SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We can formulate a problem that aims to find an  $\alpha_{\mathrm{SF}}$ -socially fair solution as a constrained optimization problem. We use a dummy objective function  $f$ . The constraint can be set up as requiring maximum clustering costs across all colors to be upper-bounded by  $\alpha_{\mathrm{SF}}$  times that of the optimal socially fair solution  $\mathrm{OPT}_{\mathrm{SF}}$ .
The constrained program can be set up as below:
$$
\min_{S:|S|\leq k}f
$$
$$
\text{s.t.} \max  _ {h \in \mathcal {H}} \frac {1}{| \mathcal {C} ^ {h} |} \sum_ {j \in C ^ {h}} d ^ {p} (j, \phi (j)) \leq \alpha_ {\mathrm {S F}} \mathrm {O P T} _ {\mathrm {S F}}
$$
For a clustering problem with  $k$  centers on the  $\ell$ -community instance with  $k = \ell$ , in a solution  $(S, \phi)$  that has one center in each subset,  $d(j, \phi(j)) = 0$  for each point  $j \in \mathcal{C}$ . Thus this solution has clustering cost for each color  $h$  as  $\sum_{j \in C^h} d^p(j, \phi(j)) = 0$ . Which implies that  $\mathrm{OPT}_{\mathrm{SF}} = 0$ .
Thus on the  $\ell$ -community instance, feasible solutions to the  $\alpha_{\mathrm{SF}}$ -socially fair problem, for finite  $\alpha_{\mathrm{SF}}$ , have  $\max_{h\in \mathcal{H}}\frac{1}{|\mathcal{C}^h|}\sum_{j\in C^h}d^p (j,\phi (j)) = 0$ . We now show  $(S_{CI},\phi_{CI})$  is the only set of solutions that have  $\max_{h\in \mathcal{H}}\sum_{j\in C^h}d^p (j,\phi (j)) = 0$ . Thus, they will be the only feasible solutions.
For any solution  $(S,\phi)$  that is not in  $(S_{CI},\phi_{CI})$ , there must be a point assigned to a center that is not in its own community. For such a point  $d(j,\phi (j)) = R$ . Thus  $\max_{h\in \mathcal{H}}\frac{1}{|\mathcal{C}^h|}\sum_{j\in \mathcal{C}^h}d^p (j,\phi (j))\geq \frac{R}{\max_{h\in\mathcal{H}}|\mathcal{C}^h|}$ . Therefore, an  $\alpha_{\mathrm{SF}}$ -fair solution in the socially fair notion for finite  $\alpha_{\mathrm{SF}}$  must be a solution in the set of optimal solutions  $(S_{CI},\phi_{CI})$ .
At this point, similar to proof of Lemma 4, we had shown that any solution that satisfies socially fair approximately do not assign points to centers not in its original community on the  $\ell$ -community instance. The remaining of the proof is the same as that part of the proof in Lemma 4. We can use the same examples for even  $k$  and odd  $k$  to show that any solution that satisfies  $\mathbf{GF}$  with  $\Omega \left(\frac{n}{k}\right)$  additive violation any  $k \geq 2$  is not in the set of solutions  $(S_{CI}, \phi_{CI})$ . Thus the set of solutions that satisfies socially fair has no overlap with the set of solutions that satisfies  $\mathbf{GF}$  with  $\Omega \left(\frac{n}{k}\right)$  additive violation.
Lemma 6. For any  $k \geq 5$ , there exist a clustering problem where no feasible solution exists that satisfies both Proportional Fairness and  $GF$  even if we allow an additive violation of  $\Omega\left(\frac{n}{k}\right)$  in the  $GF$  constraint.
Proof. For a given value of  $\alpha_{AP}$  for the proportionally fair constraint, consider Figure 8. For the GF constraints, the upper and lower bounds for each color to  $\frac{1}{2}$  and the total number of points  $n$  is always even. Consider some  $k \geq 5$ . It follows that the sum of cluster sizes assigned to centers on either the right side or the left side would be at least  $\frac{n}{2}$ , WLOG assume that it is the left side and denote the total number of points assigned to clusters on the left side by  $|C_{LS}|$  and let  $S_{LS}$  be the centers on the left side. The total number of points on the left side may not be assigned to a single center but rather distributed among the centers  $S_{LS}$ . To satisfy the GF constraints at an additive violation of  $\rho$ , it follows that the number of red points that have to be assigned to the left side is at least  $\sum_{i \in S_{L,S}} \left( \frac{1}{2} |C_i| - \rho \right) \geq \frac{n}{4} - k\rho$ . Set  $\rho = \frac{n}{4k} - \frac{n}{k^2} - 1$ , then it follows that at least  $\left\lceil \frac{n}{k} \right\rceil$  red points are assigned to a center on the left at a distance of at least  $R$ . Since the maximum distance between any two red points by the triangle inequality is  $2r < \frac{R}{\alpha_{AP}}$ , it follows that this set of red points forms a blocking coalition. I.e., these points would also have a lower distance from their assigned center if they were instead assigned to a red center.
Figure 8: Instances to show incompatibility between Proportional Fairness and GF. We always have  $n / 2$  blue points on the left and  $n / 2$  red points on the right. For even  $k$  we would have  $k / 2$  locations for the blue and red points each. For odd  $k$  we have  $\lfloor k / 2 \rfloor$  blue locations and  $\lceil k / 2 \rceil$  red locations. For each color, there is always a location at the center at a distance  $r$  from the other locations. Points of different color are at a distance of at least  $R$  from each other. For any value of  $\alpha_{AP}$  for the proportionally fair constraint, we set  $r < \frac{R}{2\alpha_{AP}}$ .
Proof. Consider a case of the  $\ell$ -community instance where the first  $\ell - 1$  communities consist of points of only blue points. And the last community contains  $\frac{n}{2\ell}$  points of red points and  $\frac{n}{2\ell}$  points of green color. This  $\ell$ -community instance is illustrated in Figure 7. Consider the clustering problem where  $k = \ell$  and DS constraint  $k_{\mathrm{blue}}, k_{\mathrm{red}}, k_{\mathrm{green}} > 0$ . We establish below two claims.
Claim 7. On the above mentioned clustering problem, an  $\alpha_{NR}$ -fair solution in the Fairness in Your Neighborhood notion for finite  $\alpha_{NR}$  is a solution in the set of optimal solutions  $(S_{CI},\phi_{CI})$ .
Claim 8. On the above mentioned clustering problem, an  $\alpha_{SF}$ -fair solution in the Socially Fair notion for finite  $\alpha_{SF}$  is a solution in the set of optimal solutions  $(S_{CI},\phi_{CI})$ .
Those two claims can be proved with the same argument as in claim 5 and claim 6.
However, satisfying DS on this with  $k_{\mathrm{blue}}$ ,  $k_{\mathrm{red}}$ ,  $k_{\mathrm{green}} > 0$  requires a center of each color be picked. Thus a solution from the set of optimal solutions  $(S_{CI}, \phi_{CI})$  does not satisfy DS because it will only pick one point from the right most subset as a center. Thus either green points or red points will not appear in the set of centers.
On the other hand, since a solution satisfying DS has at least one center of each color, it will contain two centers, one green, one red chosen from the right most subset. And there are  $k - 2$  centers allocated to the  $k - 1$  communities on the left. By pigeon hole principle, one of the communities of all blue points will have no center allocated. All blue points in this subset are then assigned to a center in a nearby community, thus a DS satisfying solution is not in the set  $(S_{CI},\phi_{CI})$ . Thus the set of DS satisfying solutions has no overlap with the set of solutions that satisfy either one of the two fairness constraints.
Below we use the same example to show incompatibility between DS and Proportional Fair.
Claim 9. On the above mentioned clustering problem, there is no feasible solution that satisfies both Proportional Fairness and DS.
Proof. We show a DS satisfying solution on above example is not proportional fair. As argued above, a solution satisfying DS can allocate  $k - 2$  centers for the  $k - 1$  communities on the left. There will be a community of size  $\frac{n}{k}$  of which all points are assigned to a nearby center not in its community. This community forms a coalition of size  $\frac{n}{k}$  and would have smaller distance if they get assigned a center in their own community. Therefore a DS satisfying solution is not proportional fair.
Remark: For each of the above proofs we constructed an example which is parametric in the number of centers  $k$ . Moreover, for these examples for the optimal unconstrained  $k$ -center objective to equal 0 at least  $k$  centers have to be used. I.e., the points are spread over at least  $k$  locations. Furthermore, it is not difficult to see in each of the above examples that an optimal solution for the unconstrained  $k$ -center objective satisfies fairness in your neighborhood, socially fair, and the proportionally fair constraints. In fact, it is easy to show that any  $k$ -center which has a radius of 0 immediately satisfies fairness in your neighborhood, socially fair, and the proportionally fair constraints. However, the same is not true for GF or DS. This indicates that the above distance-based fairness constraints can be aligned with the clustering cost whereas the same cannot be said about GF or DS.
Compatibility between GF and DS: One can easily show compatibility between GF and DS. Specifically, consider some values for the centers over the colors  $\{k_h\}_{h\in \mathcal{H}}$  that satisfies the DS constraints, i.e.  $\forall h\in \mathcal{H}:k_h^l\leq k_h\leq k_h^u$  and has  $\sum_{h\in \mathcal{H}}k_h\leq k$ . Then simply pick a set  $Q_{h}$  of  $k_{h}$  points of color  $h$ . Now if we give DIVIDE the entire dataset  $\mathcal{C}$  and the set of centers  $\cup_{h\in \mathcal{H}}Q_h$  as inputs, i.e. call DIVIDE  $(\mathcal{C},\cup_{h\in \mathcal{H}}Q_h)$ , then by the guarantees of divide each center would be active and each cluster would satisfy the GF constraints at an additive violation of 2.
Our final conclusions about the incompatibility and compatibility of the constrains are summarized in Figure 9.
E Example for Running DIVIDE:
Consider the following example running the DIVIDE subroutine. Specifically, we have a set of points  $C$  with a total of  $n = |C| = 38$  points. We have 3 colors (blue, red, and green) with the following points:  $|C^{\mathrm{blue}}| = 15$ ,  $|C^{\mathrm{red}}| = 14$ , and  $|C^{\mathrm{green}}| = 9$ . We have  $Q \subset C$  with a total size of 4  $(|Q| = 4)$ . Accordingly, we have  $T_{\mathrm{blue}} = \frac{15}{4} = 3\frac{3}{4}$ ,  $T_{\mathrm{red}} = \frac{14}{4} = 3\frac{1}{2}$ , and  $T_{\mathrm{green}} = \frac{9}{4} = 2\frac{1}{4}$ . Therefore, in the beginning of the iteration for each color  $h$  (line (8) in algorithm block 1) we have  $b_{\mathrm{blue}} = 3$ ,  $b_{\mathrm{red}} = 2$ ,  $b_{\mathrm{green}} = 1$ . Following the execution of the algorithm, the first three centers  $q = 0$  to  $q = 2$  receive
Figure 9: (In)Compatibility of clustering constraints. Red arrows indicate empty feasible set when both constraints are applied, while green arrows indicate non-empty feasibility set when both constraints are applied.
$\lceil T_{\mathrm{blue}} \rceil$  many blue points, the last  $(q = |Q| - 1)$  and first center  $(q = 0)$  receive  $\lceil T_{\mathrm{red}} \rceil$  many red points, and center  $q = 1$  receives  $\lceil T_{\mathrm{green}} \rceil$ . All other assignments would be the floor of  $T_{\mathrm{h}}$ . Figure 10 illustrates this.
Figure 10: Diagram illustrating how DIVIDE would run over the example. The "tape" has different centers (cells) starting from  $q = 0$  and ending with  $q = |Q| - 1$ . We go over the tape for each color  $h \in \mathcal{H}$ . In a given row  $h$ , centers marked with an  $\mathbf{X}$  are assigned  $\lceil T_h \rceil$  points, otherwise they are assigned  $\lfloor T_h \rfloor$  points.
F Additional Experiments Results
Here we show additional experimental results. As a reminder, the lower and upper proportion bounds for any color  $h$  to  $\alpha_h = (1 + \delta)r_h$  and  $\beta_h = (1 - \delta)r_h$  for some  $\delta \in [0,1]$ . Further, the DS constraints are set to  $k_h^l = \lceil \theta r_h k \rceil$  where  $\theta \in [0,1]$  and  $k_h^u = k$  for every color  $h \in \mathcal{H}$ .
We call our run over the Adult dataset in Section 7 as (A-Adult). In that run  $\delta = 0.2$  and  $\theta = 0.8$ . We also, run another experiment (B-Adult) over the Adult where we set  $\delta = 0.05$  and  $\theta = 0.9$ . Figure 11 shows the new results. We do not see a change qualitatively. It is perhaps noteworthy that the DS-Violation values for COLOR-BLIND and ALG-GF are even higher as well as the GF-Violation for ALG-DS. On the other hand, we find that our algorithms that satisfy  $\mathbf{GF} + \mathbf{DS}$  have very low (almost zero) values for GF-Violation and DS-Violation at a moderate PoF that is comparable to ALG-GF which satisfies only one constraint.
Figure 11: B-Adult results: (a) PoF comparison of 5 algorithms, with COLOR-BLIND as baseline; (b) GF-Violation comparison; (c) DS-Violation comparison.
Additionally, we show results over the Census1990 dataset where we use age as the color (group) membership attribute. As done in [20] we merge the 9 age groups into 3. Specifically, groups  $\{0,1,2\}$ ,  $\{4,5,6\}$ , and  $\{7,8\}$  are each merged into one group leading to total of 3 groups. Further, we subsample 6,000 records from the dataset. We run two experiments where in the first (A-Census1990) we have  $\delta = 0.05$  and  $\theta = 0.7$  whereas in the second (B-Census1990) we have  $\delta = 0.1$  and  $\theta = 0.8$ . We also use different cluster values. In terms of the 3 objective measures of PoF, GF-Violation, and DS-Violation, we do not see a qualitative change as can be seen from Figures 12 and 13. Specifically, the DS algorithm (ALG-DS) has a low PoF but high GF-Violation. Further, COLOR-BLIND and ALG-GF have significant DS-Violation values. On the other hand our algorithms for GF+DS have low values for both GF-Violation and DS-Violation and a moderate PoF.
(a)
Figure 12: A-Census1990 results: (a) PoF comparison of 5 algorithms, with COLOR-BLIND as baseline; (b) GF-Violation comparison; (c) DS-Violation comparison.
(a)
Figure 13: B-Census1990 results: (a) PoF comparison of 5 algorithms, with COLOR-BLIND as baseline; (b) GF-Violation comparison; (c) DS-Violation comparison.
Run-Time: Here we show some run-time analysis results. We first calculate the "incremental" run time over GF. Specifically, given a solution from the ALG-GF (the algorithm for the GF constraints) we see that the additional run-time to post-process it to satisfy the  $\mathbf{GF} + \mathbf{DS}$  is very small in proportion. We measure  $t_{\mathbf{GF} \rightarrow \mathbf{GF} + \mathbf{DS}} = \frac{\text{Time to process GF Solution to satisfy GF} + \text{DS}}{\text{Time to obtain GF Solution}}$  and show the results in Figure 14 over all 4 runs. We see that the additional run time is constantly at least two orders of magnitude smaller than the time required to obtain a GF solution. The fact that added run-time is small further encourages a decision maker to satisfy the DS constraint given a solution that satisfies GF only.
(A-Adult)
Figure 14:  $t_{\mathbf{GF} \to \mathbf{GF} + \mathbf{DS}}$  over the 4 runs of (A-Adult), (B-Adult), (A-Census1990), and (B-Census1990).
(B-Adult)
(A-Census1990)
(B-Census1990)
If we were to do the same using the ALG-DS we find the opposite. Specifically, we measure  $t_{\mathbf{DS} \to \mathbf{GF} + \mathbf{DS}} = \frac{\text{Time to process DS Solution to satisfy GF+DS}}{\text{Time to obtain DS Solution}}$  and find that the additional run-time required to satisfy GF+DS starting from a DS solution is orders of magnitude higher in comparison to the time required to satisfy DS as shown in Figure 15. We conjecture that the reason is that ALG-DS is highly optimized in terms of run-time since it runs in  $O(nk)$  time [37]. On the other hand, the post-processing step (post-processing a DS solution to a GF+DS) requires solving an LP which although is done in polynomial time, can be more costly in terms of run time.
Figure 15:  $t_{\mathbf{DS} \to \mathbf{GF} + \mathbf{DS}}$  over the 4 runs of (A-Adult), (B-Adult), (A-Census1990), and (B-Census1990).
Finally, we show a full run-time comparison between GFTOGFDS which starts from a GF solution and DSTOGFDS which starts from a DS solution. We find that the run-times are generally comparable with one algorithm at times being faster than the other.
Figure 16: Full run-time comparison between GFToGFDS and DSToGFDS over the 4 runs of (A-Adult), (B-Adult), (A-Census1990), and (B-Census1990).
Using a Bi-Criteria Algorithm as ALG-GF: Our implementation of GF follows Bercea et al. [10] and Bera et al. [9] which would violate the GF constraints by at most 2. Empirically, this may cause issues for the GFTOGFDS algorithm since it requires a GF algorithm with zero additive violation and therefore assumes that every cluster has at least one point from each color. However, it would not cause issues as long as the resulting solution satisfies condition of having at least one point from each color in every cluster which would be the case if  $\min_{h\in \mathcal{H},i\in \bar{S}}\beta_h|\bar{C}_i| > 2$  where  $\bar{S}$  is the GF solution and  $\bar{C}_i$  is its  $i^{\mathrm{th}}$  cluster. If the condition not met, then it is reasonable to think that the value of  $k$  was set too high or that the dataset includes outlier points since the cluster sizes are very small. Furthermore, using a GF algorithm with an additive violation of 2 lead to a final GF+DS having a GF violation of at most 4 if the condition is satisfied. However, empirically we find the GF violation to be generally smaller than 1. Finally, note that we treat the GF algorithm as a block-box and therefore it can be replaced by other algorithms such as those of [17, 40] which have no violation for GF. In our experiments, we run our algorithms over datasets and value of  $k$  where the condition is satisfied.