Sinan G. Aksoy, Ryan Bennink, Yuzhou Chen et al.
We present and discuss seven different open problems in applied combinatorics. The application areas relevant to this compilation include quantum computing, algorithmic differentiation, topological data analysis, iterative methods, hypergraph cut algorithms, and power systems.
Simon Foucart, Chunyang Liao, Nate Veldt
Learning a smooth graph signal from partially observed data is a well-studied task in graph-based machine learning. We consider this task from the perspective of optimal recovery, a mathematical framework for learning a function from observational data that adopts a worst-case perspective tied to model assumptions on the function to be learned. Earlier work in the optimal recovery literature has shown that minimizing a regularized objective produces optimal solutions for a general class of problems, but did not fully identify the regularization parameter. Our main contribution provides a way to compute regularization parameters that are optimal or near-optimal (depending on the setting), specifically for graph signal processing problems. Our results offer a new interpretation for classical optimization techniques in graph-based learning and also come with new insights for hyperparameter selection. We illustrate the potential of our methods in numerical experiments on several semi-synthetic graph signal processing datasets.
Vedangi Bengali, Nate Veldt
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.
Vicente Balmaseda, Ying Xu, Yixin Cao et al.
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.
Alex Crane, Thomas Stanley, Blair D. Sullivan et al.
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.
Nate Veldt, Thomas Stanley, Benjamin W. Priest et al.
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 $Ω(n^2)$ time to even approximate. We introduce a framework for metric MSTs that first (1) finds a forest of disconnected components using practical heuristics, and then (2) finds a small weight set of edges to connect disjoint components of the forest into a spanning tree. We prove that optimally solving the second step still takes $Ω(n^2)$ time, but we provide a subquadratic 2.62-approximation algorithm. In the spirit of learning-augmented algorithms, we then show that if the 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.
Nate Veldt, Thomas Stanley, Benjamin W. Priest et al.
We present improved learning-augmented algorithms for finding an approximate minimum spanning tree (MST) for points in an arbitrary metric space. Our work follows a recent framework called metric forest completion (MFC), where the learned input is a forest that must be given additional edges to form a full spanning tree. Veldt et al. (2025) showed that optimally completing the forest takes $Ω(n^2)$ time, but designed a 2.62-approximation for MFC with subquadratic complexity. The same method is a $(2γ+ 1)$-approximation for the original MST problem, where $γ\geq 1$ is a quality parameter for the initial forest. We introduce a generalized method that interpolates between this prior algorithm and an optimal $Ω(n^2)$-time MFC algorithm. Our approach considers only edges incident to a growing number of strategically chosen ``representative'' points. One corollary of our analysis is to improve the approximation factor of the previous algorithm from 2.62 for MFC and $(2γ+1)$ for metric MST to 2 and $2γ$ respectively. We prove this is tight for worst-case instances, but we still obtain better instance-specific approximations using our generalized method. We complement our theoretical results with a thorough experimental evaluation.
Nate Veldt
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.
Nate Veldt, Austin R. Benson, Jon Kleinberg
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.
Nate Veldt, Austin R. Benson, Jon Kleinberg
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.
Philip S. Chodrow, Nate Veldt, Austin R. Benson
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.
Nate Veldt, Anthony Wirth, David F. Gleich
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.
Nate Veldt, Austin R. Benson, Jon Kleinberg
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.
Nate Veldt, David F. Gleich, Anthony Wirth
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.
Cameron Ruggles, Nate Veldt, David F. Gleich
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.
Nate Veldt, David Gleich, Anthony Wirth et al.
We outline a new approach for solving optimization problems which enforce triangle inequalities on output variables. We refer to this as metric-constrained optimization, and give several examples where problems of this form arise in machine learning applications and theoretical approximation algorithms for graph clustering. Although these problem are interesting from a theoretical perspective, they are challenging to solve in practice due to the high memory requirement of black-box solvers. In order to address this challenge we first prove that the metric-constrained linear program relaxation of correlation clustering is equivalent to a special case of the metric nearness problem. We then developed a general solver for metric-constrained linear and quadratic programs by generalizing and improving a simple projection algorithm originally developed for metric nearness. We give several novel approximation guarantees for using our framework to find lower bounds for optimal solutions to several challenging graph clustering problems. We also demonstrate the power of our framework by solving optimizing problems involving up to 10^{8} variables and 10^{11} constraints.
Nate Veldt, Anthony Wirth, David F. Gleich
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.