# Cheeger-type approximation for sparsest st-cut

arXiv: https://arxiv.org/abs/1410.3889 · HTML: https://mlsanity.com/paper/1410.3889/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2014.

## Built on by (most cited first, 1 of 1)

- [Spectral Clustering on Large Datasets: When Does it Work? Theory from Continuous Clustering and Density Cheeger-Buser](https://mlsanity.com/paper/2305.06541/) (2023)

## Abstract (arXiv)

We introduce the st-cut version the Sparsest-Cut problem, where the goal is to find a cut of minimum sparsity among those separating two distinguished vertices s,t∈ V. Clearly, this problem is at least as hard as the usual (non-st) version. Our main result is a polynomial-time algorithm for the product-demands setting, that produces a cut of sparsity O(√OPT), where OPT denotes the optimum, and the total edge capacity and the total demand are assumed (by normalization) to be 1. Our result generalizes the recent work of Trevisan [arXiv, 2013] for the non-st version of the same problem (Sparsest-Cut with product demands), which in turn generalizes the bound achieved by the discrete Cheeger inequality, a cornerstone of Spectral Graph Theory that has numerous applications. Indeed, Cheeger's inequality handles graph conductance, the special case of product demands that are proportional to the vertex (capacitated) degrees. Along the way, we obtain an O(log n)-approximation, where n=cardV, for the general-demands setting of Sparsest st-Cut.