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Nate Veldt

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9 papers

2 author rows

ICML Conference 2025 Conference Paper

Approximate Forest Completion and Learning-Augmented Algorithms for Metric Minimum Spanning Trees

Nate Veldt
Thomas Stanley
Benjamin W. Priest
Trevor Steil
Keita Iwabuchi
T. S. Jayram
Geoffrey Sanders

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.

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ICML Conference 2025 Conference Paper

Edge-Colored Clustering in Hypergraphs: Beyond Minimizing Unsatisfied Edges

Alex Crane
Thomas Stanley
Blair D. Sullivan
Nate Veldt

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.

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ICML Conference 2024 Conference Paper

Combinatorial Approximations for Cluster Deletion: Simpler, Faster, and Better

Vicente Balmaseda
Ying Xu
Yixin Cao 0001
Nate Veldt

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.

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JMLR Journal 2023 Journal Article

Augmented Sparsifiers for Generalized Hypergraph Cuts

Nate Veldt
Austin R. Benson
Jon Kleinberg

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. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2023. ( edit, beta )

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ICML Conference 2023 Conference Paper

Optimal LP Rounding and Linear-Time Approximation Algorithms for Clustering Edge-Colored Hypergraphs

Nate Veldt

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.

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ICML Conference 2022 Conference Paper

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

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.

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NeurIPS Conference 2021 Conference Paper

Approximate Decomposable Submodular Function Minimization for Cardinality-Based Components

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 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.

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MFCS Conference 2020 Conference Paper

Graph Clustering in All Parameter Regimes

Junhao Gan
David F. Gleich
Nate Veldt
Anthony Wirth
Xin Zhang 0094

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.

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ICML Conference 2016 Conference Paper

A Simple and Strongly-Local Flow-Based Method for Cut Improvement

Nate Veldt
David F. Gleich
Michael W. Mahoney

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.

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