Sketching Cuts in Graphs and Hypergraphs

TL;DR

This paper explores sketching and streaming algorithms for cut problems in graphs and hypergraphs, establishing space complexity bounds and introducing cut-sparsifiers, with initial work on general CSPs.

Contribution

It provides new space complexity bounds for approximating Max-Cut in streaming models and introduces cut-sparsifiers for hypergraphs, along with initial steps for sketching general CSPs.

Findings

Max-Cut approximation requires near-linear space in streaming.

Hypergraph cut-sparsifiers with O(ε^{-2} n (r+log n)) edges exist.

First steps towards sketching general CSPs.

Abstract

Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that $(1-\epsilon)$-approximation for Max-Cut must use $n^{1-O(\epsilon)}$ space; moreover, beating $4/5$-approximation requires polynomial space. For the sketching model, we show that $r$-uniform hypergraphs admit a $(1+\epsilon)$-cut-sparsifier (i.e., a weighted subhypergraph that approximately preserves all the cuts) with $O(\epsilon^{-2} n (r+\log n))$ edges. We also make first steps towards sketching general CSPs (Constraint Satisfaction Problems).