SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof. Let  $(S_{\mathbf{GF} + \mathbf{DS}}^{*},\phi_{\mathbf{GF} + \mathbf{DS}}^{*})$  be an optimal solution to the  $\mathbf{GF} + \mathbf{DS}$  problem.  $\forall i\in S_{\mathbf{GF} + \mathbf{DS}}^{*}$  let  $N(i) = \arg \min_{\bar{i}\in \bar{S}}d(i,\bar{i})$ , i.e.  $N(i)$  is the nearest center in  $\bar{S}$  to center  $i$  (ties are broken using the smallest index).  $\phi_0$  is formed by assigning all points which belong to center  $i\in S_{\mathbf{GF} + \mathbf{DS}}^{*}$  to  $N(i)$ . More formally,  $\forall j\in \mathcal{C}:\phi^{*}_{\mathbf{GF} + \mathbf{DS}}(j) = i$  we set  $\phi_0(j) = N(i)$ . Note that it is possible for more than one center  $i$  in  $S_{\mathbf{GF} + \mathbf{DS}}^{*}$  to have the same nearest center in  $\bar{S}$ . We will now show that  $\phi_0$  satisfies the GF constraint exactly. Note first that if a center  $\bar{i}\in \bar{S}$  has not been assigned any points by  $\phi_0$ , then it is empty and trivially satisfies the GF constraint exactly. Therefore, we assume that  $\bar{i}$  has a non-empty cluster. Denote by  $N^{-1}(\bar{i})$  the set of centers  $i\in S_{\mathbf{GF} + \mathbf{DS}}^{*}$  for which  $\bar{i}$  is the nearest center, then using Fact A.1 and the fact that every cluster in  $(S_{\mathbf{GF} + \mathbf{DS}}^{*},\phi^{*}_{\mathbf{GF} + \mathbf{DS}})$  satisfies the GF constraint exactly we have:
$$
\beta_ {h} \leq \min  _ {i \in N ^ {- 1} (\bar {i})} \frac {\left| C _ {i} ^ {h} \right|}{\left| C _ {i} \right|} \leq \frac {\sum_ {i \in N ^ {- 1} (\bar {i})} \left| C _ {i} ^ {h} \right|}{\sum_ {i \in N ^ {- 1} (\bar {i})} \left| C _ {i} \right|} = \frac {\left| C _ {\bar {i}} ^ {h} \right|}{\left| C _ {\bar {i}} \right|} \leq \max  _ {i \in N ^ {- 1} (\bar {i})} \frac {\left| C _ {i} ^ {h} \right|}{\left| C _ {i} \right|} \leq \alpha_ {h} \tag {10}
$$
The proves guarantee (1) of the lemma. Now we prove guarantee (2), we denote by  $R_{\mathbf{DS}}^*$  the optimal clustering cost for the DS constrained problem. We can show that  $\forall j \in \mathcal{C}$ :
$$
\begin{array}{l} d (j, \phi_ {0} (j)) \leq d (j, \phi_ {\mathbf {G F} + \mathbf {D S}} ^ {*} (j)) + d (\phi_ {\mathbf {G F} + \mathbf {D S}} ^ {*} (j), \phi_ {0} (j)) \\ \leq d \left(j, \phi^ {*} _ {\mathbf {G F} + \mathbf {D S}} (j)\right) + d \left(\phi^ {*} _ {\mathbf {G F} + \mathbf {D S}} (j), N \left(\phi^ {*} _ {\mathbf {G F} + \mathbf {D S}} (j)\right)\right) \quad (\text {s i n c e} \phi_ {0} (j) = N \left(\phi^ {*} _ {\mathbf {G F} + \mathbf {D S}} (j)\right)) \\ \leq R _ {\mathbf {G F} + \mathbf {D S}} ^ {*} + \alpha_ {\mathbf {D S}} R _ {\mathbf {D S}} ^ {*} \\ \leq \left(1 + \alpha_ {\mathbf {D S}}\right) R _ {\mathbf {G F} + \mathbf {D S}} ^ {*} \\ \end{array}
$$
Where the last holds since  $R_{\mathbf{DS}}^{*} \leq R_{\mathbf{GF} + \mathbf{DS}}^{*}$  because the set of solutions constrained by DS is a subset of the set of solutions constrained by GF+DS.
Now we can prove the lemma. By the above claim, it follows that when ASSIGNMENTGF is called, the LP solution from line (3) of algorithm block 3 satisfies: (1) The GF constraints exactly and (2) Has a clustering cost of at most  $(1 + \alpha_{\mathbf{DS}})R_{\mathbf{GF} + \mathbf{DS}}^{*}$ . This is because LP (2) includes all integral assignments from  $\mathcal{C}$  to  $\bar{S}$  including  $\phi_0$ . Since this LP assignment is fed to MAXFLOWGF it follows by Theorem B.1 that the final solution satisfies: (1) The GF constraint at an additive violation of 2, (2) Has a clustering cost of at most  $(1 + \alpha_{\mathbf{DS}})R_{\mathbf{GF} + \mathbf{DS}}^{*}$ . Guarantee (3) holds since some centers may become closed (assigned no points) and therefore  $S' \subset \bar{S}$  (possibly being a proper subset).
We restate the following theorem and give its proof:
Proof. By Lemma 2 above, the set of centers  $S'$  is a subset (possibly proper) subset of  $S$  and therefore the DS constraints may no longer be satisfied. Algorithm 2 selects points from each color  $h$  so that when they are added to  $S'$ , then for each color  $h$  the set of centers is at least  $\beta_h k$ . Since these new centers are opened using the DIVIDE subroutine then it follows that they are all active (guarantee (2) of Lemma 1).
Further, by guarantee (3) of Lemma 1 for DIVIDE we have for any point  $j$  assigned to a new center  $q$  that  $d(j,q)\leq 2d(j,\phi '(j))\leq 2(1 + \alpha_{\mathbf{DS}})R_{\mathbf{GF} + \mathbf{DS}}^*$ .
Finally, by guarantee (1) of Lemma 1 DIVIDE is called over a cluster that satisfies  $\mathbf{GF}$  at an additive violation of 2 and therefore the resulting additive violation is at most  $\max \{2,\frac{2}{|Q_i|} +2\}$ . Since  $2\leq \frac{2}{|Q_i|} +2\leq \frac{2}{2} +2 = 3$ . The additive violation is at most 3.
We restate the next theorem and give its proof:
Proof. We point out the following fact:
Fact C.1. Every cluster in  $(\bar{S},\bar{\phi})$  has at least one point from each color.
Proof. This holds, since given a center  $i \in \bar{S}$  we have  $|\bar{C}_i| > 0$  and therefore  $\forall h \in \mathcal{H}: |\bar{C}_i^h| \geq \beta_h |\bar{C}_i| > 0$  and therefore  $|\bar{C}_i^h| \geq 1$  since it must be an integer.
We note that the values  $\{\beta_h, \alpha_h\}_{h \in \mathcal{H}}$  and  $k$  must lead to a feasible DS problem, i.e. there exist positive integers  $g_h$  such that  $\sum_{h \in \mathcal{H}} g_h = k$  and  $\forall h \in \mathcal{H} : \beta_h k \leq g_h \leq \alpha_h k$ . Accordingly, since lines (4-13) in algorithm 4 can always pick a point of some color  $h$  such that the upper bound  $\alpha_h k$  is not exceeded for every cluster  $i$ . Therefore the following fact must hold
Fact C.2. By the end of line (13) we have  $\forall i\in \bar{S}:|Q_i|\geq 1$
Further, the final  $s_h$  values are valid for  $\mathbf{D}\mathbf{S}$
Claim 4. By the end of line (13) the values of  $s_h$  satisfy: (1)  $\sum_{h \in \mathcal{H}} s_h \leq k$ , (2)  $\forall h \in \mathcal{H} : \beta_h k \leq s_h \leq \alpha_h k$ .
Proof. Lines (4-13) add values to  $s_h$  if the lower bound  $\beta_h k$  for color  $h$  is not satisfied. If the lower bound is satisfied for all colors, then points of some color  $h$  are added provided that adding them would not exceed the upper bound of  $\alpha_h k$  (see line 5). Therefore, by the end of line (13) for any color  $h \in \mathcal{H} : s_h \leq \alpha_h k$  and either  $s_h \geq \beta_h k$  or  $s_h < \beta_h k^3$ .
If by the end of line (13) we have  $\forall h\in \mathcal{H}:s_h\geq \beta_hk$  , then the algorithm moves to line (22). Otherwise, it will keep picking points and incrementing  $s_h$  until  $\forall h\in \mathcal{H}:s_h\geq \beta_hk$
3To see why we could have  $s_h < \beta_h k$ , consider the case where  $\bar{k} < k$  and therefore there would not be enough clusters to so that we can add points for each color.
Further, since such valid DS values exist it must be that the above satisfies  $\sum_{h\in \mathcal{H}}s_h\leq k$  and  $\forall h\in \mathcal{H}:s_h\leq \alpha_hk$ . This concludes the proof for the claim.
By Lemma 1 for DIVIDE the new centers  $S = \cup_{i\in S}Q_{i}$  are all active (guarantee 2 of DIVIDE) and since the values of  $s_h$  are valid (Claim 4 above), therefore  $S$  satisfies the DS constraints.
Since the assignment in each cluster in the new solution  $(S,\phi)$  is formed using DIVIDE over the clusters of  $(\bar{S},\bar{\phi})$  then by guarantee 1 of DIVIDE, each cluster  $(S,\phi)$  satisfies GF at an additive violation of 2. Finally, the clustering cost is at most  $R\leq 2\bar{R}$  (guarantee 3 of DIVIDE).
We restate the following corollary and give its proof:
Proof. Using the previous theorem (Theorem 4.2) the solution  $(\bar{S},\bar{\phi})$  has a cost of  $\bar{R}\leq \alpha_{\mathbf{GF}}\mathrm{OPT}_{\mathbf{GF}}$ . The post-processed solution that satisfies  $\mathbf{GF}$  at an additive violation of 2 and DS simultaneously has a cost of  $R\leq 2\bar{R}\leq 2\alpha_{\mathbf{GF}}\mathrm{OPT}_{\mathbf{GF}}\leq 2\alpha_{\mathbf{GF}}\mathrm{OPT}_{\mathbf{GF} + \mathbf{DS}}$ . The last inequality follows because  $\mathrm{OPT}_{\mathbf{GF}}\leq \mathrm{OPT}_{\mathbf{GF} + \mathbf{DS}}$  which is the case since both problems minimize the same objective, however by definition the constraint set of  $\mathbf{GF} + \mathbf{DS}$  is a subset of the constraint set of  $\mathbf{GF}$ .
Before we proceed, we define the following clustering instance which will be used in the proof:
Definition 1.  $\ell$ -Community Instance: The  $\ell$ -community instance is a clustering instance where the set of points  $\mathcal{C}$  can be partitioned into  $\ell$  communities (subsets)  $\{C_1^{CI},\ldots ,C_\ell^{CI}\}$  of coinciding points (points within the same community are separated by a distance of 0). Further, the communities are of equal size, i.e.  $\forall i\in \ell :|C_i^{CI}| = \frac{n}{\ell}$ . Moreover, the distance between any two points belonging to different communities in the partition is at least  $R > 0$ .
Figure 5: An  $\ell$ -community instance to show Price of Fairness (GF) and incompatibility between GF and other fairness constraints when  $k$  is even.
Figure 6: An  $\ell$ -community instance to show Price of Fairness (GF) and incompatibility between GF and other fairness constraints when  $k$  is odd.
Figures 5 and 6 show two examples of the  $\ell$ -community instances. When clustering with a value of  $k$ , the given  $\ell$ -community instance with  $k = \ell$  is arguably the most "natural" clustering instance where the clustering output is the communities  $\{C_1^{CI}, \ldots, C_k^{CI}\}$ .
The following fact clearly holds for any  $\ell$ -community instance:
Fact C.3. If we cluster an  $\ell$ -community instance with  $k = \ell$  then: (1) The set of optimal solutions are  $(S_{CI}, \phi_{CI})$  where  $S_{CI}$  has exactly one center from each community  $\{C_1^{CI}, \ldots, C_k^{CI}\}$ . Further, points are assigned to a center in the same community. (2) Clustering cost of  $(S_{CI}, \phi_{CI})$  is 0. (3) Any solution other than  $(S_{CI}, \phi_{CI})$  has a clustering cost of at least  $R > 0$ .
We restate the following proposition and give its proof:
Proof. Consider the case where  $k \geq 2$  is even and refer to Figure 5 where we have  $\ell = k$  communities that alternate from red to blue color. Further, by Fact C.3 the optimal solution has a clustering cost of 0. The optimal solution would have one center in each of the  $k = \ell$  communities, and assign points to its closest center.
If we set the lower and upper proportion bounds to  $\frac{1}{2}$  for both colors, then to satisfy GF each cluster should have both red and blue points. There must exists a cluster  $C_i$  of size  $|C_i| \geq \frac{n}{k}$ , it follows that to satisfy the GF constraints at an additive violation of  $\rho$ , then  $|C_i^{\mathrm{blue}}| \geq \frac{1}{2} |C_i| - \rho = \frac{n}{2k} - \rho$  and similarly we would have  $|C_i^{\mathrm{red}}| \geq \frac{n}{2k} - \rho$ . By setting  $\rho = \frac{n}{2k} - \epsilon$  for some constant  $\epsilon > 0$ , then we have  $|C_i^{\mathrm{blue}}|, |C_i^{\mathrm{red}}| > 0$ . This implies that a point will be assigned to a center at a distance  $R > 0$  and therefore the PoF is unbounded.
For a value of  $k$  that is odd, see the example of Figure 6. Here instead the last community has the same number of red and blue points. We call the cluster whose center is in the last community  $C_{\mathrm{last}}$ . If  $|C_{\mathrm{last}}| \neq \frac{n}{k}$ , then there are points assigned to the center of  $C_{\mathrm{last}}$  from other communities incurring cost  $R > 0$  or points in the last community are assigned to other centers at distance  $R > 0$ . If  $|C_{\mathrm{last}}| = \frac{n}{k}$ , then in the remaining  $k - 1$  communities with total of  $n - \frac{n}{k}$  points,  $k - 1$  centers are chosen. There must exists a cluster  $C_i$  of size  $|C_i| \geq \frac{n}{k}$ . We then follow the same argument as in the even  $k$  case, which is to satisfy GF with additive violation  $\rho = \frac{n}{2k} - \epsilon$  for both color, we must have  $|C_i^{\mathrm{blue}}|, |C_i^{\mathrm{red}}| > 0$ . This means at least a point will be assigned to a center at a distance  $R > 0$  and therefore the PoF is unbounded for the odd  $k$  case as well.
Proof. Consider a case of the  $\ell$  community instance shown in Figure 7 where  $k \geq 3$  and  $k = \ell$ . Here all communities are blue, except for the last which has  $\frac{n}{2k}$  red points and  $\frac{n}{2k}$  green points. Similar to the previous proposition since it is a community instance with  $\ell = k$ , then by Fact C.3 the optimal solution has a clustering cost of 0 and would have one center in each community and assign each point to its closest center.
Suppose for DS we set  $k_{\mathrm{blue}}^l, k_{\mathrm{red}}^l, k_{\mathrm{green}}^l > 0$ , this implies that we should pick a center of each color. This implies that we can have at most  $k - 2$  blue center, therefore there will be a community (composed of all blue points) where no point is picked as a center. Therefore, the clustering cost is  $R > 0$  and the PoF is unbounded.
Figure 7: An  $\ell$ -community instance to show Price of Fairness (DS) and incompatibility between DS and other fairness constraints.
Proof. The proof follows similarly to Proposition 5.1. For  $k \geq 2$  and  $k$  is even. Consider the same case as in Figure 5 where we have  $\ell = k$ . In this case, we set the upper and lower proportion bounds for both  $\mathbf{GF}$  and  $\mathbf{DS}$  to  $\frac{1}{2}$ . This implies to satisfy  $\mathbf{DS}$  the number of red and blue centers should each be  $\frac{k}{2}$ . Thus solutions that satisfy the  $\mathbf{DS}$  constraint are the optimal unconstrained solutions as specified in Fact C.3. The rest of the proof proceeds exactly as the proof for the even  $k$  case in Proposition 5.1.
For  $k \geq 3$  and  $k$  is odd, consider the same case as in Figure 6. In this case, we set the upper and lower proportion bounds for  $\mathbf{GF}$  to  $\frac{1}{2}$ . And we set the upper and lower bound for number of centers in  $\mathbf{DS}$  constraint as  $\frac{k - 1}{2} + 1$  and  $\frac{k - 1}{2}$  respectively for both colors. Note that an optimal solution specified in Fact C.3 which chooses either a red or blue point in the right most community as a center satisfy this DS constraint. The rest of the proof proceeds exactly as the proof for the odd  $k$  case in Proposition 5.1.
Proof. This follows from Theorem 4.2 since we can always post-process a solution that only satisfies  $\mathbf{GF}$  into one that satisfies both  $\mathbf{GF}$  at an additive violation of 2 and  $\mathbf{DS}$  simultaneously and clearly from the theorem we would have  $\mathrm{PoF} = \frac{\text{clustering cost of GF post-processed solution}}{\text{clustering cost of GF solution}} \leq \frac{2\text{clustering cost of GF solution}}{\text{clustering cost of GF solution}} \leq 2$ .
D Omitted Proofs, Additional Results, and Details for Section 6
In this section, we provide more details and proofs for theorems and facts that appeared in Section 6. We present proof for theorem 6.1. We begin by giving the full definitions of the relevant fairness constraints.
Definition 2. Neighborhood Radius [30]: For a given set of points  $\mathcal{C}$  to cluster and a given number of centers  $k$ , the neighborhood radius of a point  $j$  is the minimum radius  $r$  such that at least  $|C| / k$  of the points in  $\mathcal{C}$  are within distance  $r$  of  $j$ :  $NR_{C,k}(j) = \min \{r: |B_r(j) \cap C| \geq |C| / k\}$ , where  $B_r(j)$  is the closed ball of radius  $r$  around  $j$ .
Definition 3. Fairness in Your Neighborhood Constraint [30]: For a given set of points  $\mathcal{C}$  with metric  $d(.,.)$ , a clustering  $(S,\phi)$  is  $\alpha_{NR}$ -fair if for all  $j\in \mathcal{C}$ ,  $d(j,\phi (j))\leq \alpha_{NR}\cdot NR_{\mathcal{C},k}(j)$ .
Definition 4. Socially Fair [1, 24]: For a clustering problem with  $k$  centers on points  $\mathcal{C}$  which are from  $|\mathcal{H}|$  groups and  $\cup_{h\in \mathcal{H}}\mathcal{C}^h = \mathcal{C}$ , the socially fair clustering optimization problem is to minimize the maximum average clustering cost across all groups:  $\min_{S:|S|\leq k,\phi}\max_{h\in \mathcal{H}}\frac{1}{|\mathcal{C}^h|}\sum_{j\in C^h}d^p (j,\phi (j))^4.$
Note that socially fair does not optimize over assignment functions because it assumes assignment follows optimal rule: a point is assigned to the cluster center closest to it.
Definition 5. An  $\alpha_{SF}$ -socially fair solution to a clustering problem is a solution of cost at most  $\alpha_{SF}$  of the optimal socially fair solution.
This definition allows us to bound the clustering cost of an  $\alpha_{\mathrm{SF}}$ -Socially Fair solution  $(S_{\alpha},\phi_{\alpha})$  as:
$$
\max  _ {h \in \mathcal {H}} \frac {1}{| \mathcal {C} ^ {h} |} \sum_ {j \in C ^ {h}} d ^ {p} (j, \phi_ {\alpha} (j)) \leq \alpha_ {\mathrm {S F}} \min  _ {S: | S | \leq k} \max  _ {h \in \mathcal {H}} \frac {1}{| \mathcal {C} ^ {h} |} \sum_ {j \in C ^ {h}} d ^ {p} (j, \phi (j)).
$$
Definition 6. Approximately proportional[15]: Given a set of centers  $S \subseteq \mathcal{C}$  with  $|S| = k$ , it is  $\alpha_{AP}$ -approximately proportional ( $\alpha_{AP}$ -proportional) if  $\forall U \subseteq \mathcal{C}$  and  $|U| \geq \left\lceil \frac{n}{k} \right\rceil$  and for all  $y \in C$ , there exists  $i \in U$  with  $\alpha_{AP} \cdot d(i, y) \geq d(i, \phi(i))$ .
${}^{4}p = 1$  for the  $k$  -median and  $p = 2$  for the  $k$  -means
Proof. We show incompatibility of GF with the three Fairness notions. Recall that we consider GF and another fairness constraint at the same time, and incompatible means there are cases where no feasible solution exists that satisfies both constraints at the same time.
Lemma 4. For any  $k \geq 2$ , there exist a clustering problem where no feasible solution exists that satisfies both Fairness in Your Neighborhood and  $\mathbf{GF}$  even if we allow an additive violation of  $\Omega\left(\frac{n}{k}\right)$  in the  $\mathbf{GF}$  constraint.
Proof. Consider the case where  $k \geq 2$ , and consider the clustering problem on a  $\ell$ -community instance with  $k = \ell$ . We consider the case where lower and upper proportion bound of  $\mathbf{GF}$  are set to  $\frac{1}{2}$  for both colors.
Claim 5. 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  $(S_{CI},\phi_{CI})$ .
Proof. By Definitions 1 and 2, for any point  $j \in \mathcal{C}$ , its neighborhood radius is  $\mathrm{NR}_{C,k}(j) = \min \{r : |B_r(j) \cap C| \geq |C| / k\} = 0$ . This is because each point is in one of the  $l$  subsets, and by definition, the subset is of size  $\frac{n}{l} = \frac{n}{k}$ , and points in the same subset are separated by a distance 0.
For a solution on a  $\ell$ -community instance  $(S, \phi) \in (S_{CI}, \phi_{CI})$ , for any point  $j \in \mathcal{C}$ , because  $S$  contains a center in the community where  $j$  is,  $d(j, S) = 0$ .
By definition of  $\alpha_{NR}$ -fairness, this means on a  $\ell$ -community instance, any solution in  $(S_{CI}, \phi_{CI})$  is  $\alpha_{\mathrm{NR}}$ -fair with a finite  $\alpha_{\mathrm{NR}}$ . This is because for any  $(S, \phi) \in (S_{CI}, \phi_{CI})$ ,  $d(j, S) = 0 \leq \alpha_{\mathrm{NR}} \cdot \mathrm{NR}_{C,k}(j)$  holds for  $\alpha_{\mathrm{NR}}$  equal to any finite value.
In any solution that is not in  $(S_{CI},\phi_{CI})$ , there is at least a point which is assigned to a center not in the point's own community. Thus for such a solution  $(S,\phi)$ , there exist  $j\in \mathcal{C}$ ,  $d(i,S) = R$ . Thus for  $(S,\phi)\notin (S_{CI},\phi_{CI})$ , for some  $j\in \mathcal{C}$ , there is no finite  $\alpha_{\mathrm{NR}}$  such that  $d(j,\phi (j)) = R\leq \alpha_{\mathrm{NR}}\cdot \mathrm{NR}_{C,k}(j)$  holds.
This shows that a solution that achieves  $\alpha_{NR}$ -fairness for a finite  $\alpha_{NR}$  must be a solution from the set of solutions  $(S_{CI},\phi_{CI})$ .
Thus we have shown that any solution that satisfies the fairness in your neighborhood constraint approximately do not assign points to centers not in its original community.
To characterize the set of solutions that satisfy  $\mathbf{GF}$  with additive  $\Omega\left(\frac{n}{k}\right)$  violation, we consider two cases separately:  $k$  is even and  $k$  is odd.
Consider the case where  $k \geq 2$  is even and refer to Figure 5 where we have  $\ell = k$  communities that alternate from red to blue color.
Since the lower and upper proportion bounds are set to  $\frac{1}{2}$  for both colors, then to satisfy  $\mathbf{GF}$  each cluster should have both red and blue points. There must exist a cluster  $C_i$  of size  $|C_i| \geq \frac{n}{k}$ , it follows that to satisfy the  $\mathbf{GF}$  constraints at an additive violation of  $\rho$ , then  $|C_i^{\mathrm{blue}}| \geq \frac{1}{2} |C_i| - \rho = \frac{n}{2k} - \rho$  and similarly we would have  $|C_i^{\mathrm{red}}| \geq \frac{n}{2k} - \rho$ . By setting  $\rho = \frac{n}{2k} - \epsilon$  for some constant  $\epsilon > 0$ , then we have  $|C_i^{\mathrm{blue}}|, |C_i^{\mathrm{red}}| > 0$ . This implies that a point need be assigned to a center at a distance  $R > 0$  for the solution to satisfy  $\mathbf{GF}$  with additive  $\Omega\left(\frac{n}{k}\right)$  violation. Therefore such a solution is not in the solution set that satisfies fairness in your neighborhood.
For a value of  $k$  that is odd, see the example Figure 6. Here instead the last community has the same number of red and blue points. We call the cluster whose center is in the last community  $C_{\mathrm{last}}$ .
If  $|C_{\mathrm{last}}| \neq \frac{n}{k}$ , then there are points assigned to the center of  $C_{\mathrm{last}}$  from other communities incurring cost  $R > 0$  or points in the last community are assigned to other centers at distance  $R > 0$ . In both cases there is at least a point assigned to a center not in its community. If  $|C_{\mathrm{last}}| = \frac{n}{k}$ , then
in the remaining  $k - 1$  communities with total of  $n - \frac{n}{k}$  points,  $k - 1$  centers are chosen. A solution satisfying  $\mathbf{GF}$  has one cluster  $C_i$  with at least  $\frac{1}{k - 1}\left(n - \frac{n}{k}\right) = \frac{n}{k}$  points. Then we follow the same argument as in the even  $k$  case. That is, to satisfy  $\mathbf{GF}$  with  $\rho$  additive violation on the  $C_i$ ,  $|C_i^{\mathrm{blue}}| \geq \frac{1}{2}|C_i| - \rho = \frac{n}{2k} - \rho$ ,  $|C_i^{\mathrm{red}}| \geq \frac{n}{2k} - \rho$ , with  $\rho = \frac{n}{2k} - \epsilon$  for some constant  $\epsilon > 0$ , at least a point will be assigned to center at distance  $R > 0$ . Thus such a solution is not in the set of solutions  $(S_{CI},\phi_{CI})$ . Thus the set of solutions that satisfies fairness in your neighborhood has no overlap with the set of solutions that satisfies  $\mathbf{GF}$  with  $\Omega(\frac{n}{k})$  additive violation.
Lemma 5. For any  $k \geq 2$ , there exist a clustering problem where no feasible solution exists that satisfies both Socially Fair and GF even if we allow an additive violation of  $\Omega\left(\frac{n}{k}\right)$  to the GF constraint.
Proof. We follow a similar line of argument as in Lemma 4. Consider the case when  $k \geq 2$ , and consider the clustering problem on a  $\ell$ -community instance with  $k = \ell$ . We consider the case where lower and upper proportion bound of  $\mathbf{GF}$  are set to  $\frac{1}{2}$  for both colors.
Claim 6. 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  $(S_{CI},\phi_{CI})$ .
Proof. Denote the clustering cost of an optimal solution to the a Socially Fair clustering problem as  $\mathrm{OPT}_{\mathrm{SF}}$ . By definition,
$$
\mathrm {O P T} _ {\mathrm {S F}} = \min  _ {S: | S | \leq k} \max  _ {h \in \mathcal {H}} \frac {1}{| \mathcal {C} ^ {h} |} \sum_ {j \in C ^ {h}} d ^ {p} (j, \phi (j)).
$$