Unweighted One-Sided Code Sparsifiers and Thin Subgraphs
Shayan Oveis Gharan, Arvin Sahami

TL;DR
This paper introduces the concept of unweighted one-sided sparsifiers for linear codes and applies it to graph cut-spaces, establishing bounds on the number and size of thin subgraphs through combinatorial methods.
Contribution
It proves the existence of many unweighted one-sided 1/2-sparsifiers for any linear code and derives bounds on thin subgraphs in graphs, using purely combinatorial proofs.
Findings
At least 2^{n - k} unweighted one-sided 1/2-sparsifiers exist for any k-dimensional code.
There are at most n/2 + O(√(nk)) size sparsifiers.
A lower bound on the number of 1/2-thin subgraphs in a graph is established.
Abstract
For a linear code and , call a set an (unweighted) one-sided -sparsifier of if for all , , where is the projection of onto the coordinates in and is the Hamming weight of . \\ We show that every -dimensional linear code has at least many unweighted one-sided -sparsifiers and hence one of size at most . As an application, letting denote the cut-space of a graph , we show a lower bound of on the number of -thin subgraphs of and the existence of a -thin subgraph with at least $\lvert E \rvert /2-O(\sqrt{\lvert…
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Taxonomy
TopicsAlgorithms and Data Compression · DNA and Biological Computing · Cellular Automata and Applications
