Multiplicative error set system sparsification: A simpler proof via chain length contraction

TL;DR

This paper introduces a simpler, more optimal characterization of set system sparsifiability based on chain length, extending Karger's contraction algorithm and improving bounds for CSP sparsification.

Contribution

It provides a novel, simplified proof linking chain length to multiplicative sparsifiability, enhancing previous characterizations and bounds.

Findings

Chain length characterizes multiplicative sparsifiability of set systems.

The proof generalizes Karger's contraction algorithm and recent linear algebraic methods.

Improved bounds for weighted CSP sparsification are derived.

Abstract

The chain length of a set family $\mathcal{S} \subseteq 2^{[m]}$ is the largest ascending sequence of sets in containment order in the union-closure of $\mathcal S$. In this work, we provide a significantly simpler and more optimal characterization of the sparsifiability of set systems in terms of their chain length, improving on the work of Brakensiek and Guruswami [STOC 2025]. Our proof relies on a generalization of Karger's [SODA 1993] famous contraction algorithm and its recent linear algebraic extensions [Khanna-Putterman-Sudan SODA 2024], and our resulting bounds show that, just as VC dimension characterizes the \emph{additive sparsifiability} of a set system, chain length governs the \emph{multiplicative sparsifiability}. As a corollary, we obtain improved bounds for weighted CSP sparsification.