SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Algorithm 4 GFToGF+DS
1: Input: Points  $\mathcal{C}$ , Solution  $(\bar{S},\bar{\phi})$  with clusters  $\{\bar{C}_i,\dots ,\bar{C}_{\bar{k}}\}$  satisfying the GF constraints with  $|\bar{S}| = \bar{k}\leq k$   
2: Output: Solution  $(S,\phi)$  satisfying the GF and DS constraints simultaneously.  
3: Initialize:  $\forall h\in \mathcal{H}:s_h = 0,\forall i\in \bar{S}:Q_i = \{\}$   
4: for  $i\in \bar{S}$  do  
5: if  $\exists h\in \mathcal{H}:s_h < k_h^l$  then  
6: Let  $h_0$  be a color such that  $s_{h_0} < k_{h_0}^l$   
7: else  
8: Pick  $h_0$  such that  $s_{h_0} + 1\leq k_{h_0}^u$   
9: end if  
10: Pick a point  $j_{h_0}$  of color  $h_0$  in cluster  $\bar{C}_i$   
11: Set  $Q_{i} = \{j_{h_{0}}\}$   
12: Update  $s_{h_0} = s_{h_0} + 1$   
13: end for  
14: while  $\exists h\in \mathcal{H}:s_h < k_h^l$  do  
15: Pick a color  $h_0$  such that  $s_{h_0} < k_{h_0}^l$   
16: Pick a center  $i\in \bar{S}$  with cluster  $\bar{C}_i$  where there exists a point of color  $h_0$  not in  $Q_{i}$   
17: Pick a point  $j_{h_0}$  of color  $h_0$  in cluster  $\bar{C}_i$   
18: Set  $Q_{i} = Q_{i}\cup \{j_{h_{0}}\}$   
19: Update  $s_{h_0} = s_{h_0} + 1$   
20: end while  
21: Set  $S = \cup_{i\in \bar{S}}Q_i$   
22: for  $i\in \bar{S}$  do  
23:  $\phi_i = \mathrm{DIVIDE}(\bar{C}_i,Q_i)$   
24:  $\forall j\in \bar{C}_i:\mathrm{Set}\phi (j) = \phi_i(j)$ . {Assignment to center is updated using DIVIDE.}  
25: end for
Here we start with a solution  $(\bar{S},\bar{\phi})$  of cost  $\bar{R}$  that satisfies the GF constraints and we want to make it satisfy GF and DS simultaneously. More specifically, given any GF solution we show how it can be post-processed to satisfy  $\mathbf{GF} + \mathbf{DS}$  at a bounded increase to its clustering cost by a factor of 2 (see Theorem 4.2). This implies as a corollary that if we have an  $\alpha_{\mathrm{GF}}$ -approximation algorithm for GF then we can obtain a  $2\alpha_{\mathrm{GF}}$ -approximation algorithm for GF+DS (see Corollary 1).
The algorithm essentially first "covers" each given cluster  $\bar{C}_i$  of the given solution  $(\bar{S},\bar{\phi})$  by picking a point of some color  $h$  to be a future center given that picking a point of such a color would not violate the DS constraints (lines(4-13)). If there are still colors which do not have enough picked centers (below the lower bound  $k_h^l$ ), then more points are picked from clusters where points of such colors exist (lines(14-20)). Once the algorithm has picked correct points for each color, then the DIVIDE subroutine is called to divide the cluster among the picked points.
Now we state the main theorem:
Theorem 4.2. If we have a solution  $(\bar{S},\bar{\phi})$  of cost  $\bar{R}$  that satisfies the GF constraints where the number of non-empty clusters is  $|\bar{S}| = \bar{k} \leq k$ , then we can obtain a solution  $(S,\phi)$  that satisfies GF at an additive violation of 2 and DS simultaneously with cost  $R \leq 2\bar{R}$ .
Corollary 1. Given an  $\alpha_{GF}$ -approximation algorithm for  $GF$ , then we can have a  $2\alpha_{GF}$ -approximation algorithm that satisfies  $GF$  at an additive violation of 2 and  $DS$  simultaneously.
Remark: If the given GF solution has the number of cluster  $\bar{k} = k$ , then the output will have an additive violation of zero, i.e. satisfy the GF constraints exactly. This would happen DIVIDE would always receive  $Q_{i}$  with  $|Q_{i}| = 1$  and therefore we can use the guarantee of DIVIDE for the special case of  $|Q| = 1$ .
4.4 Solving  $\mathbf{GF} + \mathbf{DS}$  using a GF Solution
Figure 3: Figure showing the PoF relation between Unconstrained, GF, DS, and GF+DS clustering.
5 Price of (Doubly) Fair Clustering
Here we study the degradation in the clustering cost (the price of fairness) that comes from imposing the fairness constraint on the clustering objective. The price of fairness  $\mathrm{PoF}_c$  is defined as  $\mathrm{PoF}_c = \frac{\text{Clustering Cost subject to Constraint } c}{\text{Clustering Cost of Agnostic Solution}}$  [20, 9]. Note that since we have two constrains here  $\mathbf{GF}$  and  $\mathbf{DS}$ , we also consider prices of fairness of the form  $\mathrm{PoF}_{c_1 \to c_2} = \frac{\text{Clustering Cost subject to Constraints } c_1 \text{ and } c_2}{\text{Clustering Cost subject to Constraint } c_1}$  which equal the amount of degradation in the clustering cost if we were to impose constraint  $c_2$  in addition to constraint  $c_1$  which is already imposed. Note that we are concerned with the price of fairness in the worst case. Interestingly, we find that imposing the DS constraint over the GF constraint leads to a bounded PoF if we allow an additive violation of 2 for  $\mathbf{GF}$  while the reverse is not true even if we allow an additive violation of  $\Omega\left(\frac{n}{k}\right)$  for  $\mathbf{GF}$ .
We find the following:
Proposition 5.1. For any value of  $k \geq 2$ , imposing  $GF$  can lead to an unbounded PoF even if we allow an additive violation of  $\Omega\left(\frac{n}{k}\right)$ .
Proposition 5.2. For any value of  $k \geq 3$ , imposing DS can lead to an unbounded PoF.
Proposition 5.3. For any value of  $k \geq 2$ , imposing  $GF$  on a solution that only satisfies  $DS$  can lead to an unbounded increase in the clustering cost even if we allow an additive violation of  $\Omega\left(\frac{n}{k}\right)$ .
Proposition 5.4. Imposing  $DS$  on a solution that only satisfies  $GF$  leads to a bounded increase in the clustering cost of at most 2 (PoF ≤ 2) if we allow an additive violation of 2 in the  $GF$  constraints.
6 Incompatibility with Other Distance-Based Fairness Constraints
In this section, we study the incompatibility between the DS and GF constraints and a family of distance-based fairness constraints. We note that the results of this section do not take into account the clustering cost and are based only on the feasibility set. That is, we consider more than one constraints simultaneously and see if the feasibility set is empty or not. Two constraints are considered incompatible if the intersection of their feasible sets is empty. In some cases we also consider solution that could have violations to the constraints. We present two main findings here and defer the proofs and further details to the Appendix section<sup>1</sup>.
Theorem 6.1. For any value  $k \geq 2$ , the fairness in your neighborhood [30], socially fair constraint [1, 24] are each incompatible with GF even if we allow an additive violation of  $\Omega\left(\frac{n}{k}\right)$  in the GF constraint. For any value  $k \geq 5$ , the proportionally fair constraints [15] is incompatible with GF even if we allow an additive violation of  $\Omega\left(\frac{n}{k}\right)$  in the GF constraint.
Theorem 6.2. For any value  $k \geq 3$ , the fairness in your neighborhood [30], socially fair [1, 24] and proportionally fair [15] constraints are each incompatible with DS.
<sup>1</sup>Note that socially fair clustering [1, 24] is defined as an optimization problem not a constraint. However, it can be straightforwardly turned into a constraint, see the Appendix for full details.
We use Python 3.9, the CPLEX package [38] for solving linear programs and NetworkX [27] for max-flow rounding. Further, Scikit-learn is used for some standard ML related operations. We use commodity hardware, specifically a MacBook Pro with an Apple M2 chip.
We conduct experiments over datasets from the UCI repository [23] to validate our theoretical findings. Specifically, we use the Adult dataset sub-sampled to 20,000 records. Gender is used for group membership while the numeric entries are used to form a point (vector) for each record. We use the Euclidean distance. Further, for the GF constraints we set the lower and upper proportion bounds to  $\beta_h = (1 - \delta)r_h$  and  $\alpha_h = (1 + \delta)r_h$  for each color  $h$  where  $r_h$  is color  $h'$ s proportion in the dataset and we set  $\delta = 0.2$ . For the DS constraints, since we do not deal with a large number of centers we set  $k_h^l = 0.8r_hk$  and  $k_h^u = r_hk$ .
We compare the performance of 5 algorithms. Specifically, we have (1) COLOR-BLIND: An implementation of the Gonzalez  $k$ -center algorithm [25] which achieves a 2-approximation for the unconstrained  $k$ -center problem. (2) ALG-GF: A GF algorithm which follows the sketch of [9], however the final rounding step is replaced by an implementation of the MAXFLOWGF rounding subroutine. This algorithm has a 3-approximation for the GF constrained instance. (3) ALG-DS: An algorithm for the DS problem recently introduced by [37] for which also has an approximation of 3. (4) GFTOGFDS: An implementation of algorithm 4 where we simply use the GF algorithm just mentioned to obtain a GF solution. (5) DSTOGFDS: Similarly an implementation of algorithm 2 where DS algorithm is used as a starting point instead.
Throughout we measure the performance of the algorithms in terms of (1) PoF: The price of fairness of the algorithm. Note that we always calculate the price of fairness by dividing by the COLOR-BLIND clustering cost since it solves the unconstrained problem. (2) GF-Violation: Which is the maximum additive violation of the solution for the GF constraint as mentioned before. (3) DS-Violation: Which is simply the maximum value of the under-representation or over-representation across all groups in the selected centers.
(a)
Figure 4: Adult dataset results: (a) PoF comparison of 5 algorithms, with COLOR-BLIND as baseline; (b) GF-Violation comparison; (c) DS-Violation comparison.
Figure 4 shows the behaviour of all 5 algorithms. In terms of PoF, all algorithms have a significant degradation in the clustering cost compared to the COLOR-BLIND baseline except for ALG-DS. However, ALG-DS has a very large GF-Violation. In fact, the GF-Violation of ALG-DS can be more than 5 times the GF-Violation of COLOR-BLIND. This indicates that while ALG-DS has a small clustering cost, it can give very bad guarantees for the GF constraints. Finally, in terms of the DS-Violation we see that the ALG-GF and the COLOR-BLIND solution can violate the DS constraint. Note that both coincide perfectly on each other. Further, although the violation is 1, it is very significant since unlike the GF constraints the number of centers can be very small. On the other hand, we see that both GFTOGFDS and DSTOGFDS give the best of both worlds having small values for the GF-Violation and zero values for the DS-Violation and while their price of fairness can be significant, it is comparable to ALG-GF. Interestingly, the GFTOGFDS and DSTOGFDS are in agreement in terms of measures. This could be because our implementations of the “GF part” of DSTOGFDS (its handling of the GF constraints) has similarities to the GFTOGFDS algorithm. We show further experiments in the appendix.
[36] Sepideh Mahabadi and Ali Vakilian. Individual fairness for  $k$ -clustering. In International Conference on Machine Learning, 2020.
[37] Huy Lé Nguyen, Thy Nguyen, and Matthew Jones. Fair range k-center. arXiv preprint arXiv:2207.11337, 2022.
[38] Stefan Nickel, Claudius Steinhardt, Hans Schlenker, and Wolfgang Burkart. IBM ilog cplex optimization studio—a primer. In Decision Optimization with IBM ILOG CPLEX Optimization Studio: A Hands-On Introduction to Modeling with the Optimization Programming Language (OPL), pages 9–21. Springer, 2022.
[39] Manish Raghavan, Solon Barocas, Jon Kleinberg, and Karen Levy. Mitigating bias in algorithmic hiring: Evaluating claims and practices. In Proceedings of the 2020 conference on fairness, accountability, and transparency, pages 469-481, 2020.
[40] Clemens Rösner and Melanie Schmidt. Privacy preserving clustering with constraints. arXiv preprint arXiv:1802.02497, 2018.
[41] Ji Wang, Ding Lu, Ian Davidson, and Zhaojun Bai. Scalable spectral clustering with group fairness constraints. In International Conference on Artificial Intelligence and Statistics, pages 6613-6629. PMLR, 2023.
A Useful Facts and Lemmas
Fact A.1. Given non-negative numbers  $a_1, a_2, \ldots, a_n$  and positive numbers  $b_1, b_2, \ldots, b_n$ , then:
$$
\min _ {i \in [ n ]} \frac {a _ {i}}{b _ {i}} \leq \frac {\sum_ {i \in [ n ]} a _ {i}}{\sum_ {i \in [ n ]} b _ {i}} \leq \max _ {i \in [ n ]} \frac {a _ {i}}{b _ {i}}
$$
B MAXFLOWGF
We start with the following Lemma. First, note that  $x_{qj}$  is a decision variable if  $x_{qj} = 1$  then point  $j$  is assigned to center  $q$  and if  $x_{qj} = 0$  then it is not. In an integral solution  $x_{qj} \in \{0,1\}$ , but a fractional LP solution could instead have values in  $[0,1]$ . We use the bold symbol  $\mathbf{x}$  for the collection of value  $\{x_{qj}\}_{q \in Q, j \in C}$ :
Lemma 3. Given a fractional solution  $\mathbf{x}^{\text{frac}}$  that satisfies the GF constraints at an additive violation of at most  $\rho$ , then if there exists an integral solution  $\mathbf{x}^{\text{integ}}$  that satisfies:
$$
\forall q \in Q: \left\lfloor \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}} \right\rfloor \leq \sum_ {j \in C} x _ {q j} ^ {\text {i n t e g}} \leq \left[ \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}} \right] \tag {3}
$$
$$
\forall q \in Q, h \in \mathcal {H}: \left\lfloor \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} \right\rfloor \leq \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {i n t e g}} \leq \left\lceil \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} \right\rceil \tag {4}
$$
Then this integral solution  $\mathbf{x}^{\mathrm{integ}}$  satisfies the  $GF$  constraints at an additive violation of at most  $\rho + 2$ .
Proof. Since the fractional solution satisfies the GF constraints at an additive violation of  $\rho$ , then we have the following:
$$
- \rho + \left(\beta_ {h} \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}}\right) \leq \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} \leq \left(\alpha_ {h} \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}}\right) + \rho
$$
We start with the upper bound:
$$
\begin{array}{l} \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {i n t e g}} \leq \left[ \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} \right] \\ \leq \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} + 1 \\ \leq \alpha_ {h} \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}} + \rho + 1 \\ \leq \alpha_ {h} \left(\sum_ {j \in C} x _ {q j} ^ {\text {i n t e g}} + 1\right) + \rho + 1 \\ \leq \alpha_ {h} \sum_ {j \in C} x _ {q j} ^ {\text {i n t e g}} + (\alpha_ {h} + \rho + 1) \\ \leq \alpha_ {h} \sum_ {j \in C} x _ {q j} ^ {\text {i n t e g}} + (\rho + 2) \\ \end{array}
$$
Now we do the lower bound:
$$
\begin{array}{l} \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {i n t e g}} \geq \left\lfloor \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} \right\rfloor \\ \geq \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} - 1 \\ \geq \beta_ {h} \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}} - \rho - 1 \\ \geq \beta_ {h} \left(\sum_ {j \in C} x _ {q j} ^ {\text {i n t e g}} - 1\right) - (\rho + 1) \\ \geq \beta_ {h} \sum_ {j \in C} x _ {q j} ^ {\text {i n t e g}} - (\beta_ {h} + \rho + 1) \\ \geq \beta_ {h} \sum_ {j \in C} x _ {q j} ^ {\text {i n t e g}} - (\rho + 2) \\ \end{array}
$$
The LP solution given to MAXFLOWGF satisfies the GF constraints at an additive violation of  $\rho$ , we want to show that the output integral solution satisfies the above conditions of Eqs (3 and 4). The MAXFLOWGF  $(\mathbf{x}^{\mathrm{LP}}, C, Q)$  subroutine is similar to that shown in [20, 5, 10]. Specifically, given an LP solution  $\mathbf{x}^{\mathrm{LP}} = \{x^{\mathrm{LP}}\}_{q \in Q, j \in Q}$ , a set of points  $C$ , and a set of centers  $Q$  and a color assignment function  $\chi: C \to \mathcal{H}$  which assigns to each point in  $C$  exactly one color in the set of colors  $\mathcal{H}$ , we construct the flow network  $(V, A)$  according to the following:
1.  $V = \{s,t\} \cup C\cup \{(q,q^h)|q\in Q,h\in \mathcal{H}\}$
2.  $A = A_{1} \cup A_{2} \cup A_{3} \cup A_{4}$  where  $A_{1} = \{(s, j) | j \in C\}$  with upper bound of 1.  $A_{2} = \{(j, (q, q^{h})) | j \in C, x_{qj} > 0\}$  with upper bound of 1. The arc set  $A_{3} = \{((q, q^{h}), q) | q \in Q, h \in \mathcal{H}\}$  with lower bound  $\left\lfloor \sum_{j \in C^{h}} x_{qj}^{\mathrm{LP}} \right\rfloor$  and upper bound of  $\left\lceil \sum_{j \in C^{h}} x_{qj}^{\mathrm{LP}} \right\rceil$ . As for  $A_{4} = \{(q, t) | q \in Q\}$  the lower and upper bounds are  $\left\lfloor \sum_{j \in C} x_{qj}^{\mathrm{LP}} \right\rfloor$  and  $\left\lceil \sum_{j \in C} x_{qj}^{\mathrm{LP}} \right\rceil$ .
In the above all lower and upper bounds of the network are integral, therefore if we can show a feasible solution to the above then there must exist an integral flow assignment which also satisfies the constraints. By the construction of the network we have the following fact about any max flow integral solution  $\mathbf{x}^{\mathrm{integ}}$ .
Fact B.1.
$$
\forall q \in Q: \left\lfloor \sum_ {j \in C} x _ {q j} ^ {L P} \right\rfloor \leq \sum_ {j \in C} x _ {q j} ^ {\text {i n t e g}} \leq \left[ \sum_ {j \in C} x _ {q j} ^ {L P} \right] \tag {5}
$$
$$
\forall q \in Q, h \in \mathcal {H}: \left\lfloor \sum_ {j \in C ^ {h}} x _ {q j} ^ {L P} \right\rfloor \leq \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {i n t e g}} \leq \left[ \sum_ {j \in C ^ {h}} x _ {q j} ^ {L P} \right] \tag {6}
$$
Accordingly, the following theorem immediately holds:
Theorem B.1. Given an LP solution to MAXFLOWGF that satisfies the GF constraints at an additive violation of  $\rho$  and a clustering cost of  $R$ , then the output integral solution satisfies the GF constraints at an additive violation of  $\rho + 2$  and a clustering cost of at most  $R$ .
Proof. The guarantee for the additive violation of  $\mathbf{GF}$  follows immediately from Lemma 3 and Fact B.1. The guarantee for the clustering cost holds, since a point (vertex)  $j$  is not connected to a center vertex  $(q, q^h)$  unless  $x_{qj} > 0$  which can only be the case if  $d(j, q) \leq R$ .
C OMITTED PROOFS
We restate the following lemma and give its proof:
Proof. We first consider the case where  $|Q| > 1$ . We prove the following claim:
Claim 1. For the fractional assignment  $\{x_{qj}^{\text{frac}}\}_{q \in Q, j \in C}$  such that:
$$
\forall q \in Q, \forall h \in \mathcal {H}: \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} = \frac {| C ^ {h} |}{| Q |} = T _ {h}
$$
It holds that: (1)  $\forall q\in Q:\sum_{j\in C}x_{qj}^{\text{frac}}\geq 1$  (2)  $\mathbf{GF}$  constraints are satisfied at an additive violation of  $\frac{\rho}{|Q|}$ .
Proof. Now we prove the first property
$$
\forall q \in Q: \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}} = \sum_ {h \in \mathcal {H}} \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} = \frac {1}{| Q |} \sum_ {h \in \mathcal {H}} | C ^ {h} | = \frac {| C |}{| Q |} \geq 1 \quad (\text {s i n c e} Q \subset C) \tag {7}
$$
Since the GF constraints given center  $i$  are satisfied at an additive violation of  $\rho$ , then we have:
$$
\forall h \in \mathcal {H}: - \rho + \beta_ {h} | C | \leq | C ^ {h} | \leq \alpha_ {h} | C | + \rho \tag {8}
$$
Therefore, since the amount of color for each center in  $Q$  with the fractional assignment can be obtained by dividing by  $|Q|$ , then we have:
$$
\forall h \in \mathcal {H}, \forall q \in Q: - \frac {\rho}{| Q |} + \beta_ {h} \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}} \leq \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} \leq \alpha_ {h} \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}} + \frac {\rho}{| Q |} \tag {9}
$$
Therefore the GF constraints are satisfied at an additive violation of  $\frac{\rho}{|Q|}$ .
Denoting the assignment  $\phi$  resulting from DIVIDE by  $\{x_{qj}^{\mathrm{integ}}\}_{q\in Q,j\in C}$ , then the following claim holds:
Claim 2.
$$
\forall q \in Q: \left\lfloor \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}} \right\rfloor \leq \sum_ {j \in C} x _ {q j} ^ {\text {i n t e g}} \leq \left[ \sum_ {j \in C} x _ {q j} ^ {\text {f r a c}} \right]
$$
$$
\forall q \in Q, h \in \mathcal {H}: \left\lfloor \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} \right\rfloor \leq \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {i n t e g}} \leq \left\lceil \sum_ {j \in C ^ {h}} x _ {q j} ^ {\text {f r a c}} \right\rceil
$$
Proof. For any color  $h$  we have  $|C_h| = a_h|Q| + b_h$  where  $a_h$  and  $b_h$  are non-negative integers and  $b_h$  is the remainder of dividing  $|C_h|$  by  $Q$  ( $b_h \in \{0, 1, \dots, |Q| - 1\}$ ). It follows that  $\sum_{j \in C^h} x_{qq}^{\mathrm{frac}} = T_h = a_h + \frac{b_h}{|Q|}$ . DIVIDE gives each center either  $\sum_{j \in C^h} x_{qq}^{\mathrm{integ}} = a_h = \lfloor T_h \rfloor = \left\lfloor \sum_{j \in C^h} x_{qq}^{\mathrm{frac}} \right\rfloor$  or  $\sum_{j \in C^h} x_{qq}^{\mathrm{integ}} = a_h + 1 = \lceil T_h \rceil = \left\lceil \sum_{j \in C^h} x_{qq}^{\mathrm{frac}} \right\rceil$ . This proves the second condition.
For the first condition, note that  $|C| = \sum_{h\in \mathcal{H}}(a_h|Q| + b_h) = (\sum_{h\in \mathcal{H}}a_h)|Q| + a|Q| + b$  where we set  $\sum_{h\in \mathcal{H}}b_h = a|Q| + b$  with  $a$  and  $b$  being non-negative integers.  $b$  is the remainder and has values
2 In our notation  $x_{qj} \in [0,1]$  denotes the assignment of point  $j$  to center  $q$ .
in  $\{0,1,\ldots ,|Q| - 1\}$ . Accordingly, the sum of the remainders across the colors is  $a|Q| + b$ . Since the remainders are added "successively" across the centers (see Figure 2) and  $a$  is divisible by  $|Q|$ , then for any center  $q\in Q$  either  $\sum_{j\in C}x_{qj}^{\mathrm{integ}} = (\sum_{h\in \mathcal{H}}a_h) + a$  or  $\sum_{j\in C}x_{qj}^{\mathrm{integ}} = (\sum_{h\in \mathcal{H}}a_h) + a + 1$ . Note that  $\sum_{j\in C}x_{qj}^{\mathrm{frac}} = \sum_{h\in \mathcal{H}}T_h = (\sum_{h\in \mathcal{H}}a_h) + a + \frac{b}{|Q|}$ . Therefore,  $\left\lfloor \sum_{j\in C}x_{qj}^{\mathrm{frac}}\right\rfloor = (\sum_{h\in \mathcal{H}}a_h) + a$  and  $\left\lceil \sum_{j\in C}x_{qj}^{\mathrm{frac}}\right\rceil = (\sum_{h\in \mathcal{H}}a_h) + a + 1$ . This proves, the first condition.
By Claim 2 and Lemma 3 it follows that for each center  $q \in Q$  the assignment  $\{x_{qj}^{\mathrm{integ}}\}_{q \in Q, j \in C}$  satisfies the GF constraints at an additive violation of  $\frac{\rho}{|Q|} + 2$ , this proves the first guarantee.
By Claim 2 and guarantee (1) of Claim 1, then  $\forall q\in Q:\sum_{j\in C}x_{qj}^{\mathrm{integ}}\geq \left\lfloor \sum_{j\in C}x_{qj}^{\mathrm{frac}}\right\rfloor \geq 1.$  Therefore, every center  $q\in Q$  is active proving the second guarantee.
Guarantee (3) follows since  $\forall j\in C:d(j,\phi (j))\leq d(j,i) + d(i,\phi (j))\leq 2R.$
Now if  $|Q| = 1$ , then guarantee (2) follows since the cluster  $C$  is non-empty. Guarantee (3) follows similarly to the above. The additive violation in the GF constraint on the other hand is  $\rho$  since the single center  $Q$  has the exact set of points that were assigned to the original center  $i$ .
We restate the next lemma and give its proof:
Proof. We begin with the following claim which shows that there exists a solution that only uses centers from  $\tilde{S}$  to satisfy the GF constraints exactly and at a radius of at most  $(1 + \alpha_{\mathbf{DS}})R_{\mathbf{GF} + \mathbf{DS}}^{*}$ . Note that this claim has non-constructive proof, i.e. it only proves the existence of such a solution:
Claim 3. Given the set of centers  $\bar{S}$  resulting from the  $\alpha_{\mathbf{DS}}$ -approximation algorithm, then there exists an assignment  $\phi_0$  from points in  $\mathcal{C}$  to centers in  $\bar{S}$  such that the following holds: (1) The  $\mathbf{GF}$  constraint is exactly satisfied (additive violation of 0). (2) The clustering cost is at most  $(1 + \alpha_{\mathbf{DS}})R_{\mathbf{GF} + \mathbf{DS}}^{*}$ .