We revisit semi-supervised learning on hypergraphs. Same as previous approaches, our method uses a convex program whose objective function is not everywhere differentiable. We exploit the non-uniqueness of the optimal solutions, and consider confidence intervals which give the exact ranges that unlabeled vertices take in any optimal solution. Moreover, we give a much simpler approach for solving the convex program based on the subgradient method. Our experiments on real-world datasets confirm that our confidence interval approach on hypergraphs outperforms existing methods, and our subgradient method gives faster running times when the number of vertices is much larger than the number of edges. Our experiments also support that using directed hypergraphs to capture causal relationships can improve the prediction accuracy. Furthermore, our model can be readily extended to capture multiclass learning.

A coflow is a collection of related parallel flows that occur typically between two stages of a multi-stage computing task in a network, such as shuffle flows in MapReduce. The coflow abstraction allows applications to convey their semantics to the network so that application-level requirements can be better satisfied. In this paper, we study the routing and scheduling of multiple coflows to minimize the total weighted coflow completion time (CCT). We first propose a rounding-based randomized approximation algorithm, called OneCoflow, for single coflow routing and scheduling. The multiple coflow problem is more challenging as coexisting coflows will compete for the same network resources, such as link bandwidth. To minimize the total weighted CCT, we derive an online multiple coflow routing and scheduling algorithm, called OMCoflow. We then derive a competitive ratio bound of our problem and prove that the competitive ratio of OMCoflow is nearly tight. To the best of our knowledge, this is the first online algorithm with theoretical performance guarantees which considers routing and scheduling simultaneously for multi-coflows. Compared with existing methods, OMCoflow runs more efficiently and avoids frequently rerouting the flows. Extensive simulations on a Facebook data trace show that OMCoflow outperforms the state-of-the-art heuristic schemes significantly (e.g., reducing the total weighted CCT by up to 41.8% and the execution time by up to 99.2% against RAPIER).

The celebrated Cheeger’s Inequality (Alon and Milman 1985; Alon 1986) establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph. In this article, we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. In particular, the operator is induced by a diffusion process on the hypergraph, such that within each hyperedge, measure flows from vertices having maximum weighted measure to those having minimum. Since the operator is nonlinear, we have to exploit other properties of the diffusion process to recover the Cheeger’s Inequality that relates hyperedge expansion with the “second eigenvalue” of the resulting Laplacian. However, we show that higher-order spectral properties cannot hold in general using the current framework. Since higher-order spectral properties do not hold for the Laplacian operator, we instead use the concept of procedural minimizers to consider higher-order Cheeger-like inequalities. For any k ∈ N, we give a polynomial-time algorithm to compute an O (log r )-approximation to the k th procedural minimizer, where r is the maximum cardinality of a hyperedge. We show that this approximation factor is optimal under the SSE hypothesis (introduced by Raghavendra and Steurer (2010)) for constant values of k . Moreover, using the factor-preserving reduction from vertex expansion in graphs to hypergraph expansion, we show that all our results for hypergraphs extend to vertex expansion in graphs.

We revisit semi-supervised learning on hypergraphs. Same as previous approaches, our method uses a convex program whose objective function is not everywhere differentiable. We exploit the non-uniqueness of the optimal solutions, and consider confidence intervals which give the exact ranges that unlabeled vertices take in any optimal solution. Moreover, we give a much simpler approach for solving the convex program based on the subgradient method. Our experiments on real-world datasets confirm that our confidence interval approach on hypergraphs outperforms existing methods, and our sub-gradient method gives faster running times when the number of vertices is much larger than the number of edges.

We consider the edge partitioning problem that partitions the edges of an input graph into multiple balanced components, while minimizing the total number of vertices replicated (one vertex might appear in more than one partition). This problem is critical in minimizing communication costs and running time for several large-scale distributed graph computation platforms (e.g., PowerGraph, Spark GraphX). We first prove that this problem is NP-hard, and then present a new partitioning heuristic with polynomial running time. We provide a worst-case upper bound of replication factor for our heuristic on general graphs. To our knowledge, we are the first to provide such bound for edge partitioning algorithms on general graphs. Applying this bound to random power-law graphs greatly improves the previous bounds of expected replication factor. Extensive experiments demonstrated that our partitioning algorithm consistently produces much smaller replication factors on various benchmark data sets than the state-of-the-art. When deployed in the production graph engine, PowerGraph, in average it reduces replication factor, communication, and running time by 54%, 66%, and 21%, respectively.

A coflow is a collection of related parallel flows that occur typically between two stages of a multi-stage compute task in a network, such as shuffle flows in MapReduce. The coflow abstraction allows applications to convey their semantics to the network so that application-level requirements (e.g., minimizing the completion time of the slowest flow) can be better satisfied. In this paper, we study the routing and scheduling of multiple coflows to minimize the average coflow completion time (CCT). We first propose a rounding-based randomized approximation algorithm, called OneCoflow, for single coflow routing and scheduling. The multiple coflow problem is more challenging as coexisting coflows will compete for the same network resources such as link bandwidths. To minimize the average CCT, we derive an online multiple coflow routing and scheduling algorithm, called OMCoflow, and prove that it has a reasonably good competitive ratio. To the best of our knowledge, this is the first online algorithm with theoretical performance guarantees which considers routing and scheduling simultaneously for multi-coflows. Compared with existing methods, OMCoflow runs more efficiently, and it avoids the problem of frequently rerouting the flows. Extensive simulations on a Facebook data trace show that OMCoflow outperforms the state-of-the-art heuristic schemes significantly (e.g., reducing the average CCT by up to 41.8% and the execution time by up to 99.2% against RAPIER [28]).