Sensitivity Lower Bounds from Linear Dependencies
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| Title: | Sensitivity Lower Bounds from Linear Dependencies |
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| Authors: | Laplante, Sophie, Naserasr, Reza, Sunny, Anupa |
| Contributors: | Sophie Laplante and Reza Naserasr and Anupa Sunny, Javier Esparza, Daniel Kráľ |
| Publisher Information: | Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2020. |
| Publication Year: | 2020 |
| Subject Terms: | Sensitivity, Polynomial Degree, Boolean Functions, ddc:004, Theory of computation → Computational complexity and cryptography |
| Description: | Recently, using spectral techniques, H. Huang proved that every subgraph of the hypercube of dimension n induced on more than half the vertices has maximum degree at least √n. Combined with some earlier work, this completed a proof of the sensitivity conjecture. In this work we show how to derive a proof of Huang’s result using only linear dependency and independence of vectors associated with the vertices of the hypercube. Our approach leads to several improvements of the result. In particular we prove that in any induced subgraph of H_n with more than half the number of vertices, there are two vertices, one of odd parity and the other of even parity, each with at least n vertices at distance at most 2. As an application we show that for any Boolean function f, the polynomial degree of f is bounded above by s₀(f) s₁(f), a strictly stronger statement which implies the sensitivity conjecture. |
| Document Type: | Conference object Article |
| File Description: | application/pdf |
| Language: | English |
| DOI: | 10.4230/lipics.mfcs.2020.62 |
| Access URL: | https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.62 |
| Rights: | CC BY |
| Accession Number: | edsair.dedup.wf.002..7dba51c34957e538db4c049d4fb65be1 |
| Database: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstract: | Recently, using spectral techniques, H. Huang proved that every subgraph of the hypercube of dimension n induced on more than half the vertices has maximum degree at least √n. Combined with some earlier work, this completed a proof of the sensitivity conjecture. In this work we show how to derive a proof of Huang’s result using only linear dependency and independence of vectors associated with the vertices of the hypercube. Our approach leads to several improvements of the result. In particular we prove that in any induced subgraph of H_n with more than half the number of vertices, there are two vertices, one of odd parity and the other of even parity, each with at least n vertices at distance at most 2. As an application we show that for any Boolean function f, the polynomial degree of f is bounded above by s₀(f) s₁(f), a strictly stronger statement which implies the sensitivity conjecture. |
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| DOI: | 10.4230/lipics.mfcs.2020.62 |