The edit distance function: Forbidding induced powers of cycles and other questions

The edit distance between two graphs on the same labeled vertex set is defined to be the size of the symmetric difference of the edge sets. The edit distance between a graph, $G$, and a graph property, $\mathcal{H}$, is the minimum edit distance between $G$ and a graph in $\mathcal{H}$. The edit distance function of a graph property $\mathcal{H}$ is a function of $p\in [0,1]$ that measures, in the limit, the maximum normalized edit distance between a graph of density $p$ and $\mathcal{H}$. In this thesis, we address the edit distance function for the property of having no induced copy of $C_h^t$, the t^{\mbox{th}} power of the cycle of length $h$. For $h\geq 2t(t+1)+1$ and $h$ not divisible by $t+1$, we determine the function for all values of $p$. For $h\geq 2t(t+1)+1$ and $h$ divisible by $t+1$, the function is obtained for all but small values of $p$. We also obtain some results for smaller values of $h$, present alternative proofs of some important previous results using simple optimization techniques and discuss possible extension of the theory to hypergraphs

Wasserstein Soft Label Propagation on Hypergraphs: Algorithm and Generalization Error Bounds

Inspired by recent interests of developing machine learning and data mining algorithms on hypergraphs, we investigate in this paper the semi-supervised learning algorithm of propagating "soft labels" (e.g. probability distributions, class membership scores) over hypergraphs, by means of optimal transportation. Borrowing insights from Wasserstein propagation on graphs [Solomon et al. 2014], we re-formulate the label propagation procedure as a message-passing algorithm, which renders itself naturally to a generalization applicable to hypergraphs through Wasserstein barycenters. Furthermore, in a PAC learning framework, we provide generalization error bounds for propagating one-dimensional distributions on graphs and hypergraphs using 2-Wasserstein distance, by establishing the \textit{algorithmic stability} of the proposed semi-supervised learning algorithm. These theoretical results also shed new lights upon deeper understandings of the Wasserstein propagation on graphs.Comment: To appear in Proc. AAAI'1

New Algorithmic Results in Clustering and Partitioning

Clustering and partitioning tasks have found widespread applications across computing. In machine learning, clustering represents the quintessential unsupervised learning task: grouping similar data points to discover structure in data. In operations research and combinatorial optimization, one is often interested in finding bottlenecks in a network, to identify possible weakness and points of failure. In this work, we discuss recent progress in better understanding computational aspects of clustering and partitioning. Our primary goal is establishing formal mathematical guarantees on the performance of clustering algorithms, as well as proving impossibility results to determine the inherent hardness of the problems we consider. In the first part of the thesis, we discuss graph partitioning tasks, focusing on the theory behind finding small vertex separators: few vertices which, when removed, disconnect the graph into large pieces. We design approximation algorithms for this problem, based on rounding natural convex relaxations. We also outline a recently uncovered connection between this problem and the fastest mixing random walk process on a graph with a target stationary distribution. In the second part of this work we discuss some algorithmic results in partitioning hypergraphs. We introduce a new, expressive class of hypergraph cut functions. We then design approximation algorithms for hypergraph generalizations of the minimum conductance cut problem by leveraging and extending techniques from spectral graph theory to the hypergraph regime. We prove our results for all the cut functions in our newly-defined class. In the process, we also improve on a popular primal-dual algorithmic framework for graph partitioning algorithms. Finally, we address the problem of learning partitions in an interactive way, by querying a same-cluster oracle, which determines whether two points belong to the same cluster. In this context we develop and analyze novel error-resistant algorithms, and provide complementary lower bounds, showing that our algorithms achieve optimal query complexity. To this end, we develop a new analytic framework based on modeling this task as a Rényi-Ulam liar game

Graph Theory

This is the report on an Oberwolfach conference on graph theory, held 16-22 January 2005. There were three main components to the event: 5-minute presentations, lectures, and workshops. All participants were asked to give a 5-minute presentation of their interests on the first day, and subsequent days were divided into lectures and workshops. The latter ranged over many different topics, but the main three topics were: infinite graphs, topological methods and their use to prove theorems in graph theory, and Rota’s conjecture for matroids

Hypergraph Diffusions and Resolvents for Norm-Based Hypergraph Laplacians

The development of simple and fast hypergraph spectral methods has been hindered by the lack of numerical algorithms for simulating heat diffusions and computing fundamental objects, such as Personalized PageRank vectors, over hypergraphs. In this paper, we overcome this challenge by designing two novel algorithmic primitives. The first is a simple, easy-to-compute discrete-time heat diffusion that enjoys the same favorable properties as the discrete-time heat diffusion over graphs. This diffusion can be directly applied to speed up existing hypergraph partitioning algorithms. Our second contribution is the novel application of mirror descent to compute resolvents of non-differentiable squared norms, which we believe to be of independent interest beyond hypergraph problems. Based on this new primitive, we derive the first nearly-linear-time algorithm that simulates the discrete-time heat diffusion to approximately compute resolvents of the hypergraph Laplacian operator, which include Personalized PageRank vectors and solutions to the hypergraph analogue of Laplacian systems. Our algorithm runs in time that is linear in the size of the hypergraph and inversely proportional to the hypergraph spectral gap $\lambda_G$, matching the complexity of analogous diffusion-based algorithms for the graph version of the problem

On vertex independence number of uniform hypergraphs

Abstract Let H be an r-uniform hypergraph with r ≥ 2 and let α(H) be its vertex independence number. In the paper bounds of α(H) are given for different uniform hypergraphs: if H has no isolated vertex, then in terms of the degrees, and for triangle-free linear H in terms of the order and average degree.</jats:p