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

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.