SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Now, we turn to the MST-Sp criterion. Let  $\rho \coloneqq \min \left\{\frac{n / k}{L}, 2\right\}$ . Our main contribution is Algorithm 2, it obtains a  $\left(\frac{\rho(1 - \epsilon)}{2}, \frac{1}{H_{k-1}}\right)$  approximation for this criterion, where  $H_{k-1} = \sum_{i=1}^{k-1} \frac{1}{i}$  is the  $(k-1)$ -th Harmonic number. Note that  $H_{k-1}$  is  $\Theta(\log k)$ .
In high level, for each  $\ell = 2,\dots ,k$ , the algorithm calls A1goMinSp (Algorithm 1) to build a clustering  $\mathcal{A}_{\ell}^{\prime}$  with  $\ell$  groups and then it transforms  $\mathcal{A}_{\ell}^{\prime}$  (lines 5-13) into a clustering  $\mathcal{A}_{\ell}$  with  $k$  groups. In the end, it returns the clustering, among the  $k - 1$  considered, with maximum MST-Sp.
Algorithm 2 Constrained-MaxMST( $\mathcal{X}$ ; dist;  $k$ ;  $L$ ;  $\epsilon$ )
1: for  $\ell = 2,\ldots ,k$  do   
2:  $\mathcal{A}_{\ell}^{\prime}\gets \mathrm{AlgoMinSp}(\mathcal{X},\mathrm{dist},\ell ,\epsilon ,L)$    
3: NonVisited  $\leftarrow \mathcal{A}_{\ell}^{\prime}$    
4:  $\mathcal{A}_{\ell}\gets \emptyset$    
5: for each  $A^\prime$  in  $\mathcal{A}_{\ell}^{\prime}$  , iterating from the largest group to the smallest do   
6: NonVisited  $\leftarrow$  NonVisited-  $A^{\prime}$    
7: SplitNumber  $\leftarrow \left\lfloor \frac{2|A^{\prime}|}{\rho(1 - \epsilon)L}\right\rfloor$    
8: if  $|\mathcal{A}_\ell | + |\mathrm{NonVisited}| + \mathrm{SplitNumber} <   k$  then   
9: Split  $A^\prime$  into SplitNumber as balanced as possible groups and add them to  $\mathcal{A}_{\ell}$    
10: else   
11: Split  $A^\prime$  into  $k - |\mathcal{A}_{\ell}| - |\mathrm{NonVisited}|$  as balanced as possible groups; add them to  $\mathcal{A}_{\ell}$    
12: Add all groups in NonVisited to  $\mathcal{A}_{\ell}$    
13: Break   
14: Return the clustering  $A_{\ell}$  , among the  $k - 1$  obtained, that has the maximum MST-Sp
We remark that we do not need to scan the groups in  $\mathcal{A}_{\ell}^{\prime}$  by non-increasing order of their sizes to establish our guarantees presented below. However, this rule tends to avoid groups with sizes smaller than  $L$ .
Lemma 4.3. Fix  $\epsilon >0$ . Thus, for each  $\ell$ , every group in  $\mathcal{A}_{\ell}$  has at least  $\left\lfloor \frac{\rho(1 - \epsilon)L}{2} \right\rfloor$  points.
Proof. The groups that are added to  $\mathcal{A}_{\ell}$  in line 12 have at least  $(1 - \epsilon)L$  points while the number of points of those that are added at either line 11 or 9 is at least
$$
\left\lfloor \frac {| A ^ {\prime} |}{\lfloor 2 | A ^ {\prime} | / \rho (1 - \epsilon) L \rfloor} \right\rfloor \geq \left\lfloor \frac {\rho (1 - \epsilon) L}{2} \right\rfloor
$$
Moreover, if the For is not interrupted by the Break command, the total number of groups in  $\mathcal{A}_{\ell}$  is
$$
\sum_ {A ^ {\prime} \in \mathcal {A} _ {\ell} ^ {\prime}} \left\lfloor \frac {2 \left| A ^ {\prime} \right|}{\rho (1 - \epsilon) L} \right\rfloor \geq \sum_ {A ^ {\prime} \in \mathcal {A} _ {\ell} ^ {\prime}} \frac {2 \left| A ^ {\prime} \right|}{\rho (1 - \epsilon) L} - \ell \geq \frac {2 n}{\rho (1 - \epsilon) L} - k \geq \frac {2 k}{(1 - \epsilon)} - k \geq k
$$
Since the For is interrupted as soon as  $k$  groups can be obtained then,  $\mathcal{A}_{\ell}$  has  $k$  groups.
For the next results, we use  $\mathcal{C}^*$  to denote the  $(k,L)$ -clustering with maximum MST-Sp and  $w_{i}^{*}$  to denote the cost of the  $i$ -th cheapest edge in the MST for  $G_{\mathcal{C}^*}$ . Our first lemma can be seen as a generalization of Theorem 4.1. Its proof can be found in the appendix.
Lemma 4.4. For each  $\ell$ ,  $\operatorname{Min - Sp}(\mathcal{A}_{\ell}^{\prime}) \geq w_{k - \ell + 1}^{*}$ .
A simple consequence of the previous lemma is that the MST-Sp of clustering  $\mathcal{A}_{\ell}^{\prime}$  is at least  $(\ell -1)\cdot$ $w_{k - \ell +1}^{*}$ . The next lemma shows that this bound also holds for the clustering  $\mathcal{A}_{\ell}$ . The proof consists of showing that each edge of a MST for  $\mathcal{A}_{\ell}^{\prime}$  is also an edge of a MST for  $\mathcal{A}_{\ell}$ .
Lemma 4.5. For each  $\ell = 2,\ldots ,k$  we have MST-  $\operatorname {Sp}(\mathcal{A}_\ell)\geq (\ell -1)\cdot w_{k - \ell +1}^*$
Proof. Let  $T_{\ell}$  and  $T_{\ell}'$  be, respectively, the MST for  $G_{\mathcal{A}_{\ell}}$  and  $G_{\mathcal{A}_{\ell}'}$ . By the previous lemma, each of the  $(\ell - 1)$  edges of  $T_{\ell}'$  has cost at least  $\operatorname{Min-Sp}(A_{\ell}') \geq w_{k - \ell + 1}^{*}$ . Thus, to establish the result, it is enough to argue that each edge of  $T_{\ell}'$  also belongs to  $T_{\ell}$ .
We say that a group  $A \in \mathcal{A}_{\ell}$  is generated from a group  $A' \in \mathcal{A}_{\ell}'$  if  $A = A'$  or  $A$  is one of the balanced groups that is generated when  $A'$  is split in the internal For of Algorithm 2. We say that a
vertex  $x$  in  $G_{\mathcal{A}_{\ell}}$  is generated from a vertex  $x'$  in  $G_{\mathcal{A}_{\ell}'}$  if the group corresponding to  $x$  is generated by the corresponding to  $x'$ .
Let  $e' = u'v'$  be an edge in  $T_{\ell}'$  and let  $S'$  be a cut in graph  $G_{\mathcal{A}_{\ell}'}$  whose vertices are those from the connected component of  $T_{\ell}' \setminus e'$  that includes  $u'$ . We define the cut  $S$  of  $G_{\mathcal{A}_{\ell}}$  as follows:  $S = \{x \in G_{\mathcal{A}_{\ell}} | x \text{ is generated from some } x' \in S'\}$ .
Let  $u$  and  $v$  be vertices generated from  $u'$  and  $v'$ , respectively, that satisfy  $w(uv) = w(u'v')$ . It is enough to show that  $uv$  is the cheapest edge that crosses  $S$  since by the cut property [Kleinberg & Tardos (2006), Property 4.17.] this implies that  $uv \in T_{\ell}$ . We prove it by contradiction. Let us assume that there is another edge  $f = yz$  that crosses  $S$  and has weight smaller than  $w(uv)$ . Let  $y'$  and  $z'$  be vertices in  $G_{\mathcal{A}_{\ell}'}$  that generate  $y$  and  $z$ , respectively, and let  $f' = y'z'$ . Thus,  $w(f') \leq w(f) < w(uv) = w(u'v') = w(e')$ . However, this contradicts the cycle property of MST's [Kleinberg & Tardos (2006), Property 4.20] because it implies that the edge with the largest weight in the cycle of  $G_{\mathcal{A}_{\ell}'}$  comprised by edge  $f'$  and the path in  $T_{\ell}'$  the connects  $y'$  to  $z'$  belongs to the  $T_{\ell}'$ .
The next theorem is the main result of this section.
Theorem 4.6. Fix  $\epsilon >0$ . Algorithm 2 is a  $(\frac{(1 - \epsilon)\rho}{2},\frac{1}{H_{k - 1}})$ -approximation for the problem of finding the  $(k,L)$ -clustering that maximizes the MST-  $\mathbf{Sp}$  criterion.
Proof. Let  $\mathcal{C}$  be the clustering returned by Algorithm 2. Lemma 4.3 guarantees that  $\mathcal{C}$  is a  $(k, \left\lfloor \frac{(1 - \epsilon)\rho L}{2} \right\rfloor)$ -clustering.
Thus, we just need to argue about MST-Sp  $(\mathcal{C})$  . We have that
$$
\operatorname {M S T - S p} \left(\mathcal {C} ^ {*}\right) = \sum_ {i = 2} ^ {k} w _ {k - i + 1} ^ {*}
$$
and, due to Lemma 4.5,  $\mathrm{MST - Sp}(\mathcal{C})\geq \max \{(\ell -1)\cdot w_{k - \ell +1}^{*}|2\leq \ell \leq k\}$
Let  $\tilde{\ell}$  be the value  $\ell$  that maximizes  $(\ell - 1) \cdot w_{k - \ell + 1}^{*}$ . It follows that  $w_{k - i + 1}^{*} \leq ((\tilde{\ell} - 1) / (i - 1))w_{k - \tilde{\ell} + 1}^{*}$  for  $i = 2, \ldots, k$ . Thus,
$$
\frac {\operatorname {M S T - S p} \left(\mathcal {C} ^ {*}\right)}{\operatorname {M S T - S p} (\mathcal {C})} \leq \frac {\sum_ {i = 2} ^ {k} w _ {k - i + 1} ^ {*}}{(\tilde {\ell} - 1) \cdot w _ {k - \tilde {\ell} + 1} ^ {*}} \leq \frac {(\tilde {\ell} - 1) \cdot w _ {k - \tilde {\ell} + 1} ^ {*} \cdot \sum_ {i = 2} ^ {k} \frac {1}{i - 1}}{(\tilde {\ell} - 1) \cdot w _ {k - \tilde {\ell} + 1} ^ {*}} = H _ {k - 1}
$$
We end this section by showing that the optimization of MST-Sp is APX-HARD (for fixed  $k$ ) when a hard constraint on the number of points per group is imposed. The proof can be found in the appendix.
Theorem 4.7. Unless  $P = NP$ , for any  $\alpha = poly(n)$ , there is no  $(1, \frac{k - 2}{k - 1} + \frac{1}{\alpha(k - 1)})$ -approximation for the problem of finding the  $(k, L)$ -clustering that maximizes the MST-Sp criterion.
To evaluate the performance of Algorithms 1 and 2, we ran experiments with 10 different datasets, comparing the results with those of single-linkage and of the traditional  $k$ -means algorithm from Lloyd (1982) with a ++ initialization (Arthur & Vassilvitskii, 2007). For the implementation of routine MaxMinSched, employed by Algorithm 1, we used the Longest Processing Time rule. This rule has the advantage of being fast while guaranteeing a 3/4 approximation for the max-min scheduling problem (Csirik et al., 1992). The code for running the algorithms can be found at https://github.com/lmurtinho/SizeConstrainedSpacing.
Our first experiment investigates the size of the groups produced by single-linkage for the 10 datasets, whose dimensions can be found in the first two columns of Table 1. Figure 2 shows the proportion of singletons for each dataset with the growth of  $k$ . For all datasets but Vowel and Mice
Table 1: Min-Sp and MST-Sp for the different methods and datasets.
Dimensions	Min-Sp	MST-Sp
n	k	Algo 1	Algo 2	k-means	Algo 1	Algo 2	k-means
anuran	7,195	10	0.19	0.09	0.05	1.71	1.87	1.01
avila	20,867	12	0.07	0.04	0	0.77	0.81	0.66
collins	1,000	30	0.42	0.42	0.22	12.42	12.42	8.58
digits	1,797	10	19.74	19.74	13.79	178.22	178.22	145.13
letter	20,000	26	0.2	0.11	0.07	4.98	5.67	1.98
mice	552	8	0.79	0.79	0.24	5.66	5.66	2.37
newsgroups	18,846	20	1	1	0.17	19	19	8.4
pendigits	10,992	10	23.89	9.08	8.31	215.11	217.01	119.85
sensorless	58,509	11	0.13	0.08	0.03	1.31	1.36	1.29
vowel	990	11	0.49	0.49	0.11	4.94	4.94	1.84
the majority of groups are singletons, even for small values of  $k$ . This undesirable behavior motivates our constraint on the minimum size of a group.
Figure 2: Proportion of singletons for each dataset with the growth of  $k$
In our second experiment, we compare the values of Min-Sp and MST-Sp achieved by our algorithms with those of  $k$ -means. While  $k$ -means is not a particularly strong competitor in the sense that it was not designed to optimize our criteria, the motivation to include it is that  $k$ -means is a very popular choice among practitioners. Moreover, for datasets with well-separated groups, the minimization of the squared sum of errors (pursued by  $k$ -means) should also imply the maximization of inter-group criteria.
Table 1 presents the results of this experiment. The values chosen for  $k$  are the numbers of classes (dependent variable) in the datasets, while for the dist function we employed the Euclidean distance. The values associated with the criteria are averages of 10 executions, with each execution corresponding to a different seed provided to  $k$ -means. To set the value of  $L$  for the  $i$ -th execution of our algorithms, we take the size of the smallest group generated by  $k$ -means for this execution and multiply it by  $4/3$ . This way, we guarantee that the size of the smallest group produced by our methods, for each execution, is not smaller than that of  $k$ -means, which makes the comparison among the optimization criteria fairer.
With respect to the Min-Sp criterion, Algorithm 1 is at least as good as Algorithm 2 for every dataset (being superior on 6) and both Algorithm 1 and 2 outperform  $k$ -means on all datasets. On the other hand, with respect to the MST-Sp criterion, Algorithm 2 is at least as good as Algorithm 1 for every dataset (being better on 6) and, again, both algorithms outperform  $k$ -means for all datasets. In the appendix, we present additional information regarding this experiment. In particular, we show that
Table 2: Running time in seconds of single-linkage and our methods.
Dataset	single-linkage	Algo 1	Algo 2
sensorless	93.8	99.4	701.4
newsgroups	274.2	276.4	440.4
letter	4.6	5.9	116.4
avila	4.0	5.5	62.5
pendigits	1.4	2.1	16.0
the MST-Sp achieved by Algorithm 2 is on average  $81\%$  of the upper bound  $\sum_{\ell = 2}^{k}\mathrm{Min - Sp}(\mathcal{A}_{\ell}^{\prime})$  that follows from Lemma 4.4. This is much better than the  $1 / H_{k - 1}$  ratio given by the theoretical bound.
Finally, Table 2 shows the running time of our algorithms for the datasets that consumed more time. We observe that the overhead introduced by Algorithm 1 w.r.t. single-linkage is negligible while Algorithm 2, as expected, is more costly. In the appendix, we show that a strategy that only considers values of  $\ell$  that can be written as  $\lceil k / 2^t\rceil$ , for  $t = 0,\dots ,\lfloor \log k\rfloor$ , in the first loop of Algorithm 2 provides a significant gain of running time while incurring a small loss in the MST-Sp. We note that the  $\log k$  bound of Theorem 4.6 is still valid for this strategy.
We have endeavored in this paper to expand the current knowledge on clustering methods for optimizing inter-cluster criteria. We have proved that the well-known single-linkage produces partitions that maximize not only the minimum spacing between any two clusters, but also the MST spacing, a stronger guarantee. We have also studied the task of maximizing these criteria under the constraint that each group of the clustering has at least  $L$  points. We provided complexity results and algorithms with provable approximation guarantees.
One potential limitation of our proposed algorithms is their usage on massive datasets (in particular Algorithm 2) since they execute single-linkage one or many times. If the dist function is explicitly given, then the  $\Omega(n^2)$  time spent by single-linkage is unavoidable. However, if the distances can be calculated from the set of points  $\mathcal{X}$  then faster algorithms might be obtained.
The main theoretical question that remained open in our work is whether there exist constant approximation algorithms for the maximization of MST-Sp. In addition to addressing this question, interesting directions for future research include handling different inter-group measures as well as other constraints on the structure of clustering.
The authors thank BigDataCorp (https://bigdatacorp.com.br/) for providing computational power for the experiments of an initial version of the paper.
The work of the authors is partially supported by the Air Force Office of Scientific Research (award number FA9550-22-1-0475).
The work of the first author is partially supported by CNPq (grant 310741/2021-1).
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A Properties of Minimum Spanning Trees
Theorem A.1 (Cut Property, Kleinberg & Tardos (2006), Property 4.17.). Let  $G = (V, E)$  be a graph with distinct weights on its edges. Let  $S \subset V$  be a non-empty cut in  $G$ . If  $e$  is the edge with minimum cost among those that have one endpoint in  $S$  and the other one in  $V \setminus S$ , then  $e$  belongs to every MST for  $G$
Theorem A.2 (Cycle Property, Kleinberg & Tardos (2006), Property 4.20.). Let  $G = (V, E)$  be a graph with distinct weights on its edges and let  $C$  be a cycle in  $G$ . Then, the edge with the largest weight in  $C$  does not belong to any minimum spanning tree for  $G$ .
The following characterization of MST's will be useful. Its correctness follows directly from the cycle property.
Theorem A.3. Let  $G = (V, E)$  be a graph with weights on its edges. A spanning tree  $T$  for  $G$  is a MST for  $G$  if and only if for each edge  $e = uv$  in  $E$ , the weight  $w(e)$  of  $e$  satisfies  $w(e) \geq w(e')$  for every edge  $e'$  in the path that connects  $u$  to  $v$  in  $T$ .
We need the following proposition.
Proposition B.1. Let  $\mathcal{C}'$  be a  $k$ -clustering for instance  $I$  and let  $T'$  be a MST for  $G_{\mathcal{C}'}$ . Moreover, let  $C_i'$  and  $C_j'$  be groups of  $\mathcal{C}'$  such that  $\mathrm{sp}(C_i', C_j') = \mathrm{Min - Sp}(\mathcal{C}')$ . Then, the tree  $T^a$  that results from the contraction of the nodes  $C_i'$  and  $C_j'$  in  $T'$  is a MST for  $G_{\mathcal{C}^a}$ , where  $\mathcal{C}^a$  is the  $(k-1)$ -clustering obtained from  $\mathcal{C}'$  by merging  $C_i'$  and  $C_j'$ .
Proof. We show that  $T^a$  satisfies the conditions of Theorem A.3 when  $G = G_{\mathcal{C}^a}$ . For that, we will use the fact that  $T'$  satisfies the conditions of Theorem A.3 when  $G = G_{\mathcal{C}'}$ .
Let  $x$  and  $y$  be nodes of  $T^a$ . For the sake of contradiction, we assume that edge  $xy$  does not satisfy the conditions of Theorem A.3 when  $G = G_{\mathcal{C}^a}$  and  $T = T^a$ . Let  $w^*$  be the weight of edge  $xy$  and let  $e$  be an edge in the path that connects  $x$  to  $y$  in  $T^a$  such that  $w(e) > w^*$ . We have two cases:
Case 1)  $x \neq C_i' \cup C_j'$  and  $y \neq C_i' \cup C_j'$ . Then,  $e$  is also an edge in the path that connects  $x$  to  $y$  in  $T'$ . This implies that  $xy$  does not satisfy the required conditions when  $G = G_{\mathcal{C}'}$  and  $T = T'$ , which is a contradiction.
Case 2)  $x = C_i' \cup C_j'$  or  $y \neq C_i' \cup C_j'$ . Let us assume w.l.o.g. that  $x = C_i' \cup C_j'$ . Let  $w_i'$  and  $w_j'$  be, respectively, the weights of the edges  $(y, C_i')$  and  $(y, C_j')$  in  $G_{\mathcal{C}'}$ . We have that  $w^* = \min \{w_i', w_j'\}$ .
Let us assume w.l.o.g. that  $w^{*} = w_{i}^{\prime}$ . Then,  $e$  is also an edge in the path that connects  $y$  to  $C_i^\prime$  in  $T^{\prime}$ . Again, this implies that  $xy$  does not satisfy the required conditions when  $G = G_{\mathcal{C}'}$  and  $T = T'$  which is a contradiction.
It follows from the previous proposition that the tree  $T^{i-1}$  that is obtained by contracting the  $i-1$  cheapest edges of  $T$  is a MST for  $G_{\mathcal{C}^{i-1}}$ , where  $\mathcal{C}^{i-1}$  is a clustering for instance  $I$  that contains  $(k - (i-1))$  groups. The cheapest edge of  $T^{i-1}$  is exactly the  $i$ -th cheapest edge of  $T$ . Thus,  $\operatorname{Min-Sp}(\mathcal{C}^{i-1}) = w_i$ .
Similarly,  $w_{i}^{SL}$  is exactly the minimum spacing of a clustering, with  $(k - (i - 1))$  groups, that is obtained by single-linkage for instance  $I$ . Thus, it follows from Theorem 2.1 that  $w_{i}^{SL} \geq w_{i}$ .
C.1 Proof of Lemma 4.4
Proof. Let  $T^{*}$  be the MST for  $G_{\mathcal{C}^{*}}$ . If we remove the  $\ell - 1$  most expensive edges of  $T^{*}$  we obtain a forest  $F$  with  $\ell$  connected components. The clustering  $\mathcal{C}_{\ell}^{*}$  comprised by  $\ell$  groups in
which the  $i$ th group corresponds to the  $i$ th connected component of  $F$  is a  $(\ell, L)$ -clustering and  $\operatorname{Min-Sp}(\mathcal{C}_{\ell}^{*}) = w_{k - \ell + 1}^{*}$ .
Let  $OPT$  be the Min-Sp of the  $(\ell, L)$ -clustering with maximum Min-Sp. Thus, by Theorem 4.1  $\operatorname{Min-Sp}(\mathcal{A}_{\ell}^{\prime}) \geq OPT \geq \operatorname{Min-Sp}(\mathcal{C}_{\ell}^{*}) = w_{k - \ell + 1}^{*}$
Proof. We make a reduction from the  $(T/4, T/2)$ -restricted 3-PARTITION problem. Given a multiset  $S = \{s_1, \ldots, s_{3m}\}$  of positive integers that satisfies  $\sum_{i} s_i = mT$ , the 3-PARTITION problem consists of deciding whether or not there exists a partition of  $S$  into  $m$  triples such that the sum of the numbers in each one is equal to  $T$ . In the  $(T/4, T/2)$ -restricted 3-PARTITION problem, there is an additional requirement that each number of  $S$  should be in the interval  $(T/4, T/2)$ . This problem is strongly NP-COMPLETE Garey & Johnson (1979)
The instance  $I = (X, k, L, \mathrm{dist})$  for our clustering problem is built as follows: we set  $L = T$ ,  $k = m$ ; for  $i = 1, \ldots, 3m$  let  $\mathcal{X}_i$  be a set with  $s_i$  points so that the distance between points in the same group  $\mathcal{X}_i$  is 1 while the distance between points in different groups is  $\alpha + 1$ . We set  $\mathcal{X} = \mathcal{X}_1 \cup \ldots \cup \mathcal{X}_{3m}$ . Note that we are employing a pseudo-polynomial reduction but this is fine since the  $(T/4, T/2)$ -restricted 3-PARTITION problem is strongly NP-COMPLETE.
Let us assume that there is an  $(1,1 / \alpha)$ -approximation for our problem and let  $\mathcal{C}$  be the clustering returned by this algorithm for instance  $I$ . We argue that the answer to the 3-PARTITION problem is 'YES' if and only if  $\operatorname{Min-Sp}(\mathcal{C}) = \alpha + 1$ .
First, we show that if the answer is 'YES', there is a  $k$ -clustering  $\mathcal{C}^*$  for  $I$  with  $\operatorname{Min - Sp}(\mathcal{C}^*) = \alpha + 1$ . In fact, let  $S_1, \ldots, S_m$  be a solution of the 3-PARTITION problem and let  $\{s_{i_1}, s_{i_2}, s_{i_3}\}$  the numbers in  $S_i$ . Let  $\mathcal{C}^*$  be a  $k$ -clustering where the  $i$ th group is comprised by all points in  $\mathcal{X}_{i_1} \cup \mathcal{X}_{i_2} \cup \mathcal{X}_{i_3}$ . Clearly, each group has  $L$  points and the Min-Sp of this clustering is  $\alpha + 1$ . Since our algorithm is a  $(1, 1 / \alpha)$ -approximation it returns a clustering  $\mathcal{C}$  with  $\operatorname{Min - Sp}(\mathcal{C}) \geq (\alpha + 1) / \alpha$ . Since the Min-Sp of any clustering for instance  $I$  is either  $1$  or  $\alpha + 1$  we have that  $\operatorname{Min - Sp}(\mathcal{C}) = \alpha + 1$
On the other hand, if the clustering has Min-Sp  $\alpha + 1$  then all points in  $\mathcal{X}_i$ , for each  $i$ , must be in the same group. Moreover, due to the restriction that every  $|\mathcal{X}_i| = s_i \in (L/4, L/2)$ , we should have exactly 3  $\mathcal{X}_i$ 's in each of the  $m$  groups. Thus, the answer is 'YES'.
C.3 Proof of Theorem 4.7
The proof is very similar to that of Theorem 4.2.