Relaxation-Based Coarsening for Multilevel Hypergraph Partitioning

Abstract

Algorithms for many hypergraph problems, including partitioning, utilize multilevel frameworks to achieve a good trade-off between the performance and the quality of results. In this paper we introduce two novel aggregative coarsening schemes and incorporate them within state-of-the-art hypergraph partitioner Zoltan. Our coarsening schemes are inspired by the algebraic multigrid and stable matching approaches. We demonstrate the effectiveness of the developed schemes as a part of multilevel hypergraph partitioning framework on a wide range of problems.

FOBE and HOBE: First- and High-Order Bipartite Embeddings

“…We then fit embeddings to minimize the KL-Divergence between our observations and our embedding-based estimations. The second method, High-Order Bipartite Embedding (HOBE), begins by computing algebraic similarity estimates for each edge [6,25]. Using these heuristic weights, HOBE samples direct, first-and second-order relationships, to which we fit embeddings using mean-squared error.…”

“…Algebraic distance is a measure of dependence between variables popularized in algebraic multigrid (AMG) [24,5,18]. Later, it has been shown to be a reliable and fast way to capture implicit similarities between nodes in graphs [13,17] and hypergraphs that are represented as bipartite graphs [25] (which is leveraged in this paper) taking into account distant neighborhoods. Technically, it is a process of relaxing randomly initialized test vectors using stationary iterative relaxation applied on graph Laplacian homogeneous system of equations, where in the end the algebraic distance between system's variables x i and x j (that correspond to linear system's rows i and j) is defined as an maximum absolute value between the ith and jth components of the test vectors (or, depending on application, as sum or sum of squares of them).…”

“…We run JOR on R independent trials (called test vectors in AMG works, convergence proven in [6]). Formally, for rth test vector a r the update step of JOR is performed as follows, where a (t) r (i) represents node i's algebraic coordinate on iteration t ∈ {1, .., K}, and λ is a damping factor (suggested λ = 0.5 in [25]).…”

Centralities for networks with consumable resources

“…If is the set of web pages known beforehand, then centrality measure defined by such a process can be used to rank the web pages based on the ones in , giving a personalized page ranking strategy. In this class of applications, we can also mention random-walk-based similarity measures on graphs and hypergraphs (Shaydulin et al, 2017;Fouss et al, 2007;Chen & Safro, 2011) that would benefit from introducing resource consumption restrictions for the distance of a random walk.…”

Curvature of Hypergraphs via Multi-Marginal Optimal Transport

“…Roughly speaking, a hypergraph H = (V, E) consists of a finite set V of vertices and a set of hyperedges E ⊆ 2 Vjust as edges in a simple graph that can be identified with vertex pairs, hyperedge E ∈ E are subsets of V . The ubiquitous influence in modeling complex networks fostered numerous recent developments in the theory and algorithms of hypergraphs, including extensive studies of the spectral and algebraic properties such as hypergraph Laplacian [6], hypergraph partitioning [7], Cheeger's inequality for hypergraph [8], and spectrum of hypergraphs [9]. Among many tools developed for better understanding the geometry of graphs, the graph Ricci curvature [10]- [14] has attracted an increasing amount of interest in the past years.…”