We revisit the (\(f,q\))-Clustering problem that we introduced in a recent work [SODA’25]. Here, \(f\) and \(g\) are symmetric, monotone norms called inner and outer norms, respectively. The task is to partition a given set of points in a metric space into \(k\) clusters each represented by a cluster center. Each cluster is assigned a cluster cost, determined by the norm \(f\) applied to the vector of point-center distances in the cluster. The goal is to minimize the value of the norm \(g\) when applied to the vector of cluster costs. This problem subsumes fundamental clustering problems such as \(k\)-Center (i.e., \(\mathcal L_\infty, \mathcal L_\infty\)-Clustering), \(k\)-Median (i.e., \(\mathcal L_1, \mathcal L_1\)-Clustering), Min-Sum of Radii (i.e., \(\mathcal L_\infty, \mathcal L_1\)-Clustering), and Min-Load \(k\)-Clustering (i.e., \(\mathcal L_1, \mathcal L_\infty\)-Clustering). In our previous work, we focused on certain special cases of this problem for which we designed constant-factor approximation algorithms. Our bounds for more general settings left, however, large gaps to the known bounds for the basic problems they capture.

Geometric data structures have been extensively studied in the regime where the dimension is much smaller than the number of input points. But in many scenarios in Machine Learning, the dimension can be much higher than the number of points and can be so high that the data structure might be unable to read and store all coordinates of the input and query points. Inspired by these scenarios and related studies in feature selection and explainable clustering, we initiate the study of geometric data structures in this ultra-high dimensional regime. Our focus is the approximate nearest neighbor problem. In this problem, we are given a set of n points C ⊆ ℝ^d and have to produce a small data structure that can quickly answer the following query: given q ∈ ℝ^d, return a point c ∈ C that is approximately nearest to q, where the distance is under 𝓁₁, 𝓁₂, or other norms. Many groundbreaking (1+ε)-approximation algorithms have recently been discovered for 𝓁₁- and 𝓁₂-norm distances in the regime where d≪ n. The main question in this paper is: Is there a data structure with sublinear (o(nd)) space and sublinear (o(d)) query time when d≫ n? This question can be partially answered from the machine-learning literature: - For 𝓁₁-norm distances, an Õ(log(n))-approximation data structure with Õ(n log d) space and O(n) query time can be obtained from explainable clustering techniques [Dasgupta et al. ICML'20; Makarychev and Shan ICML'21; Esfandiari, Mirrokni, and Narayanan SODA'22; Gamlath et al. NeurIPS'21; Charikar and Hu SODA'22]. - For 𝓁₂-norm distances, a (√3+ε)-approximation data structure with Õ(n log(d)/poly(ε)) space and Õ(n/poly(ε)) query time can be obtained from feature selection techniques [Boutsidis, Drineas, and Mahoney NeurIPS'09; Boutsidis et al. IEEE Trans. Inf. Theory'15; Cohen et al. STOC'15]. - For 𝓁_p-norm distances, a O(n^{p-1}log²(n))-approximation data structure with O(nlog(n) + nlog(d)) space and O(n) query time can be obtained from the explainable clustering algorithms of [Gamlath et al. NeurIPS'21]. An important open problem is whether a (1+ε)-approximation data structure exists. This is not known for any norm, even with higher (e.g. poly(n)⋅ o(d)) space and query time. In this paper, we answer this question affirmatively. We present (1+ε)-approximation data structures with the following guarantees. - For 𝓁₁- and 𝓁₂-norm distances: Õ(n log(d)/poly(ε)) space and Õ(n/poly(ε)) query time. We show that these space and time bounds are tight up to poly (log n/ε) factors. - For 𝓁_p-norm distances: Õ(n² log(d) (log log(n)/ε)^p) space and Õ (n(log log(n)/ε)^p) query time. Via simple reductions, our data structures imply sublinear-in-d data structures for some other geometric problems; e.g. approximate orthogonal range search (in the style of [Arya and Mount SoCG'95]), furthest neighbor, and give rise to a sublinear O(1)-approximate representation of k-median and k-means clustering. We hope that this paper inspires future work on sublinear geometric data structures.

We give improved approximations for two metric Traveling Salesman Problem (TSP) variants. In Ordered TSP (OTSP) we are given a linear ordering on a subset of nodes o1,. .., ok. The TSP solution must have that o i+1 is visited at some point after Oi for each 1 ≤ i ≤ k. This is the special case of Precedence- Constrained TSP (PTSP) in which the precedence constraints are given by a single chain on a subset of nodes. In k-Person TSP Path (k-TSPP), we are given pairs of nodes (s1, t1), …, (sk, tk ). The goal is to find an si-ti path with minimum total cost such that every node is visited by at least one path.

We initiate the study of the following general clustering problem. We seek to partition a given set P of data points into k clusters by finding a set X of k centers and assigning each data point to one of the centers. The cost of a cluster, represented by a center x ∊ X, is a monotone, symmetric norm f (called inner norm) of the vector of distances of points assigned to x. The goal is to minimize a norm g (called outer norm) of the vector of cluster costs. This problem, which we call (f, g )-Clustering, generalizes many fundamental clustering problems such as k-Center (i.e., (𝓛∞, 𝓛∞)-Clustering), k-Median (i.e., (𝓛1, 𝓛1)-Clustering), Min-Sum of Radii (i.e., (𝓛∞, 𝓛1)-Clustering), and Min-Load k-Clustering (i.e., (𝓛1, L∞)-Clustering). A recent line of research (Byrka et al. [STOC’18], Chakrabarty, Swamy [ICALP’18, STOC’19], and Abbasi et al. [FOCS’23]) studies norm objectives that are oblivious to the cluster structure such as k-Median and k-Center. In contrast, our problem models cluster-aware objectives including Min-Sum of Radii and Min-Load k-Clustering.

We consider the well-studied Robust (k,z)-Clustering problem, which generalizes the classic k-Median, k-Means, and k-Center problems and arises in the domains of robust optimization [Anthony, Goyal, Gupta, Nagarajan, Math. Oper. Res. 2010] and in algorithmic fairness [Abbasi, Bhaskara, Venkatasubramanian, 2021 & Ghadiri, Samadi, Vempala, 2022]. Given a constant z ≥ 1, the input to Robust (k,z)-Clustering is a set P of n points in a metric space (M,δ), a weight function w: P → ℝ_{≥ 0} and a positive integer k. Further, each point belongs to one (or more) of the m many different groups S_1,S_2,…,S_m ⊆ P. Our goal is to find a set X of k centers such that max_{i ∈ [m]} ∑_{p ∈ S_i} w(p) δ(p,X)^z is minimized. Complementing recent work on this problem, we give a comprehensive understanding of the parameterized approximability of the problem in geometric spaces where the parameter is the number k of centers. We prove the following results: [(i)] 1) For a universal constant η₀ > 0.0006, we devise a 3^z(1-η₀)-factor FPT approximation algorithm for Robust (k,z)-Clustering in discrete high-dimensional Euclidean spaces where the set of potential centers is finite. This shows that the lower bound of 3^z for general metrics [Goyal, Jaiswal, Inf. Proc. Letters, 2023] no longer holds when the metric has geometric structure. 2) We show that Robust (k,z)-Clustering in discrete Euclidean spaces is (√{3/2}- o(1))-hard to approximate for FPT algorithms, even if we consider the special case k-Center in logarithmic dimensions. This rules out a (1+ε)-approximation algorithm running in time f(k,ε)poly(m,n) (also called efficient parameterized approximation scheme or EPAS), giving a striking contrast with the recent EPAS for the continuous setting where centers can be placed anywhere in the space [Abbasi et al., FOCS'23]. 3) However, we obtain an EPAS for Robust (k,z)-Clustering in discrete Euclidean spaces when the dimension is sublogarithmic (for the discrete problem, earlier work [Abbasi et al., FOCS'23] provides an EPAS only in dimension o(log log n)). Our EPAS works also for metrics of sub-logarithmic doubling dimension.

Every ReLU Neural Network (NN) tessellates its input space into activation regions. Studying this tessellation provides insights into some of the architecture’s properties. Recent research has focused on computing the tessellation generated by the output neurons, with the main objective of counting the number of generated regions. This tessellation is achieved through the encoding of each activation region using its bounding hyperplanes. In contrast, we introduce SkelEx, a novel variation of this extraction technique that encodes the extracted regions using their vertices. Next, we introduce BoundEx, which is the first algorithm designed to transform the tessellations of the output neurons into the learned decision boundary defined via the membership polytopes. We highlight the geometric perspective on forward propagation and inference introduced by SkelEx and BoundEx, allowing for more interpretable and intuitive insights. We do so by providing: 1) explanations to the impact of earlier layers; and 2) new perspective on the existence of adversarial examples together with their categorization. The code is available on https://github.com/PawPuk/SkelEx-BoundEx.

The fundamental Sparsest Cut problem takes as input a graph G together with edge capacities and demands and seeks a cut that minimizes the ratio between the capacities and demands across the cuts. For n -vertex graphs G of treewidth k , Chlamtáč, Krauthgamer, and Raghavendra (APPROX’10) presented an algorithm that yields a factor- \(2^{2^k}\) approximation in time \(2^{O(k)} \cdot n^{O(1)}\) . Later, Gupta, Talwar, and Witmer (STOC’13) showed how to obtain a 2-approximation algorithm with a blown-up runtime of \(n^{O(k)}\) . An intriguing open question is whether one can simultaneously achieve the best out of the aforementioned results, that is, a factor-2 approximation in time \(2^{O(k)} \cdot n^{O(1)}\) . In this article, we make significant progress towards this goal via the following results: (i) A factor- \(O(k^2)\) approximation that runs in time \(2^{O(k)} \cdot n^{O(1)}\) , directly improving the work of Chlamtáč et al. while keeping the runtime single-exponential in k . (ii) For any \(\varepsilon \in (0,1]\) , a factor- \(O(1/\varepsilon ^2)\) approximation whose runtime is \(2^{O(k^{1+\varepsilon }/\varepsilon)} \cdot n^{O(1)}\) , implying a constant-factor approximation whose runtime is nearly single-exponential in k and a factor- \(O(\log ^2 k)\) approximation in time \(k^{O(k)} \cdot n^{O(1)}\) . Key to these results is a new measure of a tree decomposition that we call combinatorial diameter , which may be of independent interest.

In the reconciliation $k$-median problem we ask to cluster a set of data points by picking $k$ cluster centers so as to minimize the sum of distances of the data points to their cluster centers plus the sum of pairwise distances between the centers. The problem, which is a variant of classic $k$-median, aims to find a set of cluster centers that are not too far from each other, and it has applications, or example, when selecting a committee to deliberate on a controversial topic. This problem was introduced recently (Ordozgoiti et al., 2019), and it was shown that a local-search-based algorithm is always within a factor $O(k)$ of an optimum solution and performs well in practice. In this paper, we demonstrate a close connection of reconciliation $k$-median to a variant of the $k$-facility location problem, in which each potential cluster center has an individual opening cost and we aim at minimizing the sum of client-center distances and the opening costs. This connection enables us to provide a new algorithm for reconciliation $k$-median that yields a constant-factor approximation (independent of $k$). We also provide a sparsification scheme that reduces the number of potential cluster centers to $O(k)$ in order to substantially speed up approximation algorithms. We empirically compare our new algorithms with the previous local-search approach, showing improved performance and stability. In addition, we show how our sparsification approach helps to reduce computation time without significantly compromising the solution quality.

This paper considers the well-studied algorithmic regime of designing a $(1+\epsilon)$-approximation algorithm for a k-clustering problem that runs in time $f(k,\epsilon)poly(n)$ (sometimes called an efficient parameterized approximation scheme or EPAS for short1). Notable results of this kind include EPASes in the high-dimensional Euclidean setting for k-center [Badŏiu, Har-Peled, Indyk; STOC’02] as well as k-median, and k-means [Kumar, Sabharwal, Sen; J. ACM 2010]. Our main contribution is a clean and simple EPAS that settles more than ten clustering problems (across multiple well-studied objectives as well as metric spaces) and unifies well-known EPASes. More specifically, our algorithm gives EPASes in the following settings:•Clustering objectives: k-means, k-center, k-median, priority k-center, $\ell$-centrum, ordered k-median, socially fair k-median (aka robust k-median), or any other objective that can be formulated as minimizing a monotone (not necessarily symmetric!) norm of the distances of the points from the solution (generalizing the symmetric formulation introduced by Chakrabarty and Swamy [STOC’19]).•Metric spaces: Continuous high-dimensional Euclidean spaces, metrics of bounded doubling dimension, bounded treewidth metrics, and planar metrics. Prior to our results, EPASes were only known for vanilla clustering objectives (k-means, k-median, and k-center) and each such algorithm is tailored to work for the specific input metric and clustering objective (e.g., EPASes for k means and k-center in $\mathbb{R}^{d}$ are conceptually very different). In contrast, our algorithmic framework is applicable to a wide range of well-studied objective functions in a uniform way, and is (almost) entirely oblivious to any specific metric structures and yet is able to effectively exploit those unknown structures. In particular, our algorithm is not based on the (metric- and objective-specific) technique of coresets. Key to our analysis is a new concept that we call bounded $\epsilon$-scatter dimension—an intrinsic complexity measure of a metric space that is a relaxation of the standard notion of bounded doubling dimension(often used as a source of algorithmic tractability for geometric problems). Our main technical result shows that two conditions are essentially sufficient for our algorithm to yield an EPAS on the input metric M for any clustering objective:(i)The objective is described by a monotone norm, and(ii)the $\epsilon$-scatter dimension of M is upper bounded by a function of $\epsilon$.1Quick remarks: (i) An EPAS is not comparable to polynomial time approximation schemes (PTAS), (ii) before the term EPAS was invented some researchers call this type of approximation schemes a PTAS or simply an approximation scheme (in clustering, it is often assumed that k is small) [1], [2], and (iii) both EPAS and PTAS are implied by the existence of efficient polynomial time approximation schemes (EPTAS).

Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering constraints. We present a tight approximation algorithm that for any constant $ε>0$ achieves a guarantee of $1-\frac{1}{\mathrm{e}}-ε$ while violating only the covering constraints by a multiplicative factor of $1-ε$. Our algorithm is based on a novel enumeration method, which unlike previous known enumeration techniques, can handle both packing and covering constraints. We extend the above main result by additionally handling a matroid independence constraints as well as finding (approximate) pareto set optimal solutions when multiple submodular objectives are present. Finally, we propose a novel and purely combinatorial dynamic programming approach that can be applied to several special cases of the problem yielding not only {\em deterministic} but also considerably faster algorithms. For example, for the well studied special case of only packing constraints (Kulik {\em et. al.} [Math. Oper. Res. `13] and Chekuri {\em et. al.} [FOCS `10]), we are able to present the first deterministic non-trivial approximation algorithm. We believe our new combinatorial approach might be of independent interest.

In their seminal work, Mustafa and Ray [30] showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search - this is one of the most general approaches known for such problems. Their result applies if a naturally defined "exchange graph" for two feasible solutions is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson [17]. Obtaining similar results for the related maximum coverage problem (MC) seems non-trivial due to the hard cardinality constraint. In fact, while Badanidiyuru, Kleinberg, and Lee [4] have shown (via a different analysis) that local search yields a PTAS for two-dimensional real halfspaces, they only conjectured that the same holds true for dimension three. Interestingly, at this point it was already known that local search provides a PTAS for the corresponding set cover case and this followed directly from the approach of Mustafa and Ray. In this work we provide a way to address the above-mentioned issue. First, we propose a color-balanced version of the planar separator theorem. The resulting subdivision approximates locally in each part the global distribution of the colors. Second, we show how this roughly balanced subdivision can be employed in a more careful analysis to strictly obey the hard cardinality constraint. More specifically, we obtain a PTAS for any "planarizable" instance of MC and thus essentially for all cases where the corresponding SC instance can be tackled via the approach of Mustafa and Ray. As a corollary, we confirm the conjecture of Badanidiyuru, Kleinberg, and Lee [4] regarding real halfspaces in dimension three. We feel that our ideas could also be helpful in other geometric settings involving a cardinality constraint.

We initiate the study of the following natural geometric optimization problem. The input is a set of axis-aligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every rectangle is stabbed by some line segment. A line segment stabs a rectangle if it intersects its left and its right boundary. The problem, which we call Stabbing, can be motivated by a resource allocation problem and has applications in geometric network design. To the best of our knowledge, only special cases of this problem have been considered so far. Stabbing is a weighted geometric set cover problem, which we show to be NP-hard. While for general set cover the best possible approximation ratio is Theta(log n), it is an important field in geometric approximation algorithms to obtain better ratios for geometric set cover problems. Chan et al. [SODA'12] generalize earlier results by Varadarajan [STOC'10] to obtain sub-logarithmic performances for a broad class of weighted geometric set cover instances that are characterized by having low shallow-cell complexity. The shallow-cell complexity of Stabbing instances, however, can be high so that a direct application of the framework of Chan et al. gives only logarithmic bounds. We still achieve a constant-factor approximation by decomposing general instances into what we call laminar instances that have low enough complexity. Our decomposition technique yields constant-factor approximations also for the variant where rectangles can be stabbed by horizontal and vertical segments and for two further geometric set cover problems.

We study the Ordered k-Median problem, in which the solution is evaluated by first sorting the client connection costs and then multiplying them with a predefined non-increasing weight vector (higher connection costs are taken with larger weights). Since the 1990s, this problem has been studied extensively in the discrete optimization and operations research communities and has emerged as a framework unifying many fundamental clustering and location problems such as k-Median and k-Center. Obtaining non-trivial approximation algorithms was an open problem even for simple topologies such as trees. Recently, Aouad and Segev (2017) were able to obtain an O(log n) approximation algorithm for Ordered k-Median using a sophisticated local-search approach. The existence of a constant-factor approximation algorithm, however, remained open even for the rectangular weight vector.

Abstract We study the problem of finding a minimum weight connected subgraph spanning at least k vertices on planar, node-weighted graphs. We give a (4 + ε)-approximation algorithm for this problem. We achieve this by utilizing the recent Lagrangian-multiplier preserving (LMP) primal-dual 3-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al. (SWAT’16) and adopting an approach by Chudak et al. (Math. Prog. ’04) regarding Lagrangian relaxation for the edge-weighted variant. In particular, we improve the procedure of picking additional vertices (tree merging procedure) given by Sadeghian (2013) by taking a constant number of recursive steps and utilizing the limited guessing procedure of Arora and Karakostas (Math. Prog. ’06). More generally, our approach readily gives a (4/3 ⋅ r + ε)-approximation on any graph class where the algorithm of Byrka et al. for the prize-collecting version gives an r-approximation. We argue that this can be interpreted as a generalization of an analogous result by Könemann et al. (Algorithmica ’11) for partial cover problems. Together with a lower bound construction by Mestre (STACS’08) for partial cover this implies that our bound is essentially best possible among algorithms that utilize an LMP algorithm for the Lagrangian relaxation as a black box. In addition to that, we argue by a more involved lower bound construction that even using the LMP algorithm by Byrka et al. in a non-black-box fashion could not beat the factor 4/3 ⋅ r when the tree merging step relies only on the solutions output by the LMP algorithm.

Knapsack median is a generalization of the classic k -median problem in which we replace the cardinality constraint with a knapsack constraint. It is currently known to be 32-approximable. We improve on the best known algorithms in several ways, including adding randomization and applying sparsification as a preprocessing step. The latter improvement produces the first LP for this problem with bounded integrality gap. The new algorithm obtains an approximation factor of 17.46. We also give a 3.05 approximation with small budget violation.

In the classical k-median problem the goal is to select a subset of at most k facilities in order to minimize the total cost of opened facilities and established connections between clients and opened facilities. We consider the capacitated version of the problem, where a single facility may only serve a limited number of clients. We construct approximation algorithms slightly violating the capacities based on rounding a fractional solution to the standard LP. It is well known that the standard LP (even in the case of uniform capacities) has unbounded integrality gap if we only allow violating capacities by a factor smaller than 2, or if we only allow violating the number of facilities by a factor smaller than 2. It is also known that violating capacities by a factor of 2 + ε is sufficient to obtain constant factor approximation of the connection cost in the case of uniform capacities. In this paper we substantially extend this result in the following two directions. On one hand, we obtain a 2+ε capacity violating algorithm to the more general k-facility location problem with uniform capacities, where opening facilities incurs a location specific opening cost. On the other hand, we show that violating capacities by a slightly bigger factor of 3 + ε is sufficient to obtain constant factor approximation of the connection cost also in the case of the non-uniform hard capacitated k-median problem. Our algorithms first use the clustering of Charikar et al. to partition the facilities into sets of total fractional opening at least 1 — 1/ℓ for some fixed ℓ. Then we exploit the technique of Levi, Shmoys, and Swamy developed for the capacitated facility location problem, which is to locally group the demand from clients to obtain a system of single node demand instances. Next, depending on the setting, we either work with stars of facilities (for non-uniform capacities), or we use a dedicated routing tree on the demand nodes (for non-uniform opening cost), to redistribute the demand that cannot be satisfied locally within the clusters.

In a directed graph G with non-correlated edge lengths and costs, the network design problem with bounded distances asks for a cost-minimal spanning subgraph subject to a length bound for all node pairs. We give a bi-criteria (2+\varepsilon,O(n^{0.5+\varepsilon}))-approximation for this problem. This improves on the currently best known linear approximation bound, at the cost of violating the distance bound by a factor of at most 2+\varepsilon. In the course of proving this result, the related problem of directed shallow-light Steiner trees arises as a subproblem. In the context of directed graphs, approximations to this problem have been elusive. We present the first non-trivial result by proposing a (1+\varepsilon,O(|R|^{\varepsilon}))-ap\-proximation, where R is the set of terminals. Finally, we show how to apply our results to obtain an (\alpha+\varepsilon,O(n^{0.5+\varepsilon}))-approximation for light-weight directed \alpha-spanners. For this, no non-trivial approximation algorithm has been known before. All running times depends on n and \varepsilon and are polynomial in n for any fixed \varepsilon>0.

Scatter plots are diagrams that visualize two-dimensional data as sets of points in the plane. They allow users to detect correlations and clusters in the data. Whether or not a user can accomplish these tasks highly depends on the aspect ratio selected for the plot, i.e., the ratio between the horizontal and the vertical extent of the diagram. We argue that an aspect ratio is good if the Delaunay triangulation of the scatter plot at this aspect ratio has some nice geometric property, e.g., a large minimum angle or a small total edge length. More precisely, we consider the following optimization problem. Given a set Q of points in the plane, find a scale factor s such that scaling the x-coordinates of the points in Q by s and the y-coordinates by 1=s yields a point set P(s) that optimizes a property of the Delaunay triangulation of P(s), over all choices of s. We present an algorithm that solves this problem efficiently and demonstrate its usefulness on real-world instances. Moreover, we discuss an empirical test in which we asked 64 participants to choose the aspect ratios of 18 scatter plots. We tested six different quality measures that our algorithm can optimize. In conclusion, minimizing the total edge length and minimizing what we call the 'uncompactness' of the triangles of the Delaunay triangulation yielded the aspect ratios that were most similar to those chosen by the participants in the test.

In this paper, we investigate the problem of labeling point sites in focus regions of maps or diagrams. This problem occurs, for example, when the user of a mapping service wants to see the names of restaurants or other POIs in a crowded downtown area but keep the overview over a larger area. Our approach is to place the labels at the boundary of the focus region and connect each site with its label by a linear connection, which is called a leader. In this way, we move labels from the focus region to the less valuable context region surrounding it. In order to make the leader layout well readable, we present algorithms that rule out crossings between leaders and optimize other characteristics such as total leader length and distance between labels. This yields a new variant of the boundary labeling problem, which has been studied in the literature. Other than in traditional boundary labeling, where leaders are usually schematized polylines, we focus on leaders that are either straight-line segments or Bezier curves. Further, we present algorithms that, given the sites, find a position of the focus region that optimizes the above characteristics. We also consider a variant of the problem where we have more sites than space for labels. In this situation, we assume that the sites are prioritized by the user. Alternatively, we take a new facility-location perspective which yields a clustering of the sites. We label one representative of each cluster. If the user wishes, we apply our approach to the sites within a cluster, giving details on demand.