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تسجيل مستخدم جديدImproved efficiency for covering codes matching the sphere-covering bound
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A covering code is a subset $mathcal{C} subseteq {0,1}^n$ with the property that any $z in {0,1}^n$ is close to some $c in mathcal{C}$ in Hamming distance. For every $epsilon,delta>0$, we show a construction of a family of codes with relative covering radius $delta + epsilon$ and rate $1 - mathrm{H}(delta) $ with block length at most $exp(O((1/epsilon) log (1/epsilon)))$ for every $epsilon > 0$. This improves upon a folklore construction which only guaranteed codes of block length $exp(1/epsilon^2)$. The main idea behind this proof is to find a distribution on codes with relatively small support such that most of these codes have good covering properties.
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The list-decodable code has been an active topic in theoretical computer science since the seminal papers of M. Sudan and V. Guruswami in 1997-1998. There are general result about the Johnson radius and the list-decoding capacity theorem for random c
odes. However few results about general constraints on rates, list-decodable radius and list sizes for list-decodable codes have been obtained. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new simple but strong upper bounds for list-decodable codes based on various covering codes. Then any good upper bound on the covering radius imply a good upper bound on the size of list-decodable codes. Hence the list-decodablity of codes is a strong constraint from the view of covering codes. Our covering code upper bounds for $(d,1)$ list decodable codes give highly non-trivial upper bounds on the sizes of codes with the given minimum Hamming distances. Our results give exponential improvements on the recent generalized Singleton upper bound of Shangguan and Tamo in STOC 2020, when the code lengths are very large. The asymptotic forms of covering code bounds can partially recover the list-decoding capacity theorem, the Blinovsky bound and the combinatorial bound of Guruswami-H{aa}stad-Sudan-Zuckerman. We also suggest to study the combinatorial covering list-decodable codes as a natural generalization of combinatorial list-decodable codes.
In the Metric Capacitated Covering (MCC) problem, given a set of balls $mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $mathcal{B}subseteq mathcal{B}$ and an assignment of th
e points in $P$ to the balls in $mathcal{B}$ such that each point is assigned to a ball that contains it and each ball is assigned with at most $U$ points. MCC achieves an $O(log |P|)$-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of $o(log |P|)$ even with $beta < 3$ factor expansion of the balls. Bandyapadhyay~{et al.} [SoCG 2018, DCG 2019] showed that one can obtain an $O(1)$-approximation for the problem with $6.47$ factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound $3$ and the upper bound $6.47$. In this current work, we show that it is possible to obtain an $O(1)$-approximation with only $4.24$ factor expansion of the balls. We also show a similar upper bound of $5$ for a more generalized version of MCC for which the best previously known bound was $9$.
We study several problems on geometric packing and covering with movement. Given a family $mathcal{I}$ of $n$ intervals of $kappa$ distinct lengths, and another interval $B$, can we pack the intervals in $mathcal{I}$ inside $B$ (respectively, cover $
B$ by the intervals in $mathcal{I}$) by moving $tau$ intervals and keeping the other $sigma = n - tau$ intervals unmoved? We show that both packing and covering are W[1]-hard with any one of $kappa$, $tau$, and $sigma$ as single parameter, but are FPT with combined parameters $kappa$ and $tau$. We also obtain improved polynomial-time algorithms for packing and covering, including an $O(nlog^2 n)$ time algorithm for covering, when all intervals in $mathcal{I}$ have the same length.
We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequaliti
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The multiplicity Schwartz-Zippel lemma bounds the total multiplicity of zeroes of a multivariate polynomial on a product set. This lemma motivates the multiplicity codes of Kopparty, Saraf and Yekhanin [J. ACM, 2014], who showed how to use this lemma
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