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On Semi-Algebraic Proofs and Algorithms

Authors Noah Fleming, Mika Göös, Stefan Grosser, Robert Robere

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Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • Proof Complexity
  • Extended Formulations
  • Circuit Complexity
  • Sherali-Adams

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Abstract

We give a new characterization of the Sherali-Adams proof system, showing that there is a degree-d Sherali-Adams refutation of an unsatisfiable CNF formula C if and only if there is an ε > 0 and a degree-d conical junta J such that viol_C(x) - ε = J, where viol_C(x) counts the number of falsified clauses of C on an input x. Using this result we show that the linear separation complexity, a complexity measure recently studied by Hrubeš (and independently by de Oliveira Oliveira and Pudlák under the name of weak monotone linear programming gates), monotone feasibly interpolates Sherali-Adams proofs.
We then investigate separation results for viol_C(x) - ε. In particular, we give a family of unsatisfiable CNF formulas C which have polynomial-size and small-width resolution proofs, but for which any representation of viol_C(x) - 1 by a conical junta requires degree Ω(n); this resolves an open question of Filmus, Mahajan, Sood, and Vinyals. Since Sherali-Adams can simulate resolution, this separates the non-negative degree of viol_C(x) - 1 and viol_C(x) - ε for arbitrarily small ε > 0. Finally, by applying lifting theorems, we translate this lower bound into new separation results between extension complexity and monotone circuit complexity.

Cite As Get BibTex

Noah Fleming, Mika Göös, Stefan Grosser, and Robert Robere. On Semi-Algebraic Proofs and Algorithms. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 69:1-69:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.69

Author Details

Noah Fleming
  • University of California, San Diego, CA, USA
  • Memorial University, St. John’s, Canada
Mika Göös
  • EPFL, Lausanne, Switzerland
Stefan Grosser
  • McGill University, Montreal, Canada
Robert Robere
  • McGill University, Montreal, Canada

Funding

  • Fleming, Noah: Research supported by NSERC.
  • Robere, Robert: Research supported by NSERC.

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