Incorporated a new algorithm of Chekuri and Quanrud into our algorithm and analysis. Fixed a bug in the analysis of the algorithm, and edited exposition throughout for greater clarity
We design an algorithm for computing connectivity in hypergraphs which runs in time $\hat O_r(p + \min\{λ^{\frac{r-3}{r-1}} n^2, n^r/λ^{r/(r-1)}\})$ (the $\hat O_r(\cdot)$ hides the terms subpolynomial in the main parameter and terms that depend only on $r$) where $p$ is the size, $n$ is the number of vertices, and $r$ is the rank of the hypergraph. Our algorithm is faster than existing algorithms when the the rank is constant and the connectivity $λ$ is $ω(1)$. At the heart of our algorithm is a structural result regarding min-cuts in simple hypergraphs. We show a trade-off between the number of hyperedges taking part in all min-cuts and the size of the smaller side of the min-cut. This structural result can be viewed as a generalization of a well-known structural theorem for simple graphs [Kawarabayashi-Thorup, JACM 19]. We extend the framework of expander decomposition to simple hypergraphs in order to prove this structural result. We also make the proof of the structural result constructive to obtain our faster hypergraph connectivity algorithm.
We consider the problem of approximately solving constraint satisfaction problems with arity $k > 2$ ($k$-CSPs) on instances satisfying certain expansion properties, when viewed as hypergraphs. Random instances of $k$-CSPs, which are also highly expanding, are well-known to be hard to approximate using known algorithmic techniques (and are widely believed to be hard to approximate in polynomial time). However, we show that this is not necessarily the case for instances where the hypergraph is a high-dimensional expander. We consider the spectral definition of high-dimensional expansion used by…
We show a deterministic algorithm for computing edge connectivity of a simple graph with $m$ edges in $m^{1+o(1)}$ time. Although the fastest deterministic algorithm by Henzinger, Rao, and Wang [SODA'17] has a faster running time of $O(m\log^{2}m\log\log m)$, we believe that our algorithm is conceptually simpler. The key tool for this simplication is the expander decomposition. We exploit it in a very straightforward way compared to how it has been previously used in the literature.
We introduce a notion for hierarchical graph clustering which we call the expander hierarchy and show a fully dynamic algorithm for maintaining such a hierarchy on a graph with $n$ vertices undergoing edge insertions and deletions using $n^{o(1)}$ update time. An expander hierarchy is a tree representation of graphs that faithfully captures the cut-flow structure and consequently our dynamic algorithm almost immediately implies several results including: (1) The first fully dynamic algorithm with $n^{o(1)}$ worst-case update time that allows querying $n^{o(1)}$-approximate conductance, $s$-$t$…
We consider families of codes obtained by "lifting" a base code $\mathcal{C}$ through operations such as $k$-XOR applied to "local views" of codewords of $\mathcal{C}$, according to a suitable $k$-uniform hypergraph. The $k$-XOR operation yields the direct sum encoding used in works of [Ta-Shma, STOC 2017] and [Dinur and Kaufman, FOCS 2017]. We give a general framework for list decoding such lifted codes, as long as the base code admits a unique decoding algorithm, and the hypergraph used for lifting satisfies certain expansion properties. We show that these properties are satisfied by the col…
Li and Panigrahi, in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time $o(mn)$. They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of $s$-$t$ minimum cut computations. In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connect…
The modeling flexibility provided by hypergraphs has drawn a lot of interest from the combinatorial scientific community, leading to novel models and algorithms, their applications, and development of associated tools. Hypergraphs are now a standard tool in combinatorial scientific computing. The modeling flexibility of hypergraphs however, comes at a cost: algorithms on hypergraphs are inherently more complicated than those on graphs, which sometimes translate to nontrivial increases in processing times. Neither the modeling flexibility of hypergraphs, nor the runtime efficiency of graph algo…
Partitioning the vertices of a (hyper)graph into k roughly balanced blocks such that few (hyper)edges run between blocks is a key problem for large-scale distributed processing. A current trend for partitioning huge (hyper)graphs using low computational resources are streaming algorithms. In this work, we propose FREIGHT: a Fast stREamInG Hypergraph parTitioning algorithm which is an adaptation of the widely-known graph-based algorithm Fennel. By using an efficient data structure, we make the overall running of FREIGHT linearly dependent on the pin-count of the hypergraph and the memory consum…
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)…
In the vertex connectivity problem, given an undirected $n$-vertex $m$-edge graph $G$, we need to compute the minimum number of vertices that can disconnect $G$ after removing them. This problem is one of the most well-studied graph problems. From 2019, a new line of work [Nanongkai et al.~STOC'19;SODA'20;STOC'21] has used randomized techniques to break the quadratic-time barrier and, very recently, culminated in an almost-linear time algorithm via the recently announced maxflow algorithm by Chen et al. In contrast, all known deterministic algorithms are much slower. The fastest algorithm [Gab…
Given a hypergraph $H$, the Minimum Connectivity Inference problem asks for a graph on the same vertex set as $H$ with the minimum number of edges such that the subgraph induced by every hyperedge of $H$ is connected. This problem has received a lot of attention these recent years, both from a theoretical and practical perspective, leading to several implemented approximation, greedy and heuristic algorithms. Concerning exact algorithms, only Mixed Integer Linear Programming (MILP) formulations have been experimented, all representing connectivity constraints by the means of graph flows. In th…
Many well-known, real-world problems involve dynamic data which describe the relationship among the entities. Hypergraphs are powerful combinatorial structures that are frequently used to model such data. For many of today's data-centric applications, this data is streaming; new items arrive continuously, and the data grows with time. With paradigms such as Internet of Things and Edge Computing, such applications become more natural and more practical. In this work, we assume a streaming model where the data is modeled as a hypergraph, which is generated at the edge. This data then partitioned…
Hypergraphs naturally represent group interactions, which are omnipresent in many domains: collaborations of researchers, co-purchases of items, joint interactions of proteins, to name a few. In this work, we propose tools for answering the following questions in a systematic manner: (Q1) what are structural design principles of real-world hypergraphs? (Q2) how can we compare local structures of hypergraphs of different sizes? (Q3) how can we identify domains which hypergraphs are from? We first define hypergraph motifs (h-motifs), which describe the connectivity patterns of three connected hy…
We present a shared-memory algorithm to compute high-quality solutions to the balanced $k$-way hypergraph partitioning problem. This problem asks for a partition of the vertex set into $k$ disjoint blocks of bounded size that minimizes the connectivity metric (i.e., the sum of the number of different blocks connected by each hyperedge). High solution quality is achieved by parallelizing the core technique of the currently best sequential partitioner KaHyPar: the most extreme $n$-level version of the widely used multilevel paradigm, where only a single vertex is contracted on each level. This a…
We present the first sublinear-time algorithm for a distributed message-passing network sto compute its edge connectivity $λ$ exactly in the CONGEST model, as long as there are no parallel edges. Our algorithm takes $\tilde O(n^{1-1/353}D^{1/353}+n^{1-1/706})$ time to compute $λ$ and a cut of cardinality $λ$ with high probability, where $n$ and $D$ are the number of nodes and the diameter of the network, respectively, and $\tilde O$ hides polylogarithmic factors. This running time is sublinear in $n$ (i.e. $\tilde O(n^{1-ε})$) whenever $D$ is. Previous sublinear-time distributed algorithms can…
