Unweighted One-Sided Code Sparsifiers and Thin Subgraphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912573701488640 |
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| author | Gharan, Shayan Oveis Sahami, Arvin |
| author_facet | Gharan, Shayan Oveis Sahami, Arvin |
| contents | For a linear code $\mathcal{C} \subseteq \mathbb{F}_2^n$ and $α\in [0,1]$, call a set $S \subseteq [n]$ an (unweighted) one-sided $α$-sparsifier of $\mathcal{C}$ if for all $c \in \mathcal{C}$, $\mathrm{wt}(c_S)\geq α\cdot \mathrm{wt}(c)$, where $c_S$ is the projection of $c$ onto the coordinates in $S$ and $\mathrm{wt}(c)$ is the Hamming weight of $c$. \\ We show that every $k$-dimensional linear code $\mathcal{C}\subseteq \mathbb{F}_2^n$ has at least $2^{n - k}$ many unweighted one-sided $1/2$-sparsifiers and hence one of size at most $n/2 + O(\sqrt{n k})$. As an application, letting $\mathcal{C} \subseteq \mathbb{F}_2^E$ denote the cut-space of a graph $G=(V, E)$, we show a lower bound of $2^{\lvert E \rvert- (\lvert V \rvert - 1)}$ on the number of $1/2$-thin subgraphs of $G$ and the existence of a $1/2$-thin subgraph with at least $\lvert E \rvert /2-O(\sqrt{\lvert E \rvert \cdot \lvert V \rvert})$ edges. In contrast to previous results on thin subgraphs, our proofs are purely "combinatorial". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unweighted One-Sided Code Sparsifiers and Thin Subgraphs Gharan, Shayan Oveis Sahami, Arvin Combinatorics Data Structures and Algorithms For a linear code $\mathcal{C} \subseteq \mathbb{F}_2^n$ and $α\in [0,1]$, call a set $S \subseteq [n]$ an (unweighted) one-sided $α$-sparsifier of $\mathcal{C}$ if for all $c \in \mathcal{C}$, $\mathrm{wt}(c_S)\geq α\cdot \mathrm{wt}(c)$, where $c_S$ is the projection of $c$ onto the coordinates in $S$ and $\mathrm{wt}(c)$ is the Hamming weight of $c$. \\ We show that every $k$-dimensional linear code $\mathcal{C}\subseteq \mathbb{F}_2^n$ has at least $2^{n - k}$ many unweighted one-sided $1/2$-sparsifiers and hence one of size at most $n/2 + O(\sqrt{n k})$. As an application, letting $\mathcal{C} \subseteq \mathbb{F}_2^E$ denote the cut-space of a graph $G=(V, E)$, we show a lower bound of $2^{\lvert E \rvert- (\lvert V \rvert - 1)}$ on the number of $1/2$-thin subgraphs of $G$ and the existence of a $1/2$-thin subgraph with at least $\lvert E \rvert /2-O(\sqrt{\lvert E \rvert \cdot \lvert V \rvert})$ edges. In contrast to previous results on thin subgraphs, our proofs are purely "combinatorial". |
| title | Unweighted One-Sided Code Sparsifiers and Thin Subgraphs |
| topic | Combinatorics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2502.02799 |