arXiv subjects

Nate Veldt

Publications and source records attributed to Nate Veldt.

At least 37 records · Page 2Linked to original sources

arXiv2022

Growing a Random Maximal Independent Set Produces a 2-approximate Vertex Cover

This paper presents a fast and simple new 2-approximation algorithm for minimum weighted vertex cover. The unweighted version of this algorithm is equivalent to a well-known greedy maximal independent set algorithm. We prove that this independent set algorithm produces a 2-approximate vertex cover, and we provide a principled new way to generalize it to node-weighted graphs. Our analysis is inspired by connections to a clustering objective called correlation clustering. To demonstrate the relationship between these problems, we show how a simple Pivot algorithm for correlation clustering implicitly approximates a special type of hypergraph vertex cover problem. Finally, we use implicit implementations of this maximal independent set algorithm to develop fast and simple 2-approximation algorithms for certain edge-deletion problems that can be reduced to vertex cover in an approximation preserving way.

cs.DS↗

arXiv2022

Correlation Clustering via Strong Triadic Closure Labeling: Fast Approximation Algorithms and Practical Lower Bounds

Correlation clustering is a widely studied framework for clustering based on pairwise similarity and dissimilarity scores, but its best approximation algorithms rely on impractical linear programming relaxations. We present faster approximation algorithms that avoid these relaxations, for two well-studied special cases: cluster editing and cluster deletion. We accomplish this by drawing new connections to edge labeling problems related to the principle of strong triadic closure. This leads to faster and more practical linear programming algorithms, as well as extremely scalable combinatorial techniques, including the first combinatorial approximation algorithm for cluster deletion. In practice, our algorithms produce approximate solutions that nearly match the best algorithms in quality, while scaling to problems that are orders of magnitude larger.

cs.DS↗

arXiv2021

Approximate Decomposable Submodular Function Minimization for Cardinality-Based Components

Minimizing a sum of simple submodular functions of limited support is a special case of general submodular function minimization that has seen numerous applications in machine learning. We develop fast techniques for instances where components in the sum are cardinality-based, meaning they depend only on the size of the input set. This variant is one of the most widely applied in practice, encompassing, e.g., common energy functions arising in image segmentation and recent generalized hypergraph cut functions. We develop the first approximation algorithms for this problem, where the approximations can be quickly computed via reduction to a sparse graph cut problem, with graph sparsity controlled by the desired approximation factor. Our method relies on a new connection between sparse graph reduction techniques and piecewise linear approximations to concave functions. Our sparse reduction technique leads to significant improvements in theoretical runtimes, as well as substantial practical gains in problems ranging from benchmark image segmentation tasks to hypergraph clustering problems.

cs.LG↗

arXiv2021

Generative hypergraph clustering: from blockmodels to modularity

Hypergraphs are a natural modeling paradigm for a wide range of complex relational systems. A standard analysis task is to identify clusters of closely related or densely interconnected nodes. Many graph algorithms for this task are based on variants of the stochastic blockmodel, a random graph with flexible cluster structure. However, there are few models and algorithms for hypergraph clustering. Here, we propose a Poisson degree-corrected hypergraph stochastic blockmodel (DCHSBM), a generative model of clustered hypergraphs with heterogeneous node degrees and edge sizes. Approximate maximum-likelihood inference in the DCHSBM naturally leads to a clustering objective that generalizes the popular modularity objective for graphs. We derive a general Louvain-type algorithm for this objective, as well as a a faster, specialized "All-Or-Nothing" (AON) variant in which edges are expected to lie fully within clusters. This special case encompasses a recent proposal for modularity in hypergraphs, while also incorporating flexible resolution and edge-size parameters. We show that AON hypergraph Louvain is highly scalable, including as an example an experiment on a synthetic hypergraph of one million nodes. We also demonstrate through synthetic experiments that the detectability regimes for hypergraph community detection differ from methods based on dyadic graph projections. We use our generative model to analyze different patterns of higher-order structure in school contact networks, U.S. congressional bill cosponsorship, U.S. congressional committees, product categories in co-purchasing behavior, and hotel locations from web browsing sessions, finding interpretable higher-order structure. We then study the behavior of our AON hypergraph Louvain algorithm, finding that it is able to recover ground truth clusters in empirical data sets exhibiting the corresponding higher-order structure.

cs.SI↗

arXiv2021

fauci-email: a json digest of Anthony Fauci's released emails

A collection of over 3000 pages of emails sent by Anthony Fauci and his staff were released in an effort to understand the United States government response to the COVID-19 pandemic. We describe how this email data was translated into a resource consisting of json files that make many future studies easy. Findings from our processed data include (i) successful organizational partitions using the simple mincut techniques in Zachary's karate club methodology, (ii) a natural example where the normalized cut and minimum conductance set are extremely different, and (iii) organizational groups identified by optimum modularity clusters that illustrate a working hierarchy. These example uses suggest the data will be useful for future research and pedagogical uses in terms of human and system behavioral interactions. We explain a number of ways to turn email information into a network, a hypergraph, a temporal sequence, and a tensor for subsequent analysis as well as a few examples of such analysis.

cs.SI↗

arXiv2021

Augmented Sparsifiers for Generalized Hypergraph Cuts with Applications to Decomposable Submodular Function Minimization

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 ${\bf w}_e(S) = g(|S|)$ for some concave function $g$, previous work has shown that a generalized hypergraph cut problem can be reduced to a directed graph cut problem on an augmented node set. However, existing reduction procedures often result in a dense graph, even when the hypergraph is sparse, which leads to slow runtimes for algorithms that run on the reduced graph. We introduce a new framework of sparsifying hypergraph-to-graph reductions, where a hypergraph cut defined by submodular cardinality-based splitting functions is $(1+\varepsilon)$-approximated by a cut on a directed graph. Our techniques are based on approximating concave functions using piecewise linear curves. For $\varepsilon > 0$ we need at most $O(\varepsilon^{-1}|e| \log |e|)$ edges to reduce any hyperedge $e$, which leads to faster runtimes for approximating generalized hypergraph $s$-$t$ cut problems. For the machine learning heuristic of a clique splitting function, our approach requires only $O(|e| \varepsilon^{-1/2} \log \log \frac{1}{\varepsilon})$ edges. This sparsification leads to faster approximate min $s$-$t$ graph cut algorithms for certain classes of co-occurrence graphs. Finally, we apply our sparsification techniques to develop approximation algorithms for minimizing sums of cardinality-based submodular functions.

cs.DS↗

arXiv2021

The Generalized Mean Densest Subgraph Problem

Finding dense subgraphs of a large graph is a standard problem in graph mining that has been studied extensively both for its theoretical richness and its many practical applications. In this paper we introduce a new family of dense subgraph objectives, parameterized by a single parameter $p$, based on computing generalized means of degree sequences of a subgraph. Our objective captures both the standard densest subgraph problem and the maximum $k$-core as special cases, and provides a way to interpolate between and extrapolate beyond these two objectives when searching for other notions of dense subgraphs. In terms of algorithmic contributions, we first show that our objective can be minimized in polynomial time for all $p \geq 1$ using repeated submodular minimization. A major contribution of our work is analyzing the performance of different types of peeling algorithms for dense subgraphs both in theory and practice. We prove that the standard peeling algorithm can perform arbitrarily poorly on our generalized objective, but we then design a more sophisticated peeling method which for $p \geq 1$ has an approximation guarantee that is always at least $1/2$ and converges to 1 as $p \rightarrow \infty$. In practice, we show that this algorithm obtains extremely good approximations to the optimal solution, scales to large graphs, and highlights a range of different meaningful notions of density on graphs coming from numerous domains. Furthermore, it is typically able to approximate the densest subgraph problem better than the standard peeling algorithm, by better accounting for how the removal of one node affects other nodes in its neighborhood.

cs.DS↗

arXiv2021

Strongly Local Hypergraph Diffusions for Clustering and Semi-supervised Learning

Hypergraph-based machine learning methods are now widely recognized as important for modeling and using higher-order and multiway relationships between data objects. Local hypergraph clustering and semi-supervised learning specifically involve finding a well-connected set of nodes near a given set of labeled vertices. Although many methods for local clustering exist for graphs, there are relatively few for localized clustering in hypergraphs. Moreover, those that exist often lack flexibility to model a general class of hypergraph cut functions or cannot scale to large problems. To tackle these issues, this paper proposes a new diffusion-based hypergraph clustering algorithm that solves a quadratic hypergraph cut based objective akin to a hypergraph analog of Andersen-Chung-Lang personalized PageRank clustering for graphs. We prove that, for graphs with fixed maximum hyperedge size, this method is strongly local, meaning that its runtime only depends on the size of the output instead of the size of the hypergraph and is highly scalable. Moreover, our method enables us to compute with a wide variety of cardinality-based hypergraph cut functions. We also prove that the clusters found by solving the new objective function satisfy a Cheeger-like quality guarantee. We demonstrate that on large real-world hypergraphs our new method finds better clusters and runs much faster than existing approaches. Specifically, it runs in few seconds for hypergraphs with a few million hyperedges compared with minutes for flow-based technique. We furthermore show that our framework is general enough that can also be used to solve other p-norm based cut objectives on hypergraphs. Our code is available \url{github.com/MengLiuPurdue/LHQD}.

cs.SI↗

arXiv2020

Hypergraph Clustering for Finding Diverse and Experienced Groups

When forming a team or group of individuals, we often seek a balance of expertise in a particular task while at the same time maintaining diversity of skills within each group. Here, we view the problem of finding diverse and experienced groups as clustering in hypergraphs with multiple edge types. The input data is a hypergraph with multiple hyperedge types -- representing information about past experiences of groups of individuals -- and the output is groups of nodes. In contrast to related problems on fair or balanced clustering, we model diversity in terms of variety of past experience (instead of, e.g., protected attributes), with a goal of forming groups that have both experience and diversity with respect to participation in edge types. In other words, both diversity and experience are measured from the types of the hyperedges. Our clustering model is based on a regularized version of an edge-based hypergraph clustering objective, and we also show how naive objectives actually have no diversity-experience tradeoff. Although our objective function is NP-hard to optimize, we design an efficient 2-approximation algorithm and also show how to compute bounds for the regularization hyperparameter that lead to meaningful diversity-experience tradeoffs. We demonstrate an application of this framework in online review platforms, where the goal is to curate sets of user reviews for a product type. In this context, "experience" corresponds to users familiar with the type of product, and "diversity" to users that have reviewed related products.

cs.SI↗

arXiv2020

Minimizing Localized Ratio Cut Objectives in Hypergraphs

Hypergraphs are a useful abstraction for modeling multiway relationships in data, and hypergraph clustering is the task of detecting groups of closely related nodes in such data. Graph clustering has been studied extensively, and there are numerous methods for detecting small, localized clusters without having to explore an entire input graph. However, there are only a few specialized approaches for localized clustering in hypergraphs. Here we present a framework for local hypergraph clustering based on minimizing localized ratio cut objectives. Our framework takes an input set of reference nodes in a hypergraph and solves a sequence of hypergraph minimum $s$-$t$ cut problems in order to identify a nearby well-connected cluster of nodes that overlaps substantially with the input set. Our methods extend graph-based techniques but are significantly more general and have new output quality guarantees. First, our methods can minimize new generalized notions of hypergraph cuts, which depend on specific configurations of nodes within each hyperedge, rather than just on the number of cut hyperedges. Second, our framework has several attractive theoretical properties in terms of output cluster quality. Most importantly, our algorithm is strongly-local, meaning that its runtime depends only on the size of the input set, and does not need to explore the entire hypergraph to find good local clusters. We use our methodology to effectively identify clusters in hypergraphs of real-world data with millions of nodes, millions of hyperedges, and large average hyperedge size with runtimes ranging between a few seconds and a few minutes.

cs.DS↗

arXiv2020

Parameterized Correlation Clustering in Hypergraphs and Bipartite Graphs

Motivated by applications in community detection and dense subgraph discovery, we consider new clustering objectives in hypergraphs and bipartite graphs. These objectives are parameterized by one or more resolution parameters in order to enable diverse knowledge discovery in complex data. For both hypergraph and bipartite objectives, we identify parameter regimes that are equivalent to existing objectives and share their (polynomial-time) approximation algorithms. We first show that our parameterized hypergraph correlation clustering objective is related to higher-order notions of normalized cut and modularity in hypergraphs. It is further amenable to approximation algorithms via hyperedge expansion techniques. Our parameterized bipartite correlation clustering objective generalizes standard unweighted bipartite correlation clustering, as well as bicluster deletion. For a certain choice of parameters it is also related to our hypergraph objective. Although in general it is NP-hard, we highlight a parameter regime for the bipartite objective where the problem reduces to the bipartite matching problem and thus can be solved in polynomial time. For other parameter settings, we present approximation algorithms using linear program rounding techniques. These results allow us to introduce the first constant-factor approximation for bicluster deletion, the task of removing a minimum number of edges to partition a bipartite graph into disjoint bi-cliques. In several experimental results, we highlight the flexibility of our framework and the diversity of results that can be obtained in different parameter settings. This includes clustering bipartite graphs across a range of parameters, detecting motif-rich clusters in an email network and a food web, and forming clusters of retail products in a product review hypergraph, that are highly correlated with known product categories.

cs.DS↗

arXiv2020

Clustering in graphs and hypergraphs with categorical edge labels

Modern graph or network datasets often contain rich structure that goes beyond simple pairwise connections between nodes. This calls for complex representations that can capture, for instance, edges of different types as well as so-called "higher-order interactions" that involve more than two nodes at a time. However, we have fewer rigorous methods that can provide insight from such representations. Here, we develop a computational framework for the problem of clustering hypergraphs with categorical edge labels --- or different interaction types --- where clusters corresponds to groups of nodes that frequently participate in the same type of interaction. Our methodology is based on a combinatorial objective function that is related to correlation clustering on graphs but enables the design of much more efficient algorithms that also seamlessly generalize to hypergraphs. When there are only two label types, our objective can be optimized in polynomial time, using an algorithm based on minimum cuts. Minimizing our objective becomes NP-hard with more than two label types, but we develop fast approximation algorithms based on linear programming relaxations that have theoretical cluster quality guarantees. We demonstrate the efficacy of our algorithms and the scope of the model through problems in edge-label community detection, clustering with temporal data, and exploratory data analysis.

cs.SI↗

arXiv2020

Hypergraph Cuts with General Splitting Functions

The minimum $s$-$t$ cut problem in graphs is one of the most fundamental problems in combinatorial optimization, and graph cuts underlie algorithms throughout discrete mathematics, theoretical computer science, operations research, and data science. While graphs are a standard model for pairwise relationships, hypergraphs provide the flexibility to model multi-way relationships, and are now a standard model for complex data and systems. However, when generalizing from graphs to hypergraphs, the notion of a "cut hyperedge" is less clear, as a hyperedge's nodes can be split in several ways. Here, we develop a framework for hypergraph cuts by considering the problem of separating two terminal nodes in a hypergraph in a way that minimizes a sum of penalties at split hyperedges. In our setup, different ways of splitting the same hyperedge have different penalties, and the penalty is encoded by what we call a splitting function. Our framework opens a rich space on the foundations of hypergraph cuts. We first identify a natural class of cardinality-based hyperedge splitting functions that depend only on the number of nodes on each side of the split. In this case, we show that the general hypergraph $s$-$t$ cut problem can be reduced to a tractable graph $s$-$t$ cut problem if and only if the splitting functions are submodular. We also identify a wide regime of non-submodular splitting functions for which the problem is NP-hard. We also analyze extensions to multiway cuts with at least three terminal nodes and identify a natural class of splitting functions for which the problem can be reduced in an approximation-preserving way to the node-weighted multiway cut problem in graphs, again subject to a submodularity property. Finally, we outline several open questions on general hypergraph cut problems.

cs.DS↗

arXiv2019

Graph Clustering in All Parameter Regimes

Resolution parameters in graph clustering represent a size and quality trade-off. We address the task of efficiently solving a parameterized graph clustering objective for all values of a resolution parameter. Specifically, we consider an objective we call LambdaPrime, involving a parameter $λ\in (0,1)$. This objective is related to other parameterized clustering problems, such as parametric generalizations of modularity, and captures a number of specific clustering problems as special cases, including sparsest cut and cluster deletion. While previous work provides approximation results for a single resolution parameter, we seek a set of approximately optimal clusterings for all values of $λ$ in polynomial time. In particular, we ask the question, how small a family of clusterings suffices to optimize -- or to approximately optimize -- the LambdaPrime objective over the full possible spectrum of $λ$? We obtain a family of logarithmically many clusterings by solving the parametric linear programming relaxation of LambdaPrime at a logarithmic number of parameter values, and round their solutions using existing approximation algorithms. We prove that this number is tight up to a constant factor. Specifically, for a certain class of ring graphs, a logarithmic number of feasible solutions is required to provide a constant-factor approximation for the LambdaPrime LP relaxation in all parameter regimes. We additionally show that for any graph with $n$ nodes and $m$ edges, there exists a set of $m$ or fewer clusterings such that for every $λ\in (0,1)$, the family contains an exact solution to the LambdaPrime objective. There also exists a set of $O(\log n)$ clusterings that provide a $(1+\varepsilon)$-approximate solution in all parameter regimes; we demonstrate simple graph classes for which these bounds are tight.

cs.CC↗

arXiv2019

Flow-Based Local Graph Clustering with Better Seed Set Inclusion

Flow-based methods for local graph clustering have received significant recent attention for their theoretical cut improvement and runtime guarantees. In this work we present two improvements for using flow-based methods in real-world semi-supervised clustering problems. Our first contribution is a generalized objective function that allows practitioners to place strict and soft penalties on excluding specific seed nodes from the output set. This feature allows us to avoid the tendency, often exhibited by previous flow-based methods, to contract a large seed set into a small set of nodes that does not contain all or even most of the seed nodes. Our second contribution is a fast algorithm for minimizing our generalized objective function, based on a variant of the push-relabel algorithm for computing preflows. We make our approach very fast in practice by implementing a global relabeling heuristic and employing a warm-start procedure to quickly solve related cut problems. In practice our algorithm is faster than previous related flow-based methods, and is also more robust in detecting ground truth target regions in a graph, thanks to its ability to better incorporate semi-supervised information about target clusters.

cs.SI↗

arXiv2019

Learning Resolution Parameters for Graph Clustering

Finding clusters of well-connected nodes in a graph is an extensively studied problem in graph-based data analysis. Because of its many applications, a large number of distinct graph clustering objective functions and algorithms have already been proposed and analyzed. To aid practitioners in determining the best clustering approach to use in different applications, we present new techniques for automatically learning how to set clustering resolution parameters. These parameters control the size and structure of communities that are formed by optimizing a generalized objective function. We begin by formalizing the notion of a parameter fitness function, which measures how well a fixed input clustering approximately solves a generalized clustering objective for a specific resolution parameter value. Under reasonable assumptions, which suit two key graph clustering applications, such a parameter fitness function can be efficiently minimized using a bisection-like method, yielding a resolution parameter that fits well with the example clustering. We view our framework as a type of single-shot hyperparameter tuning, as we are able to learn a good resolution parameter with just a single example. Our general approach can be applied to learn resolution parameters for both local and global graph clustering objectives. We demonstrate its utility in several experiments on real-world data where it is helpful to learn resolution parameters from a given example clustering.

cs.SI↗

arXiv2019

A Parallel Projection Method for Metric Constrained Optimization

Many clustering applications in machine learning and data mining rely on solving metric-constrained optimization problems. These problems are characterized by $O(n^3)$ constraints that enforce triangle inequalities on distance variables associated with $n$ objects in a large dataset. Despite its usefulness, metric-constrained optimization is challenging in practice due to the cubic number of constraints and the high-memory requirements of standard optimization software. Recent work has shown that iterative projection methods are able to solve metric-constrained optimization problems on a much larger scale than was previously possible, thanks to their comparatively low memory requirement. However, the major limitation of projection methods is their slow convergence rate. In this paper we present a parallel projection method for metric-constrained optimization which allows us to speed up the convergence rate in practice. The key to our approach is a new parallel execution schedule that allows us to perform projections at multiple metric constraints simultaneously without any conflicts or locking of variables. We illustrate the effectiveness of this execution schedule by implementing and testing a parallel projection method for solving the metric-constrained linear programming relaxation of correlation clustering. We show numerous experimental results on problems involving up to 2.9 trillion constraints.

cs.DC↗

arXiv2018

Correlation Clustering Generalized

We present new results for LambdaCC and MotifCC, two recently introduced variants of the well-studied correlation clustering problem. Both variants are motivated by applications to network analysis and community detection, and have non-trivial approximation algorithms. We first show that the standard linear programming relaxation of LambdaCC has a $Θ(\log n)$ integrality gap for a certain choice of the parameter $λ$. This sheds light on previous challenges encountered in obtaining parameter-independent approximation results for LambdaCC. We generalize a previous constant-factor algorithm to provide the best results, from the LP-rounding approach, for an extended range of $λ$. MotifCC generalizes correlation clustering to the hypergraph setting. In the case of hyperedges of degree $3$ with weights satisfying probability constraints, we improve the best approximation factor from $9$ to $8$. We show that in general our algorithm gives a $4(k-1)$ approximation when hyperedges have maximum degree $k$ and probability weights. We additionally present approximation results for LambdaCC and MotifCC where we restrict to forming only two clusters.

cs.CC↗