We consider a framework for clustering edge-colored hypergraphs, where the goal is to cluster (equivalently, to color) objects based on the primary type of multiway interactions they participate in. One well-studied objective is to color nodes to minimize the number of unsatisfied hyperedges—those containing one or more nodes whose color does not match the hyperedge color. We motivate and present advances for several directions that extend beyond this minimization problem. We first provide new algorithms for maximizing satisfied edges, which is the same at optimality but is much more challenging to approximate, with all prior work restricted to graphs. We develop the first approximation algorithm for hypergraphs, and then refine it to improve the best-known approximation factor for graphs. We then introduce new objective functions that incorporate notions of balance and fairness, and provide new hardness results, approximations, and fixed-parameter tractability results.

Finding a minimum spanning tree (MST) for $n$ points in an arbitrary metric space is a fundamental primitive for hierarchical clustering and many other ML tasks, but this takes $\Omega(n^2)$ time to even approximate. We introduce a framework for metric MSTs that first (1) finds a forest of trees using practical heuristics, and then (2) finds a small weight set of edges to connect disjoint components in the forest into a spanning tree. We prove that optimally solving step (2) still takes $\Omega(n^2)$ time, but we provide a subquadratic 2.62-approximation algorithm. In the spirit of learning-augmented algorithms, we then show that if the heuristic forest found in step (1) overlaps with an optimal MST, we can approximate the original MST problem in subquadratic time, where the approximation factor depends on a measure of overlap. In practice, we find nearly optimal spanning trees for a wide range of metrics, while being orders of magnitude faster than exact algorithms.

Cluster deletion is an NP-hard graph clustering objective with applications in computational biology and social network analysis, where the goal is to delete a minimum number of edges to partition a graph into cliques. We first provide a tighter analysis of two previous approximation algorithms, improving their approximation guarantees from 4 to 3. Moreover, we show that both algorithms can be derandomized in a surprisingly simple way, by greedily taking a vertex of maximum degree in an auxiliary graph and forming a cluster around it. One of these algorithms relies on solving a linear program. Our final contribution is to design a new and purely combinatorial approach for doing so that is far more scalable in theory and practice.

A recent trend in data mining has explored (hyper)graph clustering algorithms for data with categorical relationship types. Such algorithms have applications in the analysis of social, co-authorship, and protein interaction networks, to name a few. Many such applications naturally have some overlap between clusters, a nuance which is missing from current combinatorial models. Additionally, existing models lack a mechanism for handling noise in datasets. We address both of these concerns by generalizing Edge-Colored Clustering, a recent framework for categorical clustering of hypergraphs. Our generalizations allow for a budgeted number of either (a) overlapping cluster assignments or (b) node deletions. For each new model we present a greedy algorithm which approximately minimizes an edge mistake objective, as well as bicriteria approximations where the second approximation factor is on the budget. Additionally, we address the parameterized complexity of each problem, providing FPT algorithms and hardness results.

Dense subgraph discovery is a fundamental primitive in graph and hypergraph analysis which among other applications has been used for real-time story detection on social media and improving access to data stores of social networking systems. We present several contributions for localized densest subgraph discovery, which seeks dense subgraphs located nearby given seed sets of nodes. We first introduce a generalization of a recent anchored densest subgraph problem, extending this previous objective to hypergraphs and also adding a tunable locality parameter that controls the extent to which the output set overlaps with seed nodes. Our primary technical contribution is to prove when it is possible to obtain a strongly-local algorithm for solving this problem, meaning that the runtime depends only on the size of the input set. We provide a strongly-local algorithm that applies whenever the locality parameter is not too small, and show via counterexample why strongly-local algorithms are impossible below a certain threshold. Along the way to proving our results for localized densest subgraph discovery, we also provide several advances in solving global dense subgraph discovery objectives. This includes the first strongly polynomial time algorithm for the densest supermodular set problem and a flow-based exact algorithm for a heavy and dense subgraph discovery problem in graphs with arbitrary node weights. We demonstrate our algorithms on several web-based data analysis tasks.

Graph clustering is a fundamental task in network analysis where the goal is to detect sets of nodes that are well-connected to each other but sparsely connected to the rest of the graph. We present faster approximation algorithms for an NP-hard parameterized clustering framework called LambdaCC, which is governed by a tunable resolution parameter and generalizes many other clustering objectives such as modularity, sparsest cut, and cluster deletion. Previous LambdaCC algorithms are either heuristics with no approximation guarantees, or computationally expensive approximation algorithms. We provide fast new approximation algorithms that can be made purely combinatorial. These rely on a new parameterized edge labeling problem we introduce that generalizes previous edge labeling problems that are based on the principle of strong triadic closure and are of independent interest in social network analysis. Our methods are orders of magnitude more scalable than previous approximation algorithms and our lower bounds allow us to obtain a posteriori approximation guarantees for previous heuristics that have no approximation guarantees of their own.

We study the approximability of an existing framework for clustering edge-colored hypergraphs, which is closely related to chromatic correlation clustering and is motivated by machine learning and data mining applications where the goal is to cluster a set of objects based on multiway interactions of different categories or types. We present improved approximation guarantees based on linear programming, and show they are tight by proving a matching integrality gap. Our results also include new approximation hardness results, a combinatorial 2-approximation whose runtime is linear in the hypergraph size, and several new connections to well-studied objectives such as vertex cover and hypergraph multiway cut.

Many social networks and web-based datasets are characterized by multiway interactions (e.g., groups of co-purchased online retail products or group conversations in Q&A forums) and hypergraph clustering is a fundamental primitive for analyzing these types of interactions. We present an O(log n)-approximation algorithm for a broad class of hypergraph ratio cut objectives. This includes objectives involving generalized hypergraph cut functions, which allow a user to penalize cut hyperedges differently depending on the number of nodes in each cluster. Our method generalizes the cut-matching framework for graph ratio cuts, and relies only on solving maximum s-t flow problems in a special reduced graph. It is significantly faster than existing hypergraph ratio cut algorithms, while also solving a more general problem. In numerical experiments on various web-based hypergraphs, we show that it quickly finds ratio cut solutions within a small factor of optimality.

Hypergraph generalizations of many graph cut problems and algorithms have recently been introduced to better model data and systems characterized by multiway relationships. Recent work in machine learning and theoretical computer science uses a generalized cut function for a hypergraph $\mathcal{H} = (V,\mathcal{E})$ that associates each hyperedge $e \in \mathcal{E}$ with a splitting function ${\bf w}_e$, which assigns a penalty to each way of separating the nodes of $e$. When each ${\bf w}_e$ satisfies ${\bf w}_e(S) = g(\lvert S \rvert)$ for some concave function $g$, previous work has shown how to reduce the generalized hypergraph cut problem to a directed graph cut problem, although the resulting graph may be very dense. We introduce a framework for sparsifying hypergraph-to-graph reductions, where the hypergraph cut function is $(1+\varepsilon)$-approximated by a cut on a directed graph. For $\varepsilon > 0$ we need at most $O(\varepsilon^{-1}|e| \log |e|)$ edges to reduce any hyperedge $e$, while only $O(|e| \varepsilon^{-1/2} \log \log \frac{1}{\varepsilon})$ edges are needed to approximate the clique expansion, a widely used heuristic in hypergraph clustering. Our framework leads to improved results for solving cut problems in co-occurrence graphs, decomposable submodular function minimization problems, and localized hypergraph clustering problems.

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.

In forming teams or groups, one often aims to balance expertise in a main focus area while also encouraging diversity of skills in each team. In this paper we model the problem of finding diverse groups of individuals who have expertise in a given task as a clustering problem on hypergraphs with heterogeneous edge types. Here, the hyperedge types encode past experience types of groups, and the output of the clustering is groups of individuals (nodes). Unlike complementary problems that seek to find fair or balanced clusters (e.g., in terms of some protected node attributes), our model encourages diversity of past experience within these groups by striking a balance between experience and diversity with respect to node participation in edge types. We show that naive objectives lead to no diversity-experience tradeoff, which motivates our refined model based on regularizing an edge-based hypergraph clustering objective. While optimizing our objective is NP-hard, we design a 2-approximation algorithm that works for a more general class of problems where each node is allowed to have a preference for a particular cluster, and illustrate a technique for computing regularization strength bounds that reveal meaningful diversity/experience tradeoff regimes. We illustrate the utility of our framework on several real-life datasets – most notably to online review platform data – to curate sets of reviews for a given type of product which exhibit a tradeoff between reviewer experience, or familiarity with a product type, and experience, or the reviewer's tendency to also review related product types. In the setting allowing for node preferences, we show that our framework discovers sets of reviews sensitive to user preference.

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 ≥ 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 ≥ 1 has an approximation guarantee that is always at least 1/2 and converges to 1 as p ⟶ ₶. 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.

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 faster 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.

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 graph clustering exist, 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 a few seconds for hypergraphs with a few million hyperedges compared with minutes for a 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.

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.

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.

Resolution parameters in graph clustering control the size and structure of clusters formed by solving a parametric objective function. Typically there is more than one meaningful way to cluster a graph, and solving the same objective function for different resolution parameters produces clusterings at different levels of granularity, each of which can be meaningful depending on the application. In this paper, we address the task of efficiently solving a parameterized graph clustering objective for all values of a resolution parameter. Specifically, we consider a new analysis-friendly objective we call LambdaPrime, involving a parameter λ ∈ (0,1). LambdaPrime is an adaptation of LambdaCC, a significant family of instances of the Correlation Clustering (minimization) problem. Indeed, LambdaPrime and LambdaCC are closely related to other parameterized clustering problems, such as parametric generalizations of modularity. They capture a number of specific clustering problems as special cases, including sparsest cut and cluster deletion. While previous work provides approximation results for a single value of the resolution parameter, we seek a set of approximately optimal clusterings for all values of λ in polynomial time. More specifically, we show that when a graph has m edges and n nodes, there exists a set of at most m clusterings such that, for every λ ∈ (0,1), the family contains an optimal solution to the LambdaPrime objective. This bound is tight on star graphs. We obtain a family of O(log n) clusterings by solving the parametric linear programming (LP) relaxation of LambdaPrime at O(log n) λ values, and rounding each LP solution using existing approximation algorithms. We prove that this is asymptotically tight: for a certain class of ring graphs, for all values of λ, Ω(log n) feasible solutions are required to provide a constant-factor approximation for the LambdaPrime LP relaxation. To minimize the size of the clustering family, we further propose an algorithm that yields a family of solutions of a size no more than twice of the minimum LP-approximating family.

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.

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.

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.

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 Theta(log n) integrality gap for a certain choice of the parameter lambda. 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 lambda. 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.

Network alignment or graph matching is the classic problem of finding matching vertices between two graphs with applications in network de-anonymization and bioinformatics. There exist a wide variety of algorithms for it, but a challenging scenario for all of the algorithms is aligning two networks without any information about which nodes might be good matches. In this case, the vast majority of principled algorithms demand quadratic memory in the size of the graphs. We show that one such method---the recently proposed and theoretically grounded EigenAlign algorithm---admits a novel implementation which requires memory that is linear in the size of the graphs. The key step to this insight is identifying low-rank structure in the node-similarity matrix used by EigenAlign for determining matches. With an exact, closed-form low-rank structure, we then solve a maximum weight bipartite matching problem on that low-rank matrix to produce the matching between the graphs. For this task, we show a new, a-posteriori, approximation bound for a simple algorithm to approximate a maximum weight bipartite matching problem on a low-rank matrix. The combination of our two new methods then enables us to tackle much larger network alignment problems than previously possible and to do so quickly. Problems that take hours with existing methods take only seconds with our new algorithm. We thoroughly validate our low-rank algorithm against the original EigenAlign approach. We also compare a variety of existing algorithms on problems in bioinformatics and social networks. Our approach can also be combined with existing algorithms to improve their performance and speed.

Graph clustering, or community detection, is the task of identifying groups of closely related objects in a large network. In this paper we introduce a new community detection framework called LambdaCC that is based on a specially weighted version of correlation clustering. A key component in our methodology is a clustering resolution parameter, lambda, which implicitly controls the size and structure of clusters formed by our framework. We show that, by increasing this parameter, our objective effectively interpolates between two different strategies in graph clustering: finding a sparse cut and forming dense subgraphs. Our methodology unifies and generalizes a number of other important clustering quality functions including modularity, sparsest cut, and cluster deletion, and places them all within the context of an optimization problem that has been well studied from the perspective of approximation algorithms. Our approach to clustering is particularly relevant in the regime of finding dense clusters, as it leads to a 2-approximation for the cluster deletion problem. We use our approach to cluster several graphs, including large collaboration networks and social networks.

Correlation clustering is a technique for aggregating data based on qualitative information about which pairs of objects are labeled `similar' or `dissimilar.' Because the optimization problem is NP-hard, much of the previous literature focuses on finding approximation algorithms. In this paper we explore how to solve the correlation clustering objective exactly when the data to be clustered can be represented by a low-rank matrix. We prove in particular that correlation clustering can be solved in polynomial time when the underlying matrix is positive semidefinite with small constant rank, but that the task remains NP-hard in the presence of even one negative eigenvalue. Based on our theoretical results, we develop an algorithm for efficiently ``solving'' low-rank positive semidefinite correlation clustering by employing a procedure for zonotope vertex enumeration. We demonstrate the effectiveness and speed of our algorithm by using it to solve several clustering problems on both synthetic and real-world data.

Many graph-based learning problems can be cast as finding a good set of vertices nearby a seed set, and a powerful methodology for these problems is based on minimum cuts and maximum flows. We introduce and analyze a new method for locally-biased graph-based learning called SimpleLocal, which finds good conductance cuts near a set of seed vertices. An important feature of our algorithm is that it is strongly-local, meaning it does not need to explore the entire graph to find cuts that are locally optimal. This method is related to other strongly-local flow-based methods, but it enables a simple implementation. We also show how it achieves localization through an implicit l1-norm penalty term. As a flow-based method, our algorithm exhibits several advantages in terms of cut optimality and accurate identification of target regions in a graph. We demonstrate the power of SimpleLocal solving segmentation problems on a 467 million edge graph based on an MRI scan.