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Depth Lower Bounds in Stabbing Planes for Combinatorial Principles

Authors Stefan Dantchev, Nicola Galesi, Abdul Ghani, Barnaby Martin

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Author Details

Stefan Dantchev
  • Department of Computer Science, Durham University, UK
Nicola Galesi
  • Department of Computer Science, Sapienza University of Rome, Italy
Abdul Ghani
  • Department of Computer Science, Durham University, UK
Barnaby Martin
  • Department of Computer Science, Durham University, UK

Acknowledgements

While finishing the writing of this manuscript we learned about [Noah Fleming et al., 2021] from Noah Fleming. We would like to thank him for answering some questions on his paper [Paul Beame et al., 2018], and sending us the manuscript [Noah Fleming et al., 2021] and for comments on a preliminary version of this work.

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Stefan Dantchev, Nicola Galesi, Abdul Ghani, and Barnaby Martin. Depth Lower Bounds in Stabbing Planes for Combinatorial Principles. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.STACS.2022.24

Abstract

Stabbing Planes is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system established via communication complexity arguments. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas.
In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner’s Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon’s combinatorial Nullenstellensatz. 
We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Proof complexity
Keywords
  • proof complexity
  • computational complexity
  • lower bounds
  • cutting planes
  • stabbing planes

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