Showing papers by "Johan Håstad published in 2020"
TL;DR: In this paper, it was shown that a small-depth Frege refutation of the Tseitin contradiction on the grid requires subexponential size, and that polynomial size refutation must use formulas of almost logarithmic depth.
Abstract: We prove that a small-depth Frege refutation of the Tseitin contradiction on the grid requires subexponential size. We conclude that polynomial size Frege refutations of the Tseitin contradiction must use formulas of almost logarithmic depth.
10 citations
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TL;DR: An explicit construction of length-$n binary codes capable of correcting the deletion of two bits that have size $2^n/n^{4+o(1)}$ that can be list decoded from two deletions using lists of size two is given.
Abstract: We give an explicit construction of length-$n$ binary codes capable of correcting the deletion of two bits that have size $2^n/n^{4+o(1)}$. This matches up to lower order terms the existential result, based on an inefficient greedy choice of codewords, that guarantees such codes of size $\Omega(2^n/n^4)$. Our construction is based on augmenting the classic Varshamov-Tenengolts construction of single deletion codes with additional check equations. We also give an explicit construction of binary codes of size $\Omega(2^n/n^{3+o(1)})$ that can be list decoded from two deletions using lists of size two. Previously, even the existence of such codes was not clear.
7 citations
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TL;DR: In the generalization the required property is, for any set A, to be able to find $d$ sets from a family $\mathcal{F} \subseteq \binom{[n]}{n/d}$ that form a partition of $ [n]$ and such that each part is balanced on $A$.
Abstract: The Galvin problem asks for the minimum size of a family F subset of ([n]n/2) with the property that, for any set A of size n/2, there is a set S is an element of F which is balanced on A, meaning ...