New Notions and Constructions of Sparsification for Graphs and Hypergraphs

A sparsifier of a graph $G$ (Bencz\'ur and Karger; Spielman and Teng) is a sparse weighted subgraph $\tilde G$ that approximately retains the cut structure of $G$. For general graphs, non-trivial sparsification is possible only by using weighted graphs in which different edges have different weights. Even for graphs that admit unweighted sparsifiers, there are no known polynomial time algorithms that find such unweighted sparsifiers. We study a weaker notion of sparsification suggested by Oveis Gharan, in which the number of edges in each cut $(S,\bar S)$ is not approximated within a multiplicative factor $(1+\epsilon)$, but is, instead, approximated up to an additive term bounded by $\epsilon$ times $d\cdot |S| + \text{vol}(S)$, where $d$ is the average degree, and $\text{vol}(S)$ is the sum of the degrees of the vertices in $S$. We provide a probabilistic polynomial time construction of such sparsifiers for every graph, and our sparsifiers have a near-optimal number of edges $O(\epsilon^{-2} n {\rm polylog}(1/\epsilon))$. We also provide a deterministic polynomial time construction that constructs sparsifiers with a weaker property having the optimal number of edges $O(\epsilon^{-2} n)$. Our constructions also satisfy a spectral version of the additive sparsification'' property. Our construction ofadditive sparsifiers'' with $O_\epsilon (n)$ edges also works for hypergraphs, and provides the first non-trivial notion of sparsification for hypergraphs achievable with $O(n)$ hyperedges when $\epsilon$ and the rank $r$ of the hyperedges are constant. Finally, we provide a new construction of spectral hypergraph sparsifiers, according to the standard definition, with ${\rm poly}(\epsilon^{-1},r)\cdot n\log n$ hyperedges, improving over the previous spectral construction (Soma and Yoshida) that used $\tilde O(n^3)$ hyperedges even for constant $r$ and $\epsilon$.