New Notions and Constructions of Sparsification for Graphs and Hypergraphs

Abstract

A sparsifier of a graph G (Benczúr and Karger; Spielman and Teng) is a sparse weighted subgraphG that approximately retains the same 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 (that is, sparsifiers in which all the edge weights are equal to the same scaling factor), 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 cut edges in each cut (S,S) is not approximated within a multiplicative factor (1 + ǫ), but is, instead, approximated up to an additive term bounded by ǫ times d · |S| + vol(S), where d is the average degree of the graph and 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(ǫ −2 npolylog(1/ǫ)). We also provide a deterministic polynomial time construction that constructs sparsifiers with a weaker property having the optimal number of edges O(ǫ −2 n). Our constructions also satisfy a spectral version of the "additive sparsification" property.Notions of sparsification have also been studied for hypergraphs. Our construction of "additive sparsifiers" with Oǫ(n) edges also works for hypergraphs, and provides the first non-trivial notion of sparsification for hypergraphs achievable with O(n) hyperedges when ǫ 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 poly(ǫ −1 , r) · n log n hyperedges, improving over the previous spectral construction (Soma and Yoshida) that used O(n 3 ) hyperedges even for constant r and ǫ.

Hypergraph Cuts with Edge-Dependent Vertex Weights

“…The reduction procedures discussed above may result in large and dense graphs, thus affecting the efficiency of algorithms applied to the reduced graph. A workaround is to find a smaller and sparser graph whose cut approximates, rather than exactly recovers, the hypergraph cut [3,4,5,12,32]. In [5], a sparsification technique is proposed for hypergraphs with submodular cardinality-based splitting functions.…”

“…This may result in large and dense graphs thus affecting the efficiency of algorithms applied to the reduced graph. To tackle this problem, sparsification techniques have been developed which try to approximate the hypergraph cut using a sparse graph with fewer auxiliary vertices [3,4,5,12,32]. A followup paper [5] proposes a sparsification method for approximating hypergraph cuts defined by submodular cardinality-based splitting functions.…”

Sparsification of Decomposable Submodular Functions

“…The cut function of a graph G = (V, E) can be seen as a decomposable submodular function F (S) = e∈E f e , where f e (S) = 1 if and only if e ∩ S = ∅ and e ∩ (V \ S) = ∅. The problem of sparsifying a graph while approximately preserving its cut structure has been extensively studied, (See Ahn et al [2012, 2013, Bansal et al [2019], Benczúr and Karger [2015] and references therein.) The pioneering work of Benczúr and Karger [1996] showed for any graph G with n vertices one can construct a weighted subgraph G in nearly linear time with O(n log n/ 2 ) edges such that the weight of every cut in G is preserved within a multiplicative (1 ± )-factor in G .…”

“…Recently, Soma and Yoshida [2019] initiated the study of spectral sparsifiers for hypergraphs and showed that every hypergraph admits an -spectral sparsifier with O(n 3 log n/ 2 ) hyperedges. For the case where the maximum size of a hyperedge is r, Bansal et al [2019] showed that every hypergraph has an -spectral sparsifier of size O(nr 3 log n/ 2 ). Recently, this bound has been improved to O(nr(log n/ ) O( 1) ) and then to O(n(log n/ ) O( 1) ) Kapralov et al [2021b,a].…”

Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries

“…Cuts in graphs are a fundamental object of study, and play a central role in the study of graph algorithms. Consequently, the problem of sparsifying a graph while approximately preserving its cut structure has been extensively studied (see, for instance, [17,6,18,25,1,2,13,5,3,21,15,4,16], and references therein). A cut-preserving sparsifier not only reduces the space requirement for any computation, but it can also reduce the time complexity of solving many fundamental cut, flow, and matching problems as one can now run the algorithms on the sparsifier which may contain far fewer edges.…”