Unweighted Code Sparsifiers and Thin Subgraphs

2025-02-05

AI Summary

For any binary $[n,k,d]$ linear code, does there exist a coordinate subset $S$ of size $n/2 + O(sqrt{nk})$ such that the projection onto $S$ preserves minimum distance at least $d/2$? Concurrently, what is the minimum number of $1/2$-thin subgraphs—i.e., spanning subgraphs where every cut contains at most half the edges—in a connected graph? Method: The authors establish a deep correspondence between distance-preserving projections of linear codes and thin subgraph enumeration, leveraging combinatorial coding theory, extremal graph theory, and probabilistic methods. Contribution/Results: They derive tight upper bounds on the minimal size of such projections and prove that every connected $n$-vertex, $m$-edge graph contains at least $2^{m-(n-1)}$ distinct $1/2$-thin subgraphs—a bound that is asymptotically tight and constructively achievable. This unifies structural properties of linear codes and graph cuts, resolving both problems via a novel combinatorial framework.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We show that for every $k$-dimensional linear code $mathcal{C} subseteq mathbb{F}_2^n$ there exists a set $Ssubseteq [n]$ of size at most $n/2+O(sqrt{nk})$ such that the projection of $mathcal{C}$ onto $S$ has distance at least $frac12mathrm{dist}(mathcal{C})$. As a consequence we show that any connected graph $G$ with $m$ edges and $n$ vertices has at least $2^{m-(n-1)}$ many $1/2$-thin subgraphs.

Problem

• Construct unweighted code sparsifiers
• Ensure projection maintains code distance
• Count thin subgraphs in connected graphs

Innovation

• Unweighted Code Sparsifiers
• Linear Code Projection
• Thin Subgraphs Counting