to Navid Talebanfard.

Additional information

This work was completed while the 1st author was affiliated to the Computer Science Department of Sapienza University of Rome (Italy). The 1st author was funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC Grant Agreement No. 279611.

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Bonacina, I., Talebanfard, N. Strong ETH and Resolution via Games and the Multiplicity of Strategies. Algorithmica 79, 29–41 (2017). https://doi.org/10.1007/s00453-016-0228-6

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