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On the cost of essentially fair clusterings

Ioana O. Bercea, Martin Groß, Samir Khuller, Aounon Kumar, Clemens Rösner, Daniel R. Schmidt, Melanie Schmidt

arXiv:1811.10319v1cs.DS

TL;DR

Fair clustering aims to preserve protected-group representation within clusters, but existing approaches leave challenges for multiple colors and several clustering objectives. The paper combines an unconstrained integral solution with a fair fractional LP solution through weakly supervised rounding. It reports improved exact-ratio guarantees and essentially fair bicriteria approximations across five objectives.

  • Problem

    Fairlet approaches face unresolved challenges for multiple colors with ratio ranges and for objectives such as k-median, while fair clustering seeks to preserve color representation within clusters.

    Source: §1 · pp. 2–3

  • Method

    Weakly supervised LP rounding combines an unconstrained integral solution with a fair fractional LP solution to create an essentially fair clustering using the same centers.

    Source: §1 · p. 3

  • Results

    The 5-approximation for fair k-center improves the previous guarantee of 14, and essentially fair bicriteria factors are 3/5/3.488/4.675/62.856 for k-center/k-supplier/facility location/k-median/k-means.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The bicriteria framework can establish fairness after centers are fixed, with cost linear in the unconstrained solution cost and the optimal fractional fair-clustering cost.

    Source: §1 · p. 3

    For some nontrivial fairness intervals, the canonical fair-clustering LP has an integrality gap of Ω(n) for k-center, k-supplier, and facility location, and Ω(n^2) for k-means.

    Source: §5 · p. 16

Abstract

from arXiv · show

Clustering is a fundamental tool in data mining. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. However, this process may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters -- especially if the data is already biased. At NIPS 2017, Chierichetti et al. proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation for the fair $k$-center problem and a $O(t)$-approximation for the fair $k$-median problem, where $t$ is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation for fair $k$-center. We extend and improve the known results. Firstly, we give a 5-approximation for the fair $k$-center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximations for all of the classical clustering objectives $k$-center, $k$-supplier, $k$-median, $k$-means and facility location. The latter approximations are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where the centers are already fixed.

1. Introduction

Fair clustering balances color representation while optimizing clustering cost, but existing fairlet methods face unresolved challenges for multiple colors, ratio ranges, and some objectives. This paper combines an unconstrained integral solution with a fair fractional LP solution to obtain improved exact-ratio guarantees and essentially fair bicriteria approximations.

1. Introduction

  • Fair clustering seeks color-balanced clusters while optimizing distance, but fair assignment is NP-hard even for k-center with centers fixed.Finding an α-approximation for fair k-center assignment with α < 3 is NP-hard.Source: §1 · p. 2, Preliminaries · p. 5
  • Fairlet methods face two stated difficulties: capacitated clustering lacks constant-factor k-median approximations, and fairlets are unclear for multiple colors with ratio ranges.In the latter setting, subsets of very different sizes and compositions may meet the desired ratios.Source: §1 · p. 3
  • Weakly supervised LP rounding combines an unconstrained integral solution with a fair LP solution to produce an essentially fair clustering using the same centers.For the bicriteria results, any unconstrained starting solution can be used; fairness can therefore be established after centers are fixed.Source: §1 · pp. 3–4
  • Essential fairness allows each color’s point count in an integral cluster to differ by at most 1 from its mass in a fractional fair cluster.This additive violation may be proportionally large for very small clusters or color ratios.Source: §1 · p. 3
  • A 5-approximation for fair k-center improves the previous guarantee of 14, while the paper also gives essentially fair bicriteria guarantees for five clustering objectives.The bicriteria factors are 3/5/3.488/4.675/62.856 for k-center/k-supplier/facility location/k-median/k-means, respectively.Source: §1 · p. 3

Preliminaries

(I)LP formulations for fair clustering problems

  • The canonical fair clustering LP has an unboundedly growing integrality gap for some nontrivial fairness intervals: Ω(n) for k-center, k-supplier, and facility location, and Ω(n^2) for k-means.The paper states that this gap obstructs constant-factor approximation by rounding that canonical ILP; the counterexample does not extend to essential fairness.Source: (I)LP formulations for fair clustering… · p. 6

2. Black-box approximation algorithm for essentially fair clustering

The framework combines an integral solution to the unconstrained problem with a fair fractional LP solution, then rounds the combined assignment to an essentially fair clustering. It applies across the listed clustering objectives, with objective-specific cost bounds and constant-factor approximations.

2. Black-box approximation algorithm for essentially fair clustering

  • The framework first combines an unfair integral solution with a fair fractional LP solution, then rounds to an integral solution with at most one point of fairness violation per color per cluster.The rounding step introduces the potential fairness violation.Source: §2 · p. 6, §2.2 · p. 10

2.1. Step 1: Combining a fair LP solution with an integral unfair solution

  • For k-center, the combined solution costs at most cLP + c̄; for k-supplier, k-median, and facility location, at most 2cLP + c̄; and for k-means, at most 12cLP + 8c̄.cLP is the fair LP cost and c̄ is the cost of the integral unfair solution.Source: §2.1 · p. 7
  • The first step shifts fractional assignments onto centers opened by the integral solution, preserving fairness while making the combined center-opening variables integral.The resulting cost is bounded in terms of the fair LP and unfair integral solution costs.Source: §2.1 · p. 7

2.2. Step 2: Rounding the x-variables

  • For separable objectives, min-cost flow rounds the fractional assignments without increasing cost; the resulting integral solution has additive fairness violation at most one.Facility location, k-median, and k-means are separable objectives.Source: §2.2 · pp. 8–10
  • For k-center and k-supplier, transshipment rounds assignments among centers already eligible for each point, so reassignability ensures the cost does not increase.These objectives are reassignable because points can be reassigned among eligible centers without changing cost.Source: §2.2 · pp. 8, 10
  • 3, 5, 4.675, 3.488, and 62.856 are the approximation factors for k-center, k-supplier, k-median, facility location, and k-means, respectively.These are the essentially fair guarantees from the black-box framework.Source: §2.2 · p. 10

3. True approximations for fair k-center and k-supplier

For exact preservation of color ratios, the paper gives a 5-approximation for fair k-center and a 7-approximation for fair k-supplier. The algorithms use threshold-graph center selection and rounding, adapting the k-center construction to supplier locations.

3.1. 5-Approximation Algorithm for k-center

  • The k-center algorithm selects a maximal independent set in G2τ and builds a tree whose adjacent selected centers are distance three apart in Gτ.The construction is run separately on each connected component.Source: §3.1 · p. 11
  • With exact ratios, every cluster must contain a positive integer multiple of qh points of each color h, where qh is determined by the global color counts.The algorithm uses these quantities to set per-center assignment targets before rounding.Source: §3.1 · pp. 11–12
  • The rounding procedure first makes color-1 assignments integral while preserving fairness through proportional reassignment of the other colors.The resulting assignments yield an integral fair assignment within distance 5τ.Source: §3.1 · p. 12
  • 5 is the approximation factor for fair k-center with exact preservation of ratios.The algorithm chooses centers from a threshold graph and rounds a fair fractional assignment.Source: §3.1 · pp. 11, 13

3.2. 7-approximation for k-suppliers

  • For k-supplier, the threshold graph is bipartite and selected points are distance four apart in Gτ; the adjusted LP uses E4τ, and rounding uses G6τ.Because centers must lie in L, the final assignment moves temporary centers to neighboring locations in L.Source: §3.2 · p. 13
  • 7 is the approximation factor for fair k-supplier with exact preservation of ratios.Temporary centers are moved to neighboring allowed locations, increasing the assignment distance from six to seven.Source: §3.2 · p. 13

4. NP-hardness of the fair assignment problem for k-center

The fair assignment problem for k-center is NP-hard, as shown by a reduction from Exact Cover by 3-sets. The construction makes a low-cost fair assignment correspond to an exact cover.

4. NP-hardness of the fair assignment problem for k-center

  • The reduction encodes Exact Cover by 3-sets using element and set vertices, auxiliary blue vertices, red vertices, and centers at the set vertices.Element-to-center distance is 1 when the element belongs to the represented set, and 3 otherwise.Source: §4 · p. 14
  • A fair assignment of cost 1 exists when an exact cover exists, with red:blue ratio 1:3 in every cluster.The converse also holds: a fair assignment with that ratio and cost below 3 yields an exact cover.Source: §4 · p. 14
  • Each red vertex in T is assigned to a distinct center; each such center receives exactly three blue vertices from U corresponding to its represented set, so centers receiving two red vertices form an exact cover for U.Source: §4 · p. 15

5. Integrality gap of the canonical clustering LP

The canonical fair-clustering LP has large integrality gaps on a constructed instance, limiting what LP rounding can guarantee. The example’s fairness constraint restricts integral solutions to at most two centers, despite allowing a fractional solution with k centers.

5. Integrality gap of the canonical clustering LP

  • The canonical LP has integrality gap Ω(n) for fair k-center, k-supplier, and facility location, and Ω(n^2) for fair k-means.The construction uses non-trivial fairness intervals.Source: §5 · p. 15
  • The example’s fairness ratio limits any integral solution to at most two centers, even though k = r − 1.The point set has r red and r − 1 blue points, with red points alternating with blue points on a line.Source: §5 · pp. 15–16
  • For k-center, the fractional radius is 1 while the integral radius is at least ⌊(r − 1)/2⌋ = Ω(n).The k-supplier integrality gap can only be larger because k-center is a special case of k-supplier.Source: §5 · p. 16
  • In k-median and k-means, fractional assignments incur cost 1 per red point, while integral solutions must span long stretches of the line.The k-median passage gives a fractional cost of O(n); the k-means passage describes the same unit fractional cost per red point.Source: §5 · p. 16
  • This gap lower-bounds LP-rounding guarantees for the canonical fair-clustering ILP, but the counterexample breaks down under essential fairness.The passage states that no fair constant-factor approximation can be achieved by rounding that ILP.Source: §5 · p. 16

A. Notation

The table is titled “Objects and their notation.”

A. Notation

  • Table 1 is titled “Objects and their notation.”The supplied passage does not identify the listed objects or their notation.Source: Table 1 · p. 20

B. Further related work

The related-work discussion reviews approximation guarantees for classical clustering objectives and constrained variants. It also notes a k-means fact used later: restricting centers to input points increases optimum cost by at most a factor of 2.

B. Further related work

  • Classical clustering objectives have differing approximation and hardness bounds, including a 2.765-approximation for k-median and a 6.357-approximation for k-means.The section also reports a 2-approximation for k-center, a 3-approximation for k-supplier, and a 1.488-approximation for facility location.Source: B. Further related work · p. 20
  • For capacity and lower-bound variants, constant-factor approximations are not known for k-median or k-means, and natural LPs can have unbounded integrality gaps.The passage notes bicriteria approximations that violate capacity or cardinality constraints.Source: B. Further related work · p. 21

C. Facts about the k-means cost function

  • Restricting k-means centers to input points increases the optimal cost by at most a factor of 2.The cited lemma compares the best centers in Euclidean space with the best choice of centers from the point set.Source: C. Facts about the k-means cost function · p. 21

D.1. Fairlets and fairlet decompositions

Fairlets are indivisible balanced microclusters, and fair clusters can be decomposed into them. The minimum-cost fairlet decomposition has cost no greater than an optimal fair clustering with k centers across the listed objectives.

D.1. Fairlets and fairlet decompositions

  • A fairlet is a balanced set that cannot be partitioned into smaller non-empty balanced sets.The paper extends the definition to l, u-fairlets, which are balanced under specified lower and upper bounds.Source: §D.1 · p. 22
  • Every fair cluster can be partitioned into fairlets, and the union of disjoint fairlets is balanced.For a 2:3 red-to-blue balance, fairlets are atomic groups of two red and three blue points.Source: §D.1 · p. 22
  • The minimum-cost fairlet decomposition costs no more than an optimal fair clustering with k centers for k-center, k-supplier, k-median, k-means, and facility location.The decomposition is viewed as a clustering whose fairlets may share centers.Source: §D.1 · p. 22

D.2. The general fairlet approach for different problems

The fairlet approach contracts each fairlet to a representative, clusters those representatives without fairness constraints, then assigns each fairlet to its representative’s cluster. The section extends this analysis to k-supplier, k-median, and k-means.

D.2. The general fairlet approach for different problems

  • A fairlet decomposition followed by colorblind clustering yields a fair clustering; for k-center, the cost is at most (1 + γ) · OPT.The fairlet radius and representative-clustering radius are each bounded by OPT.Source: §D.2 · p. 23
  • The analysis formalizes and generalizes the fairlet approach to k-supplier, k-median, and k-means, using more advanced techniques for the latter objectives.Source: §D.2 · p. 23

D.3. Using fairlets to obtain clusterings

The section analyzes how fairlet decompositions and colorblind clusterings combine across clustering objectives, with objective-specific cost bounds and techniques. For k-median, the straightforward representative strategy has a weak guarantee, motivating a different technique.

D.3. Using fairlets to obtain clusterings

  • The analysis assumes an α-approximate fairlet decomposition and a γ-approximate colorblind clustering, each with cost bounded relative to OPT.The section defers whether the necessary black-box algorithms exist in all cases.Source: §D.3 · p. 23

k-center

  • (α + γ)OPT bounds the k-center cost after fairlet centers are clustered and each fairlet is assigned with its points.The fairlet radius and colorblind clustering cost are each bounded using OPT.Source: §D.3 · p. 23

k-supplier

  • (2α + γ)OPT bounds k-supplier cost with arbitrary fairlet representatives; the guarantee improves to (α + γ)OPT when the fairlet center is a point in the fairlet.Source: §D.3 · p. 24

k-median

  • The straightforward k-median representative method can add αOPT per point, giving at best a 2·α·|P|-approximation and requiring a different technique.The alternative assigns each fairlet through a point minimizing the distance from its fairlet center to the colorblind center, as measured via that point.Source: §D.3 · p. 24

Facility location

  • For facility location, only centers from the colorblind solution are opened, so the fairlet decomposition incurs no facility-opening costs; the resulting guarantee is (2α + γ).Source: §D.3 · p. 24

k-means

  • (4γ(α + 1) + α)OPT bounds the k-means clustering cost using fairlet centroids and a colorblind clustering.The analysis bounds the two cost sums by 4αOPT and 4OPT, respectively.Source: §D.3 · p. 25

D.4. Computing fairlet decompositions

The section turns from fairlet-based black-box guarantees to their implementation, asking what approximation ratios result from state-of-the-art methods.

D.4. Computing fairlet decompositions

  • The section examines concrete approximation ratios obtained by filling the fairlet-based black boxes with state-of-the-art methods.Source: §D.4 · p. 25

D.4.1. The happy world of k-center

The section develops fairlet constructions for fair k-center cases using threshold graphs, matchings, flows, and capacitated clustering. Approximation guarantees differ by fairness case, and no constant-factor algorithm is known for case 5.

Threshold-graph framework

  • The section’s k-center framework guesses a radius threshold and uses threshold graphs to constrain which points can belong to the same optimum cluster.For k-center, the optimum cost is one of the pairwise distances, so the algorithm can search those candidate thresholds.Source: §D.4.1 · p. 26

Case 1

  • For case 1, an improved matching-based fairlet decomposition is exact, yielding a 3-approximation for fair k-center and a 5-approximation for fair k-supplier.Unlike the earlier pair-distance approach, the improved method chooses a center for each red-blue pair, bounding its radius by OPT.Source: §D.4.1 · p. 27

Cases 2 and 4

  • For case 2 with two colors, a threshold-graph flow groups up to t blue points with each red point, giving 4- and 5-approximations for fair k-center and k-supplier.The flow gives fairlets of radius at most 2OPT.Source: §D.4.1 · pp. 27–28

Case 3

  • For case 3 with two colors, capacitated clustering and matching yield a 14-approximation for fair k-center and a 15-approximation for fair k-supplier.The fairlet radius is at most 12OPT; the construction groups s red and t blue points per fairlet.Source: §D.4.1 · p. 28

Case 5

  • For case 5, no constant-factor approximation is known because an optimal clustering is not known to admit small balanced subsets for fairlet decomposition.The difficulty is finding a good fairlet decomposition.Source: §D.4.1 · p. 29

Multiple colors

  • For case 2 across multiple colors, independent flow networks connect each non-majority color to the majority color, yielding 4- and 5-approximations for fair k-center and k-supplier.The networks use threshold edges and are solved at the smallest threshold for which every network succeeds.Source: §D.4.1 · p. 29
  • For case 3 with multiple colors, clustering the minority-color points and matching each other color to those groups gives 14- and 15-approximations for fair k-center and k-supplier.The resulting fairlet decomposition has cost at most 12OPT.Source: §D.4.1 · p. 29

Cases 2 and 4

  • The per-color generalization of case 4 cannot be expressed by the ratio-based l,u-balanced definition because that definition cannot treat colors independently.The generalization requires each cluster to meet separate lower bounds for each non-majority color.Source: §D.4.1 · p. 29

D.4.2. Fairlet decompositions for k-median, facility location and the k-means problem

The section constructs fairlet decompositions for k-median, facility location, and k-means using matching and capacitated clustering. Its stated guarantees vary by color balance and objective, with separate bounds for two-color and arbitrary-color instances.

Two colors: r = b = 1

  • When the balance is 1, minimum-cost perfect matching pairs one red and one blue point; this is the only case stated to yield constant-factor approximations so far.For k-means, pair cost is the minimum summed squared distance to a possible center; for k-median and facility location, it is the pair distance.Source: §D.4.2 · p. 29

Two colors: r > 1, b > 1

  • For two-color instances with r > 1 and b > 1, the k-median and facility-location cost is at most (2α + 1)OPT, while the k-means cost is at most 4(2α + 1)OPT.The construction clusters the majority color into sets of size b, makes r copies of each center, and matches red points to centers.Source: §D.4.2 · p. 30

Arbitrary colors

  • For arbitrary colors, the bounds are (3α + 3)OPT for k-median and facility location, and 4(3α + 3)OPT for k-means.The construction tries each color as the anchor color and takes the best resulting fairlet decomposition.Source: §D.4.2 · p. 30

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