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Revision #1 to TR15-189 | 6th December 2016 21:48

Shattered Sets and the Hilbert Function

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Revision #1
Authors: Shay Moran, Cyrus Rashtchian
Accepted on: 6th December 2016 21:48
Downloads: 203
Keywords: 


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 proves that most 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 Chv\'{a}tal conjecture in extremal combinatorics.



Changes to previous version:

Fixed minor typos throughout.


Paper:

TR15-189 | 25th November 2015 23:26

Shattered Sets and the Hilbert Function





TR15-189
Authors: Shay Moran, Cyrus Rashtchian
Publication: 26th November 2015 08:42
Downloads: 831
Keywords: 


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 proves that most 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 Chv\'{a}tal conjecture in extremal combinatorics.



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