Title: Parameterized Approximation Schemes for Fair-Range Clustering
Venue: NeurIPS 2024
Accepted: yes
Keywords: Approximation algorithms

Fixed-parameter tractability
Abstract: Fair-range clustering extends classical clustering formulations by associating each data point with one or more demographic labels. It imposes lower and upper bound constraints on the number of facilities opened for each label, ensuring fair representation of all demographic groups by the selected facilities. In this paper we focus on the fair-range $k$-median and $k$-means problems in Euclidean spaces. We give $(1+\varepsilon)$-approximation algorithms with fixed-parameter tractable running times for both problems, parameterized by the numbers of opened facilities and demographic labels. For Euclidean metrics, these are the first parameterized approximation schemes for the problems, improving upon the previously known $O(1)$-approximation ratios given by Thejaswi et al. (KDD 2022).

Parameterized Approximation Schemes for Fair-Range Clustering

Abstract

Fair-range clustering extends classical clustering formulations by associating each data point with one or more demographic labels. It imposes lower and upper bound constraints on the number of facilities opened for each label, ensuring fair representation of all demographic groups by the selected facilities. In this paper we focus on the fair-range k -median and k -means problems in Euclidean spaces. We give (1 + ε ) -approximation algorithms with fixed-parameter tractable running times for both problems, parameterized by the numbers of opened facilities and demographic labels. For Euclidean metrics, these are the first parameterized approximation schemes for the problems, improving upon the previously known O (1) -approximation ratios given by Thejaswi et al. (KDD 2022).

1 Introduction

Clustering seeks to partition a given set of clients into disjoint, cohesive clusters. Among the many formalizations of clustering, the k -median and k -means problems are perhaps the most prevalent ones, owing to the concise nature of their descriptions. Given a set of clients and facilities in a metric space along with a positive integer k , the k -median and k -means problems aim to open at most k facilities and connect each client to the nearest opened facility, such that the sum of the client-connection costs is minimized. In the k -median problem, the connection cost of each client is its distance to the corresponding facility, while in the k -means problem, it is the squared distance. Despite their seemingly simple definitions, the k -median and k -means problems are computationally challenging, and the development of their approximation algorithms continues to be a vibrant area of research. The current best approximation guarantees are the ratios of 2.613 [Gowda et al., 2023] for the k -median problem and 9 [Ahmadian et al., 2020] for the k -means problem.

The k -median and k -means problems are designed to maximize the similarity between clients and their corresponding facilities, allowing the opened facilities to be considered representative points for the client set. This understanding underscores the important roles that the k -median and k -means problems play in data summarization [Moens et al., 1999, Girdhar and Dudek, 2012]. However, algorithms developed for these problems can often yield unfair summarization of socioeconomic data, as they prioritize minimizing the clustering costs over considering the distribution of demographic labels (e.g., gender, age, race) associated with the opened facilities [Kay et al., 2015]. Driven by this

rationale, there has been considerable interest in fair-range clustering . Given a set of data points associated with demographic labels, fair-range clustering extends classical clustering formulations by imposing lower and upper bound constraints on the number of opened facilities associated with each label, thereby ensuring fairness across different demographic groups.

An instance ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ ) of the fair-range clustering problem is specified by positive integers glyph[lscript] and k , sets C of clients and F of facilities in a metric space, vectors glyph[vector] α = ( α 1 , · · · , α glyph[lscript] ) and glyph[vector] β = ( β 1 , · · · , β glyph[lscript] ) of glyph[lscript] positive integers satisfying α t ≤ β t for each t ∈ { 1 , · · · , glyph[lscript] } , and integer ρ ≥ 1 , where each f ∈ F is associated with a set τ ( f ) ⊆ { 1 , · · · , glyph[lscript] } of demographic labels. A feasible solution to the instance is specified by a subset H ⊆ F of no more than k facilities satisfying |{ f ∈ H : t ∈ τ ( f ) }| ∈ [ α t , β t ] for each t ∈ { 1 , · · · , glyph[lscript] } , and the cost of the solution is ∑ c ∈C min f ∈H δ ρ ( c, f ) , where δ is the distance function. The goal of the fair-range clustering problem is to identify a feasible solution with minimum cost.

The fair-range clustering problem is equivalent to the fair-range k -median ( FkMed ) problem when ρ = 1 , and to the fair-range k -means ( FkMeans ) problem when ρ = 2 . Despite their significance in applications requiring fair representations, the FkMed and FkMeans problems pose significantly greater computational challenges than classical clustering problems. As demonstrated by Thejaswi et al. [2021], designing polynomial-time algorithms with provable approximation guarantees for the FkMed and FkMeans problems is unlikely, as determining the existence of feasible solutions to their instances is NP-hard. For a simplified scenario where each facility is associated with a single demographic label, Thejaswi et al. [2021] showed that the FkMed and FkMeans problems can be reduced to the well known matroid clustering problem, which admits constant-factor approximation algorithms [Krishnaswamy et al., 2011, Li, 2011, Swamy, 2014, Friggstad and Zhang, 2016, Krishnaswamy et al., 2018], albeit with an O ( k ) glyph[lscript] -1 multiplicative overhead in algorithmic running time. Hotegni et al. [2023] latter gave an improved reduction to the matroid clustering problem that eliminates the O ( k ) glyph[lscript] -1 overhead. They further demonstrated that solving the FkMed and FkMeans problems in this simplified scenario can be achieved even more efficiently than solving matroid clustering problems, based on smaller-size linear programs.

In practical scenarios concerning clustering problems, the number of opened facilities (i.e., k ) is often considerably smaller than the input size. As such, assuming k to be small and treating it as a fixed parameter is a commonly used way for simplifying these problems, as exemplified in [Cohen-Addad et al., 2019, Goyal and Jaiswal, 2023, Chen et al., 2024, Jaiswal et al., 2024]. Unfortunately, the FkMed and FkMeans problems have been demonstrated to remain challenging even with this simplification: When both k and the number of demographic labels (i.e., glyph[lscript] ) are fixed parameters, Thejaswi et al. [2022] established the W[2]-hardness of the FkMed and FkMeans problems, suggesting that exactly solving them in fixed-parameter tractable time (denoted as FPT ( k, glyph[lscript] ) time, meaning n O (1) h ( k, glyph[lscript] ) for an input size of n and a positive function h ) is unlikely; Cohen-Addad et al. [2019] showed that the best possible approximation ratios of FPT ( k, glyph[lscript] ) -time algorithms, even for the case where glyph[lscript] = 1 , cannot be better than 1 + 2 e -1 for the FkMed problem and 1 + 8 e -1 for the FkMeans problem. This matches the FPT ( k, glyph[lscript] ) -time (1 + 2 e -1 + ε ) -approximation algorithm for the FkMed problem and (1 + 8 e -1 + ε ) -approximation algorithm for the FkMeans problem given by Thejaswi et al. [2022] for a simpler case considering only the lower bound constraint. Notably, the method for enumerating feasible constraint patterns given by Thejaswi et al. [2022] demonstrates that their algorithms can be effortlessly extended to accommodate the case involving both lower and upper bounds.

The negative result presented by Cohen-Addad et al. [2019] suggests that we cannot hope to approximate the FkMeans problem with a ratio better than 1 + 8 e -1 and the FkMed problem with a ratio better than 1 + 2 e -1 in FPT ( k, glyph[lscript] ) time when considering general metric spaces. However, this result does not preclude the possibility of achieving better approximations for these problems in more structured settings, such as Euclidean spaces, since the analysis in Cohen-Addad et al. [2019] is limited to general metrics. In this paper, we take the first step toward exploring the properties of Euclidean metrics for the FkMed and FkMeans problems. Our approach yields FPT ( k, glyph[lscript] ) -time approximation schemes, as stated in Theorem 1 in Section 3.4.

1.1 Other Related Work

Due to the prevalence of Euclidean data in real-world applications involving clustering, significant attention has been devoted to developing algorithms that leverage the properties of Euclidean spaces.

Exploring these properties often leads to improved approximation guarantees. One such example can be found in [Cohen-Addad et al., 2022], where a (2 . 406 + ε ) -approximation algorithm for the k -median problem and a (5 . 912 + ε ) -approximation algorithm for the k -means problem in Euclidean spaces are proposed, improving upon the state-of-the-art approximation ratios for these problems under general metrics [Gowda et al., 2023, Ahmadian et al., 2020]. Furthermore, it has been shown that the Euclidean k -median and k -means problems admit approximation schemes 2 if k is a fixed parameter and the opened facilities can be located arbitrarily [Kumar et al., 2010, Jaiswal et al., 2014, Bhattacharya et al., 2018, Ding and Xu, 2020]. These algorithms identify a subset of each client-cluster defined by an optimal solution and approximate the corresponding opened facility by the centroid of this subset. However, similar ideas are not applicable to the FkMed and FkMeans problems, as they involve finite sets of facilities and hard constraints on the labels of the opened facilities. In these cases, the centroids of the considered subsets are not guaranteed to be feasible as opened facilities.

Constraints on the number of opened facilities associated with different labels were first introduced by Hajiaghayi et al. [2010, 2012], inspired by budget considerations for the deployment of servers in content distribution networks. From then on, related clustering problems have been widely explored. When we are provided with an upper bound constraint and each facility is associated with a single label, the problems represent special cases of the matroid clustering problems and directly motivate research into the latter [Krishnaswamy et al., 2011]. For the lower-bounded case, there are FPT ( k, glyph[lscript] ) -time approximation algorithms for the k -median and k -means cost functions (given by Thejaswi et al. [2022], as previously mentioned in Section 1), and a multi-swap local-search heuristic yields an O ( glyph[lscript] ) -approximation for the k -median cost function if each facility has a single label [Thejaswi et al., 2021, Zhang et al., 2024].

In addition to imposing constraints on the distribution of labels associated with opened facilities, fair clustering has been extensively studied under various other settings that introduce different types of constraints. For example, group fairness requires each cluster to provide a fair representation of different demographic groups [Chierichetti et al., 2017, Bera et al., 2019, Bandyapadhyay et al., 2021, Dai et al., 2022, Wu et al., 2024], proportional fairness ensures that no subset of clients, of a given size, can find a closed facility that provides a lower connection cost to each of its members [Chen et al., 2019, Micha and Shah, 2020], individual fairness requires that the distance from each client to the nearest opened facility does not exceed a client-specified threshold [Jung et al., 2020, Mahabadi and Vakilian, 2020, Negahbani and Chakrabarty, 2021, Vakilian and Yalçiner, 2022, Ahmadi et al., 2022, Bateni et al., 2024], and social fairness aims to minimize the maximum clustering cost among groups of clients [Abbasi et al., 2021, Ghadiri et al., 2021, Makarychev and Vakilian, 2021, Goyal and Jaiswal, 2023, Abbasi et al., 2024].

1.2 Preliminaries

From now on, we consider an instance I = ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ ) of the fair-range clustering problem satisfying ρ ∈ { 1 , 2 } , |C ∪ F| = n , and C ∪ F ⊂ R d , along with a constant glyph[epsilon1] ∈ (0 , 0 . 5) . Given an integer i ≥ 1 and a set D , define [ i ] = { 1 , · · · , i } , and let [ D ] i be the Cartesian product D×··· × D .

︸

︷︷

︸

i

Given a point x and a set P of points in an Euclidean space, let δ ( x, P ) = min p ∈P || x -p || denote the distance from x to its nearest point in P , and let δ i ( x, P ) = min p ∈P || x -p || i for each i ≥ 1 .

The following algebraic fact will be utilized in the analysis of the running times of our algorithms.

Lemma 1 Given two real numbers s and t greater than 1, we have log t s ≤ max { s, t O ( t ) } .

The following lemma extends triangle inequality 3 to squared Euclidean metrics.

Lemma 2 Given three points x , y , and z in an Euclidean space and a real number γ ∈ (0 , 1] , we have || x -z || 2 ≤ (1 + γ -1 ) || x -y || 2 +(1 + γ ) || y -z || 2 .

We will also consider the weighted version of the fair-range clustering problem, which can be defined as follows.

f

∗

f

f

∗

(

a

)

(

b

)

Definition 1 (weighted fair-range clustering) An instance ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ ) of the fair-range clustering problem can be extended to its weighted version ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ, w ) by associating each client c ∈ C with a weight w ( c ) ≥ 1 . This extension modifies the cost of a feasible solution H ⊆ F from ∑ c ∈C δ ρ ( c, H ) to ∑ c ∈C w ( c ) δ ρ ( c, H ) .

2 An Overview of Our Algorithms

The FPT ( k, glyph[lscript] ) -time approximation algorithms for the FkMed and FkMeans problems given by Thejaswi et al. [2022] follow the framework outlined in [Cohen-Addad et al., 2019]. This framework identifies the nearest client to each facility opened in the considered optimal solution as a 'leader' and introduces a set of annuli centered at each leader. Each annulus is defined such that its outer radius is 1 + ε times its inner radius. The framework then enumerates the annuli to identify those that contain the facilities corresponding to the leaders and selects the opened facilities within these annuli, as illustrated in Figure 1-( a ) . Intuitively, the definition of the annuli and triangle inequality imply an upper bound on the distances from the selected facilities to the optimal ones. Building upon this insight, Thejaswi et al. [2022] utilized a submodular maximization method to select facilities to be opened within the annuli and demonstrated constant-factor approximation ratios.

We similarly base our algorithms on the framework proposed by Cohen-Addad et al. [2019]. Our approach focuses on exploring the properties of Euclidean metrics to further refine the selection range of opened facilities. Specifically, we partition each annulus into a set of smaller cells; for each facility opened in the optimal solution, we select the center point of the cell containing it as the facility to be opened, as shown in Figure 1-( b ) . This process involves carefully balancing the number of cells, which affects the time required to identify the desired cells, against the sizes of the cells, which influence the distance from each facility opened in the optimal solution to the center point of the cell containing it, as well as our approximation ratio. We achieve this trade-off by constructing nets as defined below.

Definition 2 ( γ -net [Gupta et al., 2003]) Given a density parameter γ > 0 , a set P ⊂ R d , and a subset R ⊆ P , we call R a γ -net of P if each p ∈ R satisfies δ ( p, R\{ p } ) ≥ γ and each p ∈ P satisfies δ ( p, R ) ≤ γ .

We partition the annular search space into cells using a set of nets for the facilities. The trade-off between the number and sizes of the cells can be managed by adjusting the density parameters (i.e., the parameter λ in Definition 2). For each annulus centered around a leader and containing its corresponding facility (the one opened in the optimal solution), we estimate the demographic labels associated with this facility and identify the subset of facilities within the annulus that share these labels. A net is then constructed for this subset, using a density parameter carefully determined by the radius of the annulus. Given the Voronoi diagram defined by the net, Definition 2 suggests that the facility opened in the optimal solution is close to the center point of its corresponding Voronoi cell. By considering each member of all constructed nets as a candidate for an opened facility, we can ensure that the candidate set includes a subset closely approximating the optimal solution.

It remains to consider how to bound the running time within FPT ( k, glyph[lscript] ) time. The algorithms given by Cohen-Addad et al. [2019] and Thejaswi et al. [2022] start with constructing a coreset, that is, a small weighted subset of the client set whose distribution closely approximates that of the full set.

This facilitates the efficient identification of leaders by enumerating the coreset. In this paper, we face additional challenges in bounding the running time. For instance, when partitioning the annuli into cells, the number of cells can depend exponentially on the spatial dimension. To ensure that we can deal with the FkMed and FkMeans problems in high-dimensional Euclidean spaces within FPT ( k, glyph[lscript] ) time, we map the considered instance to a space of O (log k +log log n ) dimensions using a combination of the method for constructing coresets given by Chen [2009] and Johnson-Lindenstrauss transform. Combining this data-reduction technique with our net-based approach for selecting opened facilities, we give FPT ( k, glyph[lscript] ) -time (1 + ε ) -approximation algorithms for the FkMed and FkMeans problems.

3 The Algorithms

Wenowpresent our algorithms for the FkMed and FkMeans problems. In Section 3.1, we introduce our data-reduction method for decreasing the size of the considered instance. In Section 3.2, we construct annular search spaces for the facilities to be opened, using the leaders from the client set. Section 3.3 details the construction of nets for the facilities, based on which we provide approximation schemes in low-dimensional spaces. Finally, in Section 3.4, we show how to combine the data-reduction method with the algorithms designed for low-dimensional spaces to deal with high-dimensional instances of the FkMed and FkMeans problems.

3.1 Data Reduction

In this section we map instance I to a smaller weighted instance in a low-dimensional space. As mentioned in Section 2, we achieve this using the coreset-construction method given by Chen [2009] and Johnson-Lindenstrauss transform, which are detailed in the following two lemmata.

Lemma 3 (Chen [2009]) Given a constant glyph[epsilon1] ∈ (0 , 0 . 5) , a set P ⊂ R d , an integer t > 0 , and an integer ρ ∈ { 1 , 2 } , a subset P † ⊆ P with a weight function w : P † → [1 , + ∞ ) satisfying |P † | ≤ d ( tε -1 log |P| ) O (1) and ∑ p ∈P † w ( p ) = |P| can be constructed in O ( |P| dt ) time, such that each H ⊂ R d with |H| ≤ t satisfies ∑ p ∈P † w ( p ) δ ρ ( p, H ) ∈ [1 -glyph[epsilon1], 1 + glyph[epsilon1] ] ∑ p ∈P δ ρ ( p, H ) .

Lemma 4 (Johnson and Lindenstrauss [1984], Ailon and Chazelle [2009]) Given a constant glyph[epsilon1] ∈ (0 , 0 . 5) and a set P ⊂ R d , we can construct a mapping ϕ : R d → R ˜ d satisfying ˜ d = O ( glyph[epsilon1] -2 log |P| ) and || ϕ ( p 1 ) -ϕ ( p 2 ) || ∈ [1 , 1 + glyph[epsilon1] ] || p 1 -p 2 || for each p 1 , p 2 ∈ P in O ( d log d ) + ( glyph[epsilon1] -1 log |P| ) O (1) time.

The following lemma is a stronger version of Johnson-Lindenstrauss transform, which preserves distances over a broader range through terminal embedding. Specifically, it modifies the condition 'for each p 1 , p 2 ∈ P ' in Lemma 4 to 'for each p 1 ∈ P and p 2 ∈ R d '.

Lemma 5 (Narayanan and Nelson [2019]) Given a constant glyph[epsilon1] ∈ (0 , 0 . 5) and a set P ⊂ R d , we can construct a mapping ϕ : R d → R ˜ d satisfying ˜ d = O ( glyph[epsilon1] -2 log |P| ) and || ϕ ( p 1 ) -ϕ ( p 2 ) || ∈ [1 , 1 + glyph[epsilon1] ] || p 1 -p 2 || for each p 1 ∈ P and p 2 ∈ R d in ( |P| dglyph[epsilon1] -1 ) O (1) time.

It can be assumed that each mapping ϕ : R d → R ˜ d constructed by Lemma 4 and Lemma 5 is injective. Such an assumption is made without loss of generality: We can create duplicates of the points in R ˜ d that have multiple preimages under ϕ . This ensures that we can always differentiate ϕ ( x ) and ϕ ( y ) for any two distinct points x and y in R d , even if ϕ ( x ) and ϕ ( y ) have identical values across all dimensions. Distinguishing the images of the points from R d is essential in fair-range clustering problems because points with the same dimensional values can have different demographic labels.

Our data-reduction method, which combines Lemma 3, Lemma 4, and Lemma 5, is presented in Algorithm 1 and illustrated in Figure 2 (this figure outlines the processing flow for the clients). This algorithm first leverages Lemma 3 within the O (log n ) -dimensional space constructed by Lemma 4, such that the client set can be replaced with a coreset of size logarithmically dependent on n and independent of d . Next, to reduce dimensions while preserving the distances between each client in the coreset and any facility, Algorithm 1 uses Lemma 5 with the coreset as input to construct an O (log k +loglog n ) -dimensional space. The following lemma provides the performance guarantees of Algorithm 1.

Algorithm 1: The data-reduction method

Input: A constant glyph[epsilon1] ∈ (0 , 0 . 5) and an instance ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ ) of the fair-range clustering problem satisfying C ∪ F ⊂ R d

Output: A mapping ϕ : R d → R ˜ d and an instance ( glyph[lscript], k, ˜ C , ˜ F , glyph[vector] α, glyph[vector] β, ρ, τ, w ) of the weighted fair-range clustering problem satisfying ρ ∈ { 1 , 2 } , ˜ C ∪ ˜ F ⊂ R ˜ d ,

1 Let ϕ 1 : R d → R d † be the mapping constructed by Lemma 4 with ( glyph[epsilon1], C ∪ F ) as the input;

2 Let C † be the weighted set constructed by Lemma 3 with ( glyph[epsilon1], { ϕ 1 ( c ) : c ∈ C} , k, ρ ) as the input, and let w † : C † → [1 , + ∞ ) be the corresponding weight function;

3 Let ϕ 2 : R d † → R ˜ d be the mapping constructed by Lemma 5 with ( glyph[epsilon1], C † ) as the input;

4 ϕ ⇐ ϕ 2 ◦ ϕ 1 , ˜ C ⇐ { ϕ 2 ( c ) : c ∈ C † } , ˜ F ⇐ { ϕ ( f ) : f ∈ F} ;

5 return ϕ : R d → R ˜ d , ( glyph[lscript], k, ˜ C , ˜ F , glyph[vector] α, glyph[vector] β, ρ, τ ◦ ϕ -1 , w † ◦ ϕ -1 2 ) .

C

{

ϕ

1

(

c

) :

c

∈ C}

C †

{

ϕ

2

(

c

) :

c

∈ C † }

Lemma 6 Given a constant glyph[epsilon1] ∈ (0 , 0 . 5) and an instance ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ ) of the fair-range clustering problem with ρ ∈ { 1 , 2 } , |C ∪F| = n , and C ∪F ⊂ R d , Algorithm 1 constructs a mapping ϕ : R d → R ˜ d and an instance ( glyph[lscript], k, ˜ C , ˜ F , glyph[vector] α, glyph[vector] β, ρ, τ, w ) of the weighted fair-range clustering problem in O ( d log d ) + ( nkglyph[epsilon1] -1 ) O (1) time, which satisfy the following properties:

(i) ∑ c ∈ ˜ C w ( c ) = |C| ,

(ii) w ( c ) ≥ 1 for each c ∈ ˜ C ,

(iii) | ˜ C| ≤ ( kglyph[epsilon1] -1 log n ) O (1) ,

(iv) ˜ d = glyph[epsilon1] -O (1) (log k +loglog n ) , and

(v) ∑ c ∈ ˜ C w ( c ) δ ρ ( c, { ϕ ( f ) : f ∈ H} ) ∈ [1 -glyph[epsilon1], (1 + glyph[epsilon1] ) 2 ρ +1 ] ∑ c ∈C δ ρ ( c, H ) for each H ⊆ F with |H| ≤ k .

3.2 The Annular Search Spaces

In this section we construct k annular search spaces, each corresponding to one of the k facilities to be opened. We first introduce some notations. Let ϕ : R d → R ˜ d be the mapping and ˜ I = ( glyph[lscript], k, ˜ C , ˜ F , glyph[vector] α, glyph[vector] β, ρ, τ, w ) be the weighted instance constructed by Algorithm 1 with ( glyph[epsilon1], I ) as the input, where ˜ C ∪ ˜ F ⊂ R ˜ d , ˜ F = { ϕ ( f ) : f ∈ F} , and each f ∈ F satisfies τ ( ϕ ( f )) = τ ( f ) . Let ˜ H ∗ = { f ∗ 1 , · · · , f ∗ k } be an optimal solution to ˜ I , and let opt = ∑ c ∈ ˜ C w ( c ) δ ρ ( c, ˜ H ∗ ) denote its cost. For each i ∈ [ k ] , let L i = { f ∈ ˜ F : τ ( f ) = τ ( f ∗ i ) } denote the set of facilities that have the same set of demographic labels as f ∗ i , and let ˜ C ∗ i = { c ∈ ˜ C : arg min f ∈ ˜ H ∗ || f -c || = f ∗ i } be the cluster of clients defined by f ∗ i . Given the lower bound constraint on the number of opened facilities, it may be the case that some facilities in ˜ H ∗ correspond to empty clusters. We thus define

˜ H ∗ 0 = { f ∈ ˜ H ∗ : ˜ C ∗ i = ∅} and ˜ H ∗ 1 = ˜ H ∗ \ ˜ H ∗ 0 . Let k ∗ = | ˜ H ∗ 1 | . Without loss of generality, we can assume that ˜ H ∗ 1 = { f ∗ 1 , · · · , f ∗ k ∗ } .

Following the framework given by Cohen-Addad et al. [2019], we select opened facilities from a set of annuli centered around a group of leaders from ˜ C . For each i ∈ [ k ∗ ] , let c i denote the client from ˜ C nearest to f ∗ i , that is, the leader corresponding to f ∗ i . Let δ ρ max = max i ∈ [ k ∗ ] δ ρ ( c i , f ∗ i ) . For each i ∈ [ k ∗ ] and j ∈ [ glyph[ceilingleft] glyph[epsilon1] -2 log n glyph[ceilingright] ] , let A ∗ ( i, j ) = { f ∈ L i : || f -c i || ρ ∈ ( glyph[epsilon1] (1 + glyph[epsilon1] ) j -1 δ ρ max n -1 , glyph[epsilon1] (1 + glyph[epsilon1] ) j δ ρ max n -1 ] } be the set of facilities from L i located in an annulus centered around c i , and let A ∗ ( i, 0) = { f ∈ L i : || f -c i || ρ ≤ glyph[epsilon1]δ ρ max n -1 } . The definitions of A ∗ ( i, j ) and δ ρ max imply the existence of an integer j ∈ { 0 , · · · , glyph[ceilingleft] glyph[epsilon1] -2 log n glyph[ceilingright]} satisfying f ∗ i ∈ A ∗ ( i, j ) . Denote by A ∗ i such a set A ∗ ( i, j ) containing f ∗ i .

Our method for constructing annular search spaces is presented in Algorithm 2. Since the collection {A ∗ (1 , 0) , · · · , A ∗ ( k ∗ , glyph[ceilingleft] glyph[epsilon1] -2 log n glyph[ceilingright] ) } can be determined based on the values of {L 1 , · · · , L k } , k ∗ , δ ρ max , and { c 1 , · · · , c k ∗ } , Algorithm 2 enumerates all possible values of these parameters in step 5 to ensure that the collection can be captured. Given an integer i ∈ [ k ∗ ] and the sets A ∗ ( i, 0) , · · · , A ∗ ( i, glyph[ceilingleft] glyph[epsilon1] -2 log n glyph[ceilingright] ) , Algorithm 2 enumerates [ glyph[ceilingleft] glyph[epsilon1] -2 log n glyph[ceilingright] ] ∪ { 0 } in step 12 to find the integer j with A ∗ i = A ∗ ( i, j ) . To avoid the case where the search spaces for the k opened facilities intersect and the set of selected facilities contains duplicate elements, Algorithm 2 employs a color-coding technique to eliminate any potential intersections. Specifically, Algorithm 2 associates each facility f ∈ ˜ F with a random integer η ( f ) ∈ [ k ] in step 4, and only selects facilities with η ( f ) = i when constructing the i -th search space for each i ∈ [ k ] in steps 14 and 17. The performance guarantees of this algorithm are presented in the following lemma.

glyph[negationslash]

glyph[negationslash]

(i) With probability no less than k -k , there exists a collection {A 1 , · · · , A k } ∈ A satisfying f ∗ i ∈ A i ⊆ A ∗ i for each i ∈ [ k ∗ ] and A i = ∅ for each i ∈ [ k ] \ [ k ∗ ] ;

glyph[negationslash]

(ii) Given a collection {A 1 , · · · , A k } ∈ A , with A i = ∅ for each i ∈ [ k ] , and a tuple ( f 1 , · · · , f k ) ∈ A 1 × A 2 × · · · × A k , we have |{ i ∈ [ k ] : t ∈ τ ( f i ) }| ∈ [ α t , β t ] for each t ∈ [ glyph[lscript] ] .

3.3 The Algorithm in Low-Dimensional Spaces

As outlined in Section 2, solutions are constructed by extracting nets from the set of facilities. The following lemma presents a method for generating nets in low-dimensional Euclidean spaces.

Lemma 8 (Har-Peled and Mendel [2006]) Given a density parameter γ > 0 and a set P ⊂ R d , a γ -net of P of size at most min {|P| , γ -d max p 1 ,p 2 ∈P || p 1 -p 2 || d } can be constructed in |P| log |P| 2 O ( d ) time.

Our approach for solving the low-dimensional weighted instance ˜ I is built upon Algorithm 2 and Lemma 8, and is outlined in Algorithm 3. Since Algorithm 2 yields the desired search spaces with probability k -k (as given by the first property stated in Lemma 7), Algorithm 3 iteratively invokes it k k times, allowing the probability of successfully constructing the desired search spaces in at least one of the iterations to be boosted to a constant. Given k sets A 1 , · · · , A k satisfying the first property stated in Lemma 7, Algorithm 3 constructs a net for each set of size greater than 1, and adds the members of the net to the candidate set of opened facilities. Finally, the algorithm constructs a set of feasible solutions to ˜ I based on the candidates for opened facilities, and returns the one with the minimum cost among them.

The following lemma says that Algorithm 3 yields a (1 + O ( glyph[epsilon1] 1 ρ )) -approximation solution to ˜ I with high probability.

Lemma 9 The following event occurs with probability no less than 1 -e -1 : Algorithm 3 yields a feasible H † to ˜ I satisfying ∑ c ∈ ˜ C w ( c ) δ ρ ( c, H † ) < (1+4 glyph[epsilon1] ) opt if ρ = 1 and ∑ c ∈ ˜ C w ( c ) δ ρ ( c, H † ) < (1 + 9 √ glyph[epsilon1] ) opt if ρ = 2 .

By analyzing the time Algorithm 3 takes to construct the set of candidate solutions, as well as the size of this set, we can establish the following upper bound on the running time of Algorithm 3.

Lemma 10 Algorithm 3 runs in no more than 2 ( kglyph[epsilon1] -1 ) O (1) + kglyph[lscript] n O (1) time.

Input: A constant glyph[epsilon1] ∈ (0 , 0 . 5) and an instance I = ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ ) of the fair-range clustering problem satisfying C ∪ F ⊂ R d

Output: A solution H ‡ to I

1 Let ϕ : R d → R ˜ d be the mapping and ˜ I = ( glyph[lscript], k, ˜ C , ˜ F , glyph[vector] α, glyph[vector] β, ρ, τ, w ) be the weighted instance constructed by Algorithm 1 with ( glyph[epsilon1], I ) as the input;

2 Let H † be the solution to ˜ I constructed by Algorithm 3 with ( glyph[epsilon1], ˜ I , |C ∪ F| ) as the input;

3 return H ‡ ⇐{ ϕ -1 ( f ) : f ∈ H † } .

3.4 Extensions to High-Dimensional Spaces

We combine the data-reduction method given in Section 3.1 with the low-dimensional algorithm given in Section 3.3 to solve the FkMed and FkMeans problems in high-dimensional spaces, as detailed in Algorithm 4. Given a constant glyph[epsilon1] ∈ (0 , 0 . 5) and an instance I = ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ ) , the algorithm starts with constructing a mapping ϕ : R d → R ˜ d and a small weighted instance ˜ I = ( glyph[lscript], k, ˜ C , ˜ F , glyph[vector] α, glyph[vector] β, ρ, τ, w ) , where ˜ C ∪ ˜ F ⊂ R ˜ d , ˜ F = { ϕ ( f ) : f ∈ F} , and τ ( ϕ ( f )) = τ ( f ) for each f ∈ F . It then constructs a solution to ˜ I and returns the set of preimages of the facilities opened in this solution under ϕ . The analysis of the performance guarantees of Algorithm 4 leads to the main result of this paper, as stated in Theorem 1.

Theorem 1 Given an instance ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ ) of fair-range clustering with C ∪ F ⊂ R d and ρ ∈ { 1 , 2 } along with a real number ε ∈ (0 , 1) , there is a randomized (1 + ε ) -approximation algorithm running in O ( d log d ) + 2 ( kε -1 ) O (1) + kglyph[lscript] n O (1) time, where n = |C ∪ F| .

4 Conclusions

In this paper, we consider the FkMed and FkMeans problems for the case where the numbers of opened facilities and demographic labels are fixed parameters. Based on a combination of a datareduction method and a space-partitioning approach for selecting opened facilities, we introduce (1 + ε ) -approximation algorithms in Euclidean spaces, representing the first parameterized approximation schemes for the problems. Given that coreset-construction methods were known for many constrained clustering problems incorporating additional constraints on instances, such as those related to capacities [Cohen-Addad and Li, 2019], group fairness [Bandyapadhyay et al., 2021], and robustness [Huang et al., 2023], an interesting direction for future work is to extend our techniques to deal with the FkMed and FkMeans problems in similar constrained settings. Another promising avenue for exploration is to accelerate heuristic algorithms for the fair-range clustering problem, such as those given in [Thejaswi et al., 2022], using the data-reduction method proposed in this work.

5 Broader Impact

Our work deals with the fair-range clustering problem, providing algorithmic insights that can facilitate fair decision-making. While our algorithms have been shown to be 'fair' according to specific definitions, it is essential to recognize that this fairness does not automatically warrant indiscriminate application. This underscores the need for careful consideration when implementing the algorithms proposed in this paper in real-world scenarios that prioritize fairness.

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A Proof of Lemma 1

Lemma 1 Given two real numbers s and t greater than 1, we have log t s ≤ max { s, t O ( t ) }

.

Proof If t ≥ log s log log s , then we have log s ≤ O ( t log t ) , and thus log t s ≤ t O ( t ) . If t < log s log log s , then we have log t s < log log s log log s s = s . Thus, Lemma 1 is true. glyph[square]

B Proof of Lemma 2

Lemma 2 Given three points x , y , and z in an Euclidean space and a real number γ ∈ (0 , 1] , we have || x -z || 2 ≤ (1 + γ -1 ) || x -y || 2 +(1 + γ ) || y -z || 2 .

Proof Triangle inequality implies that || x -z || ≤ || x -y || + || y -z || , and thus we have

This completes the proof of Lemma 2.

C Proof of Lemma 6

Lemma 6 Given a constant glyph[epsilon1] ∈ (0 , 0 . 5) and an instance ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ ) of the fair-range clustering problem with ρ ∈ { 1 , 2 } , |C ∪F| = n , and C ∪F ⊂ R d , Algorithm 1 constructs a mapping ϕ : R d → R ˜ d and an instance ( glyph[lscript], k, ˜ C , ˜ F , glyph[vector] α, glyph[vector] β, ρ, τ, w ) of the weighted fair-range clustering problem in O ( d log d ) + ( nkglyph[epsilon1] -1 ) O (1) time, which satisfy the following properties:

(i) ∑ c ∈ ˜ C w ( c ) = |C| ,

(ii) w ( c ) ≥ 1 for each c ∈ ˜ C ,

(iii) | ˜ C| ≤ ( kglyph[epsilon1] -1 log n ) O (1) ,

(iv) ˜ d = glyph[epsilon1] -O (1) (log k +loglog n ) , and

(v) ∑ c ∈ ˜ C w ( c ) δ ρ ( c, { ϕ ( f ) : f ∈ H} ) ∈ [1 -glyph[epsilon1], (1 + glyph[epsilon1] ) 2 ρ +1 ] ∑ c ∈C δ ρ ( c, H ) for each H ⊆ F with |H| ≤ k .

Proof Algorithm 1 constructs a mapping ϕ 1 : R d → R d † using Lemma 4 in step 1, a coreset C † along with the corresponding weight function w † : C † → [1 , + ∞ ) using Lemma 3 in step 2, and a mapping ϕ 2 : R d † → R ˜ d using Lemma 5 in step 3.

We begin by examining the first property of the output of Algorithm 1 stated in Lemma 6. We have

where the first step follows from the fact that ˜ C = { ϕ 2 ( c ) : c ∈ C † } (due to step 4 of Algorithm 1) and w is the composite mapping w † ◦ ϕ -1 2 (due to step 5 of Algorithm 1), the second step follows from the fact that C † is the weighted set constructed by Lemma 3 with { ϕ 1 ( c ) : c ∈ C} as the input set, and the last step is due to the assumption that the mappings constructed by Lemma 4 and Lemma 5 are injective. Equality (1) implies that the first property stated in Lemma 6 is true.

The second property stated in Lemma 6 follows directly from the fact that w ( c ) = w † ( ϕ -1 2 ( c )) for each c ∈ ˜ C (as established in step 5 of Algorithm 1) and w † is a mapping to [1 , + ∞ ) (due to Lemma 3).

We now consider the third property stated in Lemma 6. This can be verified by

glyph[square]

where the first step is due to the fact that ˜ C = { ϕ 2 ( c ) : c ∈ C † } (as established in step 4 of the algorithm) and the assumption that each mapping constructed by Lemma 5 is injective, the second step follows from Lemma 3, and the third step is due to the fact that ϕ 1 : R d → R d † is the mapping constructed by Lemma 4 with ( glyph[epsilon1], C ∪ F ) as the input.

Using inequality (2) and the fact that ϕ 2 : R d † → R ˜ d is the mapping constructed by Lemma 5 with ( glyph[epsilon1], C † ) as the input, we have

which implies that the fourth property stated in Lemma 6 is true.

Given a subset H ⊆ F with |H| ≤ k , we have

where the first step follows from Lemma 5 and the fact that ˜ C = { ϕ 2 ( c ) : c ∈ C † } and ϕ ( f ) = ϕ 2 ( ϕ 1 ( f )) for each f ∈ F (due to step 4 of Algorithm 1), the second step follows from the fact that C † is a coreset of { ϕ 1 ( c ) : c ∈ C} constructed by Lemma 3, and the last step is due to Lemma 4. This completes the proof of the last property stated in Lemma 6.

It remains to show the running time of Algorithm 1. Recall that the algorithm invokes Lemma 4 with ( glyph[epsilon1], C ∪ F ) , invokes Lemma 3 with ( glyph[epsilon1], { ϕ 1 ( c ) : c ∈ C} , k, ρ ) , and invokes Lemma 5 with ( glyph[epsilon1], C † ) , where C † ⊆ { ϕ 1 ( c ) : c ∈ C} ⊂ R d † . Combining this with inequality (2), we can express the running time of Algorithm 1 as

D Proof of Lemma 7

Lemma 7 The collection A constructed by Algorithm 2 satisfies the following two properties:

glyph[negationslash]

(i) With probability no less than k -k , there exists a collection {A 1 , · · · , A k } ∈ A satisfying f ∗ i ∈ A i ⊆ A ∗ i for each i ∈ [ k ∗ ] and A i = ∅ for each i ∈ [ k ] \ [ k ∗ ] ;

glyph[negationslash]

(ii) Given a collection {A 1 , · · · , A k } ∈ A , with A i = ∅ for each i ∈ [ k ] , and a tuple ( f 1 , · · · , f k ) ∈ A 1 × A 2 × · · · × A k , we have |{ i ∈ [ k ] : t ∈ τ ( f i ) }| ∈ [ α t , β t ] for each t ∈ [ glyph[lscript] ] .

Proof Algorithm 2 enumerates all possible values of A ∗ i for each i ∈ [ k ∗ ] and all possible values of L i for each i ∈ [ k ] \ [ k ∗ ] . Consequently, the k sets A ∗ 1 , · · · , A ∗ k ∗ , L k ∗ +1 , · · · , L k are guaranteed to be captured by the algorithm. Observe that Algorithm 2 associates each facility f with a random integer η ( f ) ∈ [ k ] . It can be shown that equality η ( f ∗ i ) = i ∀ i ∈ [ k ] holds with probability k -k . When this equality is satisfied, and the k sets A ∗ 1 , · · · , A ∗ k ∗ , L k ∗ +1 , · · · , L k are provided, the algorithm is able to construct a set A i satisfying f ∗ i ∈ A i ⊆ A ∗ i for each i ∈ [ k ∗ ] by extracting the facilities f ∈ A ∗ i with η ( f ) = i in step 14, and find a facility f ∈ L i with η ( f ) = i to construct a singleton set A i ⊆ L i for each i ∈ [ k ] \ [ k ∗ ] in step 17. Thus, the first property stated in Lemma 7 is true.

glyph[negationslash]

Now we consider the second property. Given a collection {A 1 , · · · , A k } ∈ A with A i = ∅ for each i ∈ [ k ] and a tuple ( f 1 , · · · , f k ) ∈ A 1 ×A 2 ×··· × A k , the fact that facilities in different sets are associated with distinct integers implies that { f 1 , · · · , f k } is a distinct set. Combining this with the decision condition employed in step 6 of Algorithm 2, we have

for each t ∈ [ glyph[lscript] ] , where D 1 , · · · , D k are the k sets of demographic labels used for constructing {A 1 , · · · , A k } . This completes the proof of Lemma 7. glyph[square]

E Proof of Lemma 9

Lemma 9 The following event occurs with probability no less than 1 -e -1 : Algorithm 3 yields a feasible H † to ˜ I satisfying ∑ c ∈ ˜ C w ( c ) δ ρ ( c, H † ) < (1+4 glyph[epsilon1] ) opt if ρ = 1 and ∑ c ∈ ˜ C w ( c ) δ ρ ( c, H † ) < (1 + 9 √ glyph[epsilon1] ) opt if ρ = 2 .

Proof The second property stated in Lemma 7 immediately implies that all candidate solutions constructed by Algorithm 3 are feasible solutions to ˜ I . It remains to consider the approximation ratio of the algorithm. Given that Algorithm 3 iteratively invokes Algorithm 2 k k times to construct a collection A , the probability that there exists an element {A 1 , · · · , A k } ∈ A satisfying the first property stated in Lemma 7 is at least 1 -(1 -k -k ) k k > 1 -e -1 . For the purpose of our analysis, we assume that the collection A indeed contains such an element {A 1 , · · · , A k } , and let S i be the subset of A i constructed by Algorithm 3 in step 8 or step 10 for each i ∈ [ k ] .

We consider an integer i ∈ [ k ∗ ] . Lemma 7 indicates that

if |A i | = 1 . For the case where |A i | > 1 , S i is a max x,y ∈A i glyph[epsilon1] || x -y || -net of A i . In this scenario, it can be concluded that

where the first step follows from the fact that f ∗ i ∈ A i (due to Lemma 7) and the definition of nets, the second step follows from triangle inequality (for ρ = 1 ) and Lemma 2 (for ρ = 2 , with γ = 1 ), the third step follows from the fact that an integer j ∈ { 0 , · · · , glyph[ceilingleft] glyph[epsilon1] -2 log n glyph[ceilingright]} satisfies A i ⊆ A ∗ ( i, j ) (due to Lemma 7), along with the fact that each j ∈ [ glyph[ceilingleft] glyph[epsilon1] -2 log n glyph[ceilingright] ] and { x, y } ⊆ A ∗ ( i, j ) satisfy || x -c i || ρ ≤ (1 + glyph[epsilon1] ) || y -c i || ρ and each f ∈ A ∗ ( i, 0) satisfies || f -c i || ρ ≤ glyph[epsilon1]δ ρ max n -1 (due to the definitions of A ∗ ( i, j ) and A ∗ ( i, 0) ), and the last step follows from the definition of δ ρ max and the fact that w ( c ) ≥ 1 for each c ∈ ˜ C (due to Lemma 6).

Let H ′ be the set of candidate solution constructed using the Cartesian product S 1 ×S 2 ×··· × S k in step 11 of Algorithm 3. Equality (3) and inequality (4) suggest the existence of a candidate solution { f 1 , · · · , f k } ∈ H ′ satisfying

for each i ∈ [ k ∗ ] . Thus, it can be shown that

where the first step is due to the fact that ˜ C ∗ i = ∅ for each i ∈ [ k ] \ [ k ∗ ] , the second step is due to inequality (5), the fourth step follows from the fact that ∑ k i =1 ∑ c ∈ ˜ C ∗ i w ( c ) = ∑ c ∈ ˜ C w ( c ) = | ˜ C| < n (due to Lemma 6), the fifth step is due to the definition of c i , the sixth step is due to the definition of ˜ C ∗ i , and the last step follows from the fact that glyph[epsilon1] ∈ (0 , 0 . 5) . Consequently, for the case where ρ = 2 , we have

where the third step is due to Lemma 2 (with γ = √ glyph[epsilon1] ), the fourth step follows from the definition of ˜ C ∗ i , the fifth step follows from inequality (6), and the last step is due to the fact that glyph[epsilon1] ∈ (0 , 0 . 5) .

Replacing Lemma 2 used in the third step of inequality (7) with triangle inequality, we get

for the case where ρ = 1 .

Using inequality (7) and inequality (8), along with the fact that Algorithm 3 returns the candidate solution with the minimum cost, we complete the proof of Lemma 9. glyph[square]

F Proof of Lemma 10

Lemma 10 Algorithm 3 runs in no more than 2 ( kglyph[epsilon1] -1 ) O (1) + kglyph[lscript] n O (1) time.

Proof Let H be the set of candidate solutions constructed by Algorithm 3, and let A be the collection formed by iteratively invoking Algorithm 2 k k times. Observe that Algorithm 2 enumerates all possible values of k ∗ , { c 1 , · · · , c k } , δ ρ max , and {L 1 , · · · , L k } to guess the sets {A ∗ (1 , 0) , · · · , A ∗ ( k ∗ , glyph[ceilingleft] glyph[epsilon1] -2 log n glyph[ceilingright] ) } and {L k ∗ +1 , · · · , L k } . Additionally, it enumerates [[ glyph[ceilingleft] glyph[epsilon1] -2 log n glyph[ceilingright] ] ∪ { 0 } ] k to determine the set {A ∗ 1 , · · · , A ∗ k ∗ } . Analyzing the possible values of these parameters yields

where the second step follows from the fact that ˜ C = ( kglyph[epsilon1] -1 log n ) O (1) (due to Lemma 6) and | ˜ F| < n , and the third step follows from Lemma 1 (with s = n and t = k ).

Let {A 1 , · · · , A k } ∈ A be the collection considered in one of the iterations of steps 6-12 of Algorithm 3. Let S 1 , · · · , S k be the subsets constructed in step 8 or step 10 and H ′ be the set of candidate solutions constructed in step 11 during this iteration. For each i ∈ [ k ] with |A i | > 1 , S i is a max x,y ∈A i glyph[epsilon1] || x -y || -net of A i , which is constructed based on the maximum distance between the facilities from A i in step 10. Lemma 8 implies that constructing these nets takes no more than

time. Moreover, the upper bound on the size of a net exhibited in Lemma 8 suggests that

for each i ∈ [ k ] . Inequality (11) leads to

Given the set H of candidate solutions, Algorithm 3 takes | ˜ C| ˜ dk | H | time to identify the solution with the minimum cost. Combining this with the time required for constructing nets in each iteration exhibited in inequality (10), we know that the running time of Algorithm 3 is upper-bounded by

where the first step follows from inequality (12), the second step follows from the fact that | ˜ C| ≤ ( kglyph[epsilon1] -1 log n ) O (1) (due to Lemma 3) and inequality (9), the third step follows from the fact that ˜ d = glyph[epsilon1] -O (1) (log k +loglog n ) (due to Lemma 3), and the last step is due to Lemma 1 (with s = n and t = ( kglyph[epsilon1] -1 ) O (1) ). This completes the proof of Lemma 10. glyph[square]

G Proof of Theorem 1

Theorem 1 Given an instance ( glyph[lscript], k, C , F , glyph[vector] α, glyph[vector] β, ρ, τ ) of fair-range clustering with C ∪ F ⊂ R d and ρ ∈ { 1 , 2 } along with a real number ε ∈ (0 , 1) , there is a randomized (1 + ε ) -approximation algorithm running in O ( d log d ) + 2 ( kε -1 ) O (1) + kglyph[lscript] n O (1) time, where n = |C ∪ F| .

Proof Let H ‡ be the solution to I returned by Algorithm 4, and let H † be the solution to ˜ I constructed in step 2 of Algorithm 4. Since each facility in H ‡ shares the same set of demographic labels as its corresponding image in R ˜ d , it follows that H ‡ is a feasible solution to I . Let H ∗ be an optimal solution to I and ˜ H ∗ be an optimal solution to ˜ I .

The optimality of ˜ H ∗ for ˜ I and Lemma 6 imply that

and thus we have

For the case where ρ = 1 , we can derive that inequality

holds with probability no less than 1 -e -1 , where the first step is due to the fact that H † = { ϕ ( f ) : f ∈ H ‡ } and Lemma 6, the second step follows from Lemma 9, the third step is due to inequality (13), and the last step follows from the fact that glyph[epsilon1] ∈ (0 , 0 . 5) . Similarly, we can conclude that inequality

holds with the same probability when ρ = 2 , where the second step is due to Lemma 9.

Using inequality (14) and inequality (15), we know that Algorithm 4 is a randomized (1 + 39 glyph[epsilon1] ) -approximation algorithm for the FkMed problem and a (1 + 83 √ glyph[epsilon1] ) -approximation algorithm for the FkMeans problem. Moreover, Lemma 6 and Lemma 10 imply that this algorithm runs in O ( d log d ) + 2 ( kglyph[epsilon1] -1 ) O (1) + kglyph[lscript] n O (1) time.

Given a constant ε ∈ (0 , 1) , let glyph[epsilon1] = ε 39 for the FkMed problem and glyph[epsilon1] = ( ε 83 ) 2 for the FkMeans problem, then the argument above implies the existence of (1 + ε ) -approximation algorithms with running time O ( d log d ) + 2 ( kε -1 ) O (1) + kglyph[lscript] n O (1) for both problems, as desired. glyph[square]

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Answer: [Yes]

Justification: The paper's contributions and scope have been accurately claimed in the abstract and introduction.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: The limitation of the algorithms in the paper is that they consider k and glyph[lscript] as fixed parameters, which has been clearly discussed in the paper.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors study the fair range clustering problem, where facilities are associated with multiple demographic labels, forming intersecting groups. They impose both lower and upper bounds on the number of cluster centers chosen from each label. For both $k$-median and $k$-means clustering objectives, they present a $1 + \\epsilon$ approximation algorithm when the underlying metric space is Euclidean.\n\nThe key contribution of this work is leveraging the properties of Euclidean metric spaces to improve approximation ratios while maintaining similar running times as for general metric spaces, specifically fixed-parameter tractable (FPT) in parameters $k$ and $\\ell$. \n\nThe authors make use of the techniques and results that are known the literature, but it is still challenging to stitch the pieces together to obtain a solution and formally prove all the claims, which the authors have successfully managed to do. I have verified proofs in sufficient detail and cannot find anything wrong or incorrect. It is possible that I may have missed something. \n\nThe writing can be simplified by clearly explaining the figures and explicitly stating that Figure 2 applies only to clients. It took me some time to realize that the facility set does not undergo the same transformation, as the coreset applies only to clients. Also it was difficult to understand precisely what the authors refer to as the annular region in Figure 1a (this was only clear after reading Section 4.2, and discretisation of distances).\n\nIn a nutshell, the approach works by first reducing the high-dimensional space to lower-dimensions, present $1 + \\epsilon)$ FPT algorithm in lower-dimensions and show that this solution translates to a $ 1 + O(\\epsilon)$ FPT approximation in higher dimensions.\n\nSTRENGTHS:\nThis work has significant theoretical contributions, offering new insights into a known difficult clustering problem variant, where earlier works have extensively investigated and established the computational complexity of the problem.\n\nWEAKNESSES:\nSimilar to earlier work by Thejaswi et al. (2022), the presented FPT algorithms may not scale practically, given that the exponential factors are large. In theory, dimension reduction and coreset constructions are expected to introduce a minimal $\\epsilon$ factor of distortion in distances. However, in practice, creating smaller-sized coresets often requires a larger $\\epsilon$, which limits the practical scalability of algorithms. Also the achieved approximation factors in practical applications may be significantly larger than the theoretical claims. Even though the theoretical contributions are good, I would suggest that the authors put an effort to perform (at least some) experimental evaluations when submitting to an applied conference such as NeurIPS.\n\nQUESTIONS:\nThe extension from considering only lower bound requirements to including both lower and upper bound requirements is straightforward in the work of Thejaswi et al. 2022. In their work, see Lemma~5.2, which exhaustively lists/enumerates all feasible constraint patterns satisfying the lower bound requirements, and this can be extended to enumerate feasible constrained patterns that satisfy both lower and upper bound requirements. Consequently, all algorithmic findings of diversity-aware clustering (with lower bound requirements) would extend from this adaptation to fair-range clustering (with lower and upper bound requirements), although the authors themselves do not explicitly assert this claim, the extension is immediate. Do you agree with this, or do you hold a different viewpoint? \n\nLine 52-54: This statement is not accurate. The reduction presented by Thejaswi et al. (2021) from diversity-aware $k$-median to matroid median problem shows that any algorithm for matroid median can effectively handle the scenario with disjoint facility groups. Consequently, this reduction yields a $7.081 + \\epsilon$ approximation for the disjoint facility groups case, as further detailed in Theorem~7.1 of Thejaswi et al. (2024) preprint on ArXiv this also yields a FPT($k$) algorithm with 1 + 2 e^{-1} approximation. Do you hold a different viewpoint on this?\n\nFigure 1(a): The caption does not clearly explain the significance of the two circles around the client. It would be beneficial to explicitly specify these details. Are these circles representing the distance defined by the authors in Lines 228-229 on Page 6 (as a consequence of discretisation of distances)? please clarify.\n\nLIMITATIONS:\nIn the broader impact section authors have briefly mentioned the potential social impact of their work.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-05"}

comments: SUMMARY:
The paper presents fixed-parameter approximation schemes for the fair-range k-median and k-means problems in Euclidean spaces, parameterized by both the number of facilities and labels. The results improve on existing results, which could only achieve constant-ratio approximation. The main technique used is a data-reduction technique to reduce the dimensionality, combined with an algorithm for low-dimensional spaces.

STRENGTHS:
1. The paper presents an FPT APX-scheme for very important clustering problems.
2. The approximation results improve on the current best results.
3. The results are theoretically solid and technical.

WEAKNESSES:
1. The techniques are not novel. 
2. The results are purely theoretical, with no experimental results provided.

QUESTIONS:
Did you consider any other notions of fairness, based on other constraints (e.g., related to clients)?

LIMITATIONS:
Lack of experimental work.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-11

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n- This paper studies fair range clustering, which aims to ensure that cluster centers are not dominated by specific demographic groups.\n- It focuses on fair range clustering using the 1-norm and 2-norm distance metrics.\n- The paper proposes an algorithm with three internal steps: (i) data reduction to low-dimensional space, (ii) obtaining fair centers in the low-dimensional space, and (iii) transforming the centers back to the original high-dimensional space.\n- The authors theoretically demonstrate that the algorithm can be computed in Fixed Parameter Tractable (FPT) time.\n\nSTRENGTHS:\n- The problem considered in this paper allows each center (facility) to include multiple demographics, whereas a previous work considers a single demographic label for each center.\n- The proposed algorithm can be computed in FPT-time.\n- The algorithm is theoretically applicable to high-dimensional data.\n\nWEAKNESSES:\n1. The writing and presentation are complex, making it difficult for non-experts to follow. It assumes readers are very familiar with fair range clustering.\n2. The paper focuses too much on techniques used to derive the algorithms, lacking intuitive explanations about how the algorithm works.\n3. The introduction should better justify the necessity of fair range clustering. It should include related works on (conventional) fair clustering [A, B, C] and their key differences to highlight the significance of fair \"range\" clustering.\n4. The data reduction mechanism lacks novelty, appearing to be a direct consequence of combining Lemmas 3, 4, and 5, which were not developed by the authors.\n5. The link mapping $\\phi$ between low-dimensional space and high-dimensional space, which is a key component of the proposed algorithm, is theoretically valid (through Lemmas 4 and 5), but practical construction methods are not discussed.\n6. Numerical analysis is missing. Given existing works on fair clustering [A, B, C] or fair range clustering [D] with numerical results on real data, appropriate experiments are necessary.\n7. (Minor) line 451: second -> third, line 454 : third -> fourth\n\n[A] Fair Clustering Through Fairlets\nhttps://dl.acm.org/doi/pdf/10.5555/3295222.3295256 \n\n[B] Fair Algorithms for Clustering\nhttps://proceedings.neurips.cc/paper_files/paper/2019/file/fc192b0c0d270dbf41870a63a8c76c2f-Paper.pdf\n\n[C] Variational Fair Clustering\nhttps://arxiv.org/abs/1906.08207 \n\n[D] Fair k-Center Clustering for Data Summarization\nhttps://proceedings.mlr.press/v97/kleindessner19a/kleindessner19a.pdf\n\nQUESTIONS:\n1. What is the intuitive definition of “core set”?\n2. What is the role of the core set in data reduction? \n3. The authors mention that Thejaswi et al. [2022] showed limits of approximation order in a general metric space (lines 73-77). How is the approximation order improved when considering Euclidean space rather than a general metric space?\n4. Is $\\tilde{\\mathcal{S}^*}$ in line 504 defined? If not, what is its definition?\n\nLIMITATIONS:\n1. It is not confirmed that the proposed algorithm can be applied to real scenarios.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper deals with the problem of fair-range clustering in Euclidean metric spaces. In fair-range clustering, one is given a set of clients and a set of possible facilities. Every facility is associated with a subset of $\\ell$ many classes. The goal is to pick up to $k$ many facilities such that a given clustering objective is minimized with respect to the chosen facilities as centers under the side constraint that the number of facilities that are associated with a certain class lies $i$ within a given interval $[\\alpha_i,\\beta_i]$, for all $i\\leq \\ell$. \nThe clustering objectives considered here are the $k$-Median and the $k$-Means objective.\nThe authors propose FPT $(1+\\epsilon)$-approximations for both problems with parameters $k$ and $\\ell$.\nTheir approach essentially works by first mapping the input to an Euclidean space of lower dimension and computing a coreset. For such a low-dimensional weighted instance, they give an algorithm that computes a near-optimal approximation on the modified instance. In the end, they translate the solution back to a solution for the original instance by returning the pre-image of the injective mapping from the transformation step.\n\nSTRENGTHS:\nThis is a well-written and clear paper with a good contribution. There already existed optimal FPT($k$,$\\ell$)-algorithms for the two considered problems matching known lower bounds. In this paper, the authors manage to break the barriers for general metric spaces when restricting to Euclidean metric spaces by giving near-optimal $(1+O(\\epsilon))$-approximation algorithms for the Euclidean setting.\n\nWEAKNESSES:\nApart from the Questions mentioned later, I only have minor comments. I am willing to increase my score to \"accept\" when addressing the questions.\n\n_Comments to the authors:_\n- please provide a citation for the results mentioned in the abstract (lines 9-10)\n- please define $\\tau$ and $\\ell$ in the preliminaries\n- are Lemma 1 and Lemma 2 needed somewhere in the main body of the paper? Otherwise, they could be transferred to the appendix\n- figure 1: please provide an explanation of the different symbols (small circles = clients, squares = facilities?)\n- l. 171: what do you mean by \"distance parameter\"? \n- why is the upper bound stated in equation (2) meaningful? How does it help in bounding $\\delta_{\\max}^{\\rho}$ if it contains $\\delta_{\\max}^{\\rho}$?\n- how are the annuli guessed if the possible radii depend on the maximal distance within an optimal solution in lines 229,230? When looking at the proof of Lemma 8, it becomes obvious that the value of $\\delta_{\\max}^{\\rho}$ is guessed. Please also provide this information in the text.\n- In general, is a bit difficult to recognize which are the parts that are guessed in Algorithm 2. If I understood correctly, the rings $A_i$ are guessed (which includes guessing the leader $c_i$, the radius of the annulus, which is based on guessing $\\delta_{\\max}^{\\rho}$, and the set $L_i$), the color coding is done randomly, and the feasible solution of lowest cost is again guessed (lines 15-16 in Algorithm 2). It would be helpful to specifically state which things have to be guessed in the text.\n- It is not specified how the values are chosen randomly in line 9 of Algorithm 2, resp. lines 249-253 (Uniformly at random?). Please make this more formal.\n- it does not become entirely clear to me how the feasible solution is chosen in lines 15-16 in Algorithm 2. Are you iterating over *all* possible subsets of $S$ and only then check whether the subset contains at most $k$ elements? This seems a bit wasteful. Why would it not be enough to choose one facility from every ring? It would be helpful if you write this part in lines 258-259 a bit clearer. \n\n\n\n_Language, Typos & Layout_\n- 112: \"number\" -> \"numbers\"\n- 216: I guess that \"see Appendix [...]\" belongs in the next line\n- 223: \"denotes\" -> \"denote\"\n- 229, 230: Why do you use $\\|f-c_i\\|^{\\rho}$ instead of $\\delta^{\\rho}(f,c_i)$ here?\n- it might increase readability if you restate Theorem 1 where it is proven\n\nQUESTIONS:\n- line 9 of Algorithm 2 assigns every facility an integer at random. This makes your algorithm a randomized algorithm and influences the result of Lemma 9. As Lemma 9 is used in the proof of Theorem 1, Theorem 1 should also mention that this guarantee holds *with high probability*. Please clarify.\n- What is the motivation for using first the original Lindenstrauss transform and then the stronger version in the second step? Why do you not need the stronger transform in the first step?\n\nLIMITATIONS:\nYes, the authors clearly state the limitations (fixed-parameter tractability with parameters $k$ and $\\ell$, restriction to Euclidean metric spaces) in the abstract. In Section 6, they also state considerations to make when using their results for practical applications.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}