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Shattered Sets and the Hilbert Function

Authors Shay Moran, Cyrus Rashtchian

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Shay Moran
Cyrus Rashtchian

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Shay Moran and Cyrus Rashtchian. Shattered Sets and the Hilbert Function. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.MFCS.2016.70

Abstract

We study complexity measures on subsets of the boolean hypercube and exhibit connections between algebra (the Hilbert function) and combinatorics (VC theory). These connections yield results in both directions.   Our main complexity-theoretic result demonstrates that a large and natural family of linear program feasibility problems cannot be computed by polynomial-sized constant-depth circuits. Moreover, our result applies to a stronger regime in which the hyperplanes are fixed and only the directions of the inequalities are given as input to the circuit. We derive this result by proving that a rich class of extremal functions in VC theory cannot be approximated by low-degree polynomials. We also present applications of algebra to combinatorics.  We provide a new algebraic proof of the Sandwich Theorem, which is a generalization of the well-known Sauer-Perles-Shelah Lemma. 
Finally, we prove a structural result about downward-closed sets, related to the Chvatal  conjecture in extremal combinatorics.

Subject Classification

Keywords
  • VC dimension
  • shattered sets
  • sandwich theorem
  • Hilbert function
  • polynomial method
  • linear programming
  • Chvatal's conjecture
  • downward-closed sets

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