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 of ``additive 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 $\epsilon$.Comment: 31 page

New Notions and Constructions of Sparsification for Graphs and Hypergraphs

A sparsifier of a graph G (Benczu´r and Karger; Spielman and Teng) is a sparse weighted subgraph ˜ G 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

New notions and constructions of sparsification for graphs and hypergraphs

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Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries

The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Bencz\'ur and Karger (1996) showed that given any $n$-vertex undirected weighted graph $G$ and a parameter $\varepsilon \in (0,1)$, there is a near-linear time algorithm that outputs a weighted subgraph $G'$ of $G$ of size $\tilde{O}(n/\varepsilon^2)$ such that the weight of every cut in $G$ is preserved to within a $(1 \pm \varepsilon)$-factor in $G'$. The graph $G'$ is referred to as a {\em $(1 \pm \varepsilon)$-approximate cut sparsifier} of $G$. Subsequent recent work has obtained a similar result for the more general problem of hypergraph cut sparsifiers. However, all known sparsification algorithms require $\Omega(n + m)$ time where $n$ denotes the number of vertices and $m$ denotes the number of hyperedges in the hypergraph. Since $m$ can be exponentially large in $n$, a natural question is if it is possible to create a hypergraph cut sparsifier in time polynomial in $n$, {\em independent of the number of edges}. We resolve this question in the affirmative, giving the first sublinear time algorithm for this problem, given appropriate query access to the hypergraph.Comment: ICALP 202

Vertex Sparsifiers for Hyperedge Connectivity

Recently, Chalermsook et al. [SODA'21(arXiv:2007.07862)] introduces a notion of vertex sparsifiers for $c$-edge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic algorithms for $c$-edge st-connectivity [Jin and Sun FOCS'21(arXiv:2004.07650)]. We study a natural extension called vertex sparsifiers for $c$-hyperedge connectivity and construct a sparsifier whose size matches the state-of-the-art for normal graphs. More specifically, we show that, given a hypergraph $G=(V,E)$ with $n$ vertices and $m$ hyperedges with $k$ terminal vertices and a parameter $c$, there exists a hypergraph $H$ containing only $O(kc^{3})$ hyperedges that preserves all minimum cuts (up to value $c$) between all subset of terminals. This matches the best bound of $O(kc^{3})$ edges for normal graphs by [Liu'20(arXiv:2011.15101)]. Moreover, $H$ can be constructed in almost-linear $O(p^{1+o(1)} + n(rc\log n)^{O(rc)}\log m)$ time where $r=\max_{e\in E}|e|$ is the rank of $G$ and $p=\sum_{e\in E}|e|$ is the total size of $G$, or in $\text{poly}(m, n)$ time if we slightly relax the size to $O(kc^{3}\log^{1.5}(kc))$ hyperedges.Comment: submitted to ESA 202

Cut Sparsification and Succinct Representation of Submodular Hypergraphs

In cut sparsification, all cuts of a hypergraph $H=(V,E,w)$ are approximated within $1\pm\epsilon$ factor by a small hypergraph $H'$. This widely applied method was generalized recently to a setting where the cost of cutting each $e\in E$ is provided by a splitting function, $g_e: 2^e\to\mathbb{R}_+$. This generalization is called a submodular hypergraph when the functions $\{g_e\}_{e\in E}$ are submodular, and it arises in machine learning, combinatorial optimization, and algorithmic game theory. Previous work focused on the setting where $H'$ is a reweighted sub-hypergraph of $H$, and measured size by the number of hyperedges in $H'$. We study such sparsification, and also a more general notion of representing $H$ succinctly, where size is measured in bits. In the sparsification setting, where size is the number of hyperedges, we present three results: (i) all submodular hypergraphs admit sparsifiers of size polynomial in $n=|V|$; (ii) monotone-submodular hypergraphs admit sparsifiers of size $O(\epsilon^{-2} n^3)$; and (iii) we propose a new parameter, called spread, to obtain even smaller sparsifiers in some cases. In the succinct-representation setting, we show that a natural family of splitting functions admits a succinct representation of much smaller size than via reweighted subgraphs (almost by factor $n$). This large gap is surprising because for graphs, the most succinct representation is attained by reweighted subgraphs. Along the way, we introduce the notion of deformation, where $g_e$ is decomposed into a sum of functions of small description, and we provide upper and lower bounds for deformation of common splitting functions