Cut and spectral sparsification of graphs have numerous applications, including e.g. speeding up algorithms for cuts and Laplacian solvers. These powerful notions have recently been extended to hypergraphs, which are much richer and may offer new applications. However, the current bounds on the size of hypergraph sparsifiers are not as tight as the corresponding bounds for graphs.
Our first result is a polynomial-time algorithm that, given a hypergraph on $n$ vertices with maximum hyperedge size $r$, outputs an $ε$-spectral sparsifier with $O^*(nr)$ hyperedges, where $O^*$ suppresses $(ε^{-1} \log n)^{O(1)}$ factors. This size bound improves the two previous bounds: $O^*(n^3)$ [Soma and Yoshida, SODA'19] and $O^*(nr^3)$ [Bansal, Svensson and Trevisan, FOCS'19]. Our main technical tool is a new method for proving concentration of the nonlinear analogue of the quadratic form of the Laplacians for hypergraph expanders.
We complement this with lower bounds on the bit complexity of any compression scheme that $(1+ε)$-approximates all the cuts in a given hypergraph, and hence also on the bit complexity of every $ε$-cut/spectral sparsifier. These lower bounds are based on Ruzsa-Szemerédi graphs, and a particular instantiation yields an $Ω(nr)$ lower bound on the bit complexity even for fixed constant $ε$. This is tight up to polylogarithmic factors in $n$, due to recent hypergraph cut sparsifiers of [Chen, Khanna and Nagda, FOCS'20].
Finally, for directed hypergraphs, we present an algorithm that computes an $ε$-spectral sparsifier with $O^*(n^2r^3)$ hyperarcs, where $r$ is the maximum size of a hyperarc. For small $r$, this improves over $O^*(n^3)$ known from [Soma and Yoshida, SODA'19], and is getting close to the trivial lower bound of $Ω(n^2)$ hyperarcs.
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, Benczúr 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 $(1 \pm \varepsilon)$-approxima…
Let $H=(V,E)$ be an edge-weighted hypergraph of rank $r$. Kogan and Krauthgamer extended Benczúr and Karger's random sampling scheme for cut sparsification from graphs to hypergraphs. The sampling requires an algorithm for computing the approximate strengths of edges. In this note we extend the algorithm for graphs to hypergraphs and describe a near-linear time algorithm to compute approximate strengths of edges; we build on a sparsification result for hypergraphs from our recent work. Combined with prior results we obtain faster algorithms for finding $(1+ε)$-approximate mincuts when the rank…
Graph sparsification has been studied extensively over the past two decades, culminating in spectral sparsifiers of optimal size (up to constant factors). Spectral hypergraph sparsification is a natural analogue of this problem, for which optimal bounds on the sparsifier size are not known, mainly because the hypergraph Laplacian is non-linear, and thus lacks the linear-algebraic structure and tools that have been so effective for graphs. Our main contribution is the first algorithm for constructing $ε$-spectral sparsifiers for hypergraphs with $O^*(n)$ hyperedges, where $O^*$ suppresses $(ε^{…
In a hypergraph on $n$ vertices where $D$ is the maximum size of a hyperedge, there is a weighted hypergraph spectral $\varepsilon$-sparsifier with at most $O(\varepsilon^{-2} \log(D) \cdot n \log n)$ hyperedges. This improves over the bound of Kapralov, Krauthgamer, Tardos and Yoshida (2021) who achieve $O(\varepsilon^{-4} n (\log n)^3)$, as well as the bound $O(\varepsilon^{-2} D^3 n \log n)$ obtained by Bansal, Svensson, and Trevisan (2019). The same sparsification result was obtained independently by Jambulapati, Liu, and Sidford (2022).
We present an algorithm that given any $n$-vertex, $m$-edge, rank $r$ hypergraph constructs a spectral sparsifier with $O(n \varepsilon^{-2} \log n \log r)$ hyperedges in nearly-linear $\widetilde{O}(mr)$ time. This improves in both size and efficiency over a line of work (Bansal-Svensson-Trevisan 2019, Kapralov-Krauthgamer-Tardos-Yoshida 2021) for which the previous best size was $O(\min\{n \varepsilon^{-4} \log^3 n,nr^3 \varepsilon^{-2} \log n\})$ and runtime was $\widetilde{O}(mr + n^{O(1)})$. Independent Result: In an independent work, Lee (Lee 2022) also shows how to compute a spectral hy…
A sparsifier of a graph $G$ (Benczúr 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 multiplicat…
For an undirected/directed hypergraph $G=(V,E)$, its Laplacian $L_G\colon\mathbb{R}^V\to \mathbb{R}^V$ is defined such that its ``quadratic form'' $\boldsymbol{x}^\top L_G(\boldsymbol{x})$ captures the cut information of $G$. In particular, $\boldsymbol{1}_S^\top L_G(\boldsymbol{1}_S)$ coincides with the cut size of $S \subseteq V$, where $\boldsymbol{1}_S \in \mathbb{R}^V$ is the characteristic vector of $S$. A weighted subgraph $H$ of a hypergraph $G$ on a vertex set $V$ is said to be an $ε$-spectral sparsifier of $G$ if $(1-ε)\boldsymbol{x}^\top L_H(\boldsymbol{x}) \leq \boldsymbol{x}^\top …
A motif is a frequently occurring subgraph of a given directed or undirected graph $G$. Motifs capture higher order organizational structure of $G$ beyond edge relationships, and, therefore, have found wide applications such as in graph clustering, community detection, and analysis of biological and physical networks to name a few. In these applications, the cut structure of motifs plays a crucial role as vertices are partitioned into clusters by cuts whose conductance is based on the number of instances of a particular motif, as opposed to just the number of edges, crossing the cuts. In this …
Pioneered by Benczur and Karger for cuts in graphs [STOC'96], sparsification is a fundamental topic with wide-ranging applications that has been studied, e.g., for graphs and hypergraphs, in a combinatorial and a spectral setting, and with additive and multiplicate error bounds. Rafiey and Yoshida recently considered sparsification of decomposable submodular functions [AAAI'22]. We extend their work by presenting an efficient algorithm for a sparsifier for monotone $k$-submodular functions of low curvature.
In recent years, hypergraph generalizations of many graph cut problems have been introduced and analyzed as a way to better explore and understand complex systems and datasets characterized by multiway relationships. Recent work has made use of a generalized hypergraph cut function which for a hypergraph $\mathcal{H} = (V,E)$ can be defined by associating each hyperedge $e \in E$ with a splitting function ${\bf w}_e$, which assigns a penalty to each way of separating the nodes of $e$. When each ${\bf w}_e$ is a submodular cardinality-based splitting function, meaning that ${\bf w}_e(S) = g(|S|…
In this paper we give a f-approximation algorithm for the minimum unweighted Vertex Cover problem with Hard Capacity constraints (VCHC) on f-hypergraphs. This problem generalizes standard vertex cover for which the best known approximation ratio is also f and cannot be improved assuming the unique game conjecture. Our result is therefore essentially the best possible. This improves over the previous 2.155 (for f=2) and 2f-approximation algorithms by Cheung, Goemans and Wong (CGW). At the heart of our approach is to apply iterative rounding to the problem with ideas coming from several previous…
We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph $H=(V,E)$ with $n = |V|$, $m = |E|$ and $p = \sum_{e \in E} |e|$ the best known algorithm to compute a global minimum cut in $H$ runs in time $O(np)$ for the uncapacitated case and in $O(np + n^2 \log n)$ time for the capacitated case. We show the following new results. 1. Given an uncapacitated hypergraph $H$ and an integer $k$ we describe an algorithm that runs in $O(p)$ time to find a subhypergraph $H'$ with sum of degrees $O(kn)$ that preserves all edge-connectivities up to $k$ (a $k$-sparsifier)…
Given an unweighted tree $T=(V,E)$ with terminals $K \subset V$, we show how to obtain a $2$-quality vertex flow and cut sparsifier $H$ with $V_H = K$. We prove that our result is essentially tight by providing a $2-o(1)$ lower-bound on the quality of any cut sparsifier for stars. In addition we give improved results for quasi-bipartite graphs. First, we show how to obtain a $2$-quality flow sparsifier with $V_H = K$ for such graphs. We then consider the other extreme and construct exact sparsifiers of size $O(2^{k})$, when the input graph is unweighted.
A valued constraint satisfaction problem (VCSP) instance $(V,Π,w)$ is a set of variables $V$ with a set of constraints $Π$ weighted by $w$. Given a VCSP instance, we are interested in a re-weighted sub-instance $(V,Π'\subset Π,w')$ such that preserves the value of the given instance (under every assignment to the variables) within factor $1\pmε$. A well-studied special case is cut sparsification in graphs, which has found various applications. We show that a VCSP instance consisting of a single boolean predicate $P(x,y)$ (e.g., for cut, $P=\mbox{XOR}$) can be sparsified into $O(|V|/ε^2)$ const…
We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize O(n^{2-e}) for e > 0, unless NP is in coNP/poly and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot …
Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot…
We give almost tight conditional lower bounds on the running time of the kHyperPath problem. Given an $r$-uniform hypergraph for some integer $r$, kHyperPath seeks a tight path of length $k$. That is, a sequence of $k$ nodes such that every consecutive $r$ of them constitute a hyperedge in the graph. This problem is a natural generalization of the extensively-studied kPath problem in graphs. We show that solving kHyperPath in time $O^*(2^{(1-γ)k})$ where $γ>0$ is independent of $r$ is probably impossible. Specifically, it implies that Set Cover on $n$ elements can be solved in time $O^*(2^{(1 …
We develop a framework for incorporating edge-dependent vertex weights (EDVWs) into the hypergraph minimum s-t cut problem. These weights are able to reflect different importance of vertices within a hyperedge, thus leading to better characterized cut properties. More precisely, we introduce a new class of hyperedge splitting functions that we call EDVWs-based, where the penalty of splitting a hyperedge depends only on the sum of EDVWs associated with the vertices on each side of the split. Moreover, we provide a way to construct submodular EDVWs-based splitting functions and prove that a hype…
The hypergraph Moore bound is an elegant statement that characterizes the extremal trade-off between the girth - the number of hyperedges in the smallest cycle or even cover (a subhypergraph with all degrees even) and size - the number of hyperedges in a hypergraph. For graphs (i.e., $2$-uniform hypergraphs), a bound tight up to the leading constant was proven in a classical work of Alon, Hoory and Linial [AHL02]. For hypergraphs of uniformity $k>2$, an appropriate generalization was conjectured by Feige [Fei08]. The conjecture was settled up to an additional $\log^{4k+1} n$ factor in the size…
