Better Algorithms for Individually Fair $k$-Clustering

1 Introduction . As machine learning algorithms are widely used in practice for making high-stakes decisions affecting human lives , there has been a huge body of work on FAIR-ML trying to ensure ‘ fairness ’ in the solutions returned by these algorithms . There are two large intersecting bodies of work : one body ’ s main focus in to understand what ‘ fairness ’ means ( e.g . [ 26 , 39 , 38 , 36 , 14 , 25 ] ) in various different contexts , and the second body ’ s focus has been on addressing the algorithmic challenges brought forth by these considerations ( e.g. , [ 24 , 9 , 33 , 6 , 29 , 5 ] ) . This paper falls in the second class . In particular , we consider an individual fairness model proposed by Jung , Kannan , and Lutz [ 25 ] for a k-clustering problem . Given points ( clients ) X in a space with metric distance d , find k points ( facilities ) S ⊆ X , minimizing ( ∑ v∈X d ( v , S ) p ) 1/p where d ( v , S ) is v ’ s distance to the closest point in S. This includes k-CENTER , k-MEDIAN and the popular k-MEANS objective for p =∞ , p = 1 and p = 2 respectively , problems which have been extensively studied [ 12 , 23 , 27 , 4 , 2 ] in the algorithms literature . Jung et . al . [ 25 ] proposed that in this context a solution would be deemed individually fair , if for every client v ∈ X there is an open facility not too far from it . More precisely , if there is a facility within distance r ( v ) which is the smallest radius ∗Supported by NSF grant # 2041920 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . around v that contains n/k points . The rationale behind n/k is that if all the points had equal chance of getting picked as a facility , each client v would expect a facility to be open within distance r ( v ) . Furthermore , this formulation encourages a solution to open more facilities in dense areas of the metric which is motivated by the facility location application of clustering . Jung et al . [ 25 ] gave a solution where every client v was served within a radius of 2r ( v ) , which as a jargon is called 2-approximate fair solution . However , their solution did not explicitly consider the “ objective ” function ( k-MEANS/k-MEDIAN , for instance ) in the clustering problem , which is often used as a proxy to measure the quality of the clustering . This was addressed in a follow up paper by Mahabadi and Vakilian [ 29 ] who gave a ( 7 , O ( p ) ) -approximation with respect to the ℓp-norm objective . That is , they give a local-search based solution which is 7-approximately fair , but the objective is violated by some Cp-factor where the constant C is rather large ( for p = 1 , the factor is 84 ) . The theoretical running time of their algorithm is Õ ( pk5n4 ) . 1.1 Our Contributions . The main contribution of our paper is to give improved algorithms for this problem using linear programming rounding . Our study stems from two observations : one , that the problem at the core of Jung et al . [ 25 ] was in fact studied as “ weighted/priority k-CENTER problem ” by Plesník [ 32 ] , and that if all the r ( v ) ’ s were the same ( which may not at all be the case ) , then the clustering problem has also been studied under the guise of centridian/ordered median problem [ 3 , 8 , 10 ] . Combining ideas from these two bodies of work , we design an ( 8 , 8 ) algorithm for the FAIR-k-MEDIAN problem , which obtains an 8-approximation for both cost and fairness ( our cost guarantees improve as p grows ) . Result 1 . There is an ( 8 , 21+2/p ) -approximation algorithm for FAIR- ( p , k ) -CLUSTERING that runs in LP solving time plus Õ ( n2 ) , overall Õ ( kn4 ) . In particular , we have ( 8 , 8 ) -approximation and ( 8 , 4 ) -approximation algorithms for FAIR-k-MEDIAN and FAIR-k-MEANS , respectively . Remark . The approximation factor compares itself with the optimal solution which opens centers at the points in X . This is also the notion considered by [ 29 ] and [ 25 ] . If the optimal solution is allowed to open centers anywhere in the real space , then one has an additional factor 2 loss in the approximation [ 15 ] . Whether this extra hit of factor 2 is necessary , we leave as an interesting algorithmic question . Although solving an LP may seem prohibitive in practice , we can obtain a much faster running time by implementing a sparsification routine ( inspired by [ 32 , 21 ] ) with a marginal hit in the fairness and clustering cost ( see Lemma 5 for details ) . Empirically , this greatly decreases the running time , and is often faster than the [ 29 ] implementation . In our experiments , we also find that our theoretical bounds are too pessimistic . Indeed , we show that our algorithm ’ s cost is at most % 1 more than the optimal clustering cost ( which does not have any fairness violation ) , almost always , and never more than % 15 in the rest . Furthermore , our maximum fairness violation is at most a factor of 1.27 which is much better than our theoretical guarantee of 8 . We also do a more fine-grained analysis of the fairness violation : consider a vector where each coordinate stands for a clients “ unfairness ” indicating the ratio of its distance to r ( v ) . When we plot this as a histogram , we find that most of the mass is shifted to the “ left ” , that is , the percentage of clients who satisfy their fairness constraints is significantly larger than in the [ 29 ] solution . This seems to suggest the linear program , which is trying to minimize the cost , itself tries to increase the number of fairly treated clients . We leave a theoretical investigation of this phenomenon for future work . Our experiments also demonstrate the price of fairness . We find that our linear programs , which maintains absolute fairness , have objective value considerably larger than that of [ 29 ] . On the other hand , if we tune the “ fairness violation ” of the linear program to match that of [ 29 ] , then the objective value of our algorithm drops . We run experiments to further elaborate on the inherent cost of fairness in our datasets by demonstrating how the optimal cost changes with respect to varying degrees of fairness relaxation . It is worth noting that our algorithm works for arbitrary values of r ( v ) ≥ 0 for points v ( and this may be true for [ 29 ] as well ) , and we present our results thus . This setting might be of interest in applications where the “ fair radius ” may not be n/k but something more nuanced . 1.2 Other related work . k-CENTER has a 2-approximation due to Gonzales , and Hochbaum and Shmoys [ 17 , 21 ] and they prove it is NP-hard to get better approximations . k-MEDIAN , and k-MEANS are hard to approximate within factors better than 1.73 and 3.94 [ 19 ] respectively with current best approximations being 2.67 by [ 7 ] and 9 by [ 2 ] . Also recently , there has been an improvement on lower-bounds for approximating Continuous k-MEDIAN and k-MEANS where centers can be picked anywhere in the real space . By Cohen-Addad , Karthik , and Lee [ 15 ] , it is NP-hard to approximate Continuous k-MEDIAN and k-MEANS within factors 2− o ( 1 ) and 4− o ( 1 ) respectively . FAIR-k-CENTER is a special case of Priority k-CENTER where the radii r ( v ) in which a point v demands a center at that distance , are general values . [ 32 ] introduced this problem and gave a best possible 2-approximation . Gørtz and Wirth [ 18 ] study the problem for asymmetric metrics and prove that it is NP-hard to obtain any non-trivial approximation . [ 5 ] give a 9-approximation for the problem in presence of outliers and further generalize to constant approximations for general constraints on the solution centers . Another very closely related problem is Chance-k-Coverage introduced in [ 20 ] in which for any point v , in addition to r ( v ) , a probability p ( v ) is given and the goal is to find a distribution on possible solutions such that a solution drawn from this distribution covers v with probability at least p ( v ) . This also has a 9-approximation by [ 20 ] . A clustering problem related to FAIR-k-MEDIAN and FAIR-k-MEANS is the Simultaneous kClustering in which the goal is to find a solution with approximation guarantees with respect to any monotone , symmetric norm . This problem has an O ( 1 ) -approximation due to Chakrabarty and Swamy [ 11 ] with a line of previous work including [ 3 , 10 , 8 ] . Another similar notion of individual fairness is introduced by Chen et al . [ 13 ] in which a solution is fair if there is no group of size at least n/k for which there exists a facility that would reduce the connection cost of all members of the group if opened . [ 13 ] give a ( 1 + √ 2 ) -approximation for ℓ1 , ℓ2 , and ℓ∞ norm distances for the setting where facilities can be places anywhere in the real space . Micha and Shah [ 30 ] modified the approach to give close to 2-approximation for ℓ2 and proved the previous results for ℓ1 and ℓ∞ are indeed tight . Recently , Abbasi et al . [ 1 ] introduced a new notion of fairness centered around providing equitable group representations . In this problem , the points are partitioned into groups and the goal is to minimize the maximum clustering cost a group can have in the solution . The idea is that the cluster centers should represent each group fairly , and this is captured by ensuring that each group ’ s clustering cost is comparable with the others . This notion is orthogonal to the fairness criteria we study in this paper ; Although we could tune the maximum distance the points in a group can have to our solution , that does not give any guarantees with respect to the objective cost and vice versa . 2 Preliminaries . In this section , we formally define our problems , establish some notations , and describe a classic clustering routine due to Hochbaum and Shmoys [ 21 ] with modifications by Plesník [ 32 ] . Given a subset S ⊆ X , we use d ( v , S ) to denote v ’ s distance to the closest point in S. Definition 1 ( FAIR- ( p , k ) -CLUSTERING Problem ) . The input is a metric space ( X , d ) , radius function r : X → R+ , and integers p , k ≥ 1 . The goal is to find S ⊆ X of size at most k such that d ( v , S ) ≤ r ( v ) for all v ∈ X and the clustering cost ( ∑ v∈X d ( v , S ) p ) 1/p is minimized . Let opt be the clustering cost of an optimal solution . For α , β ≥ 1 , an ( α , β ) -approximate solution is S ⊆ X of size at most k with d ( v , S ) ≤ αr ( v ) for all v ∈ X and ( ∑ v∈X d ( v , S ) p ) 1/p≤ βopt . In plain English , the fairness approximation is α while the objective/cost approximation is β . Our main result is the following . Theorem 1 . There is an ( 8 , 21+2/p ) -approximation algorithm for FAIR- ( p , k ) -CLUSTERING that runs in LP solving time plus Õ ( n2 ) , overall Õ ( kn4 ) . The algorithm relies on rounding a solution to the following LP for FAIR- ( p , k ) -CLUSTERING2 where the optimal LP objective is at most optp . The variable yu denotes the amount by which u ∈ X 2The LP and its rounding is slightly different for p = ∞ ; we omit this from this version . is open as a center . xvu for v , u ∈ X is the amount by which v is assigned to u . The constraints respectively capture the conditions : every client must connect to someone , k centers are opened , no client can connect to an unopened center , and crucially that a client can not travel to a center further than r ( v ) . optp ≥ min ∑ v , u∈X d ( v , u ) pxvu ( LP ) ∑ u∈X xvu = 1 , ∀v ∈ X ( LP1 ) ∑ u∈X yu = k ( LP2 ) xvu ≤ yu , ∀v , u ∈ X ( LP3 ) xvu = 0 , ∀v , u : d ( v , u ) > r ( v ) ( LP4 ) 0 ≤ xvu , yu ≤ 1 , ∀v , u ∈ X . From here on , we use the notation B ( u , r ) for u ∈ X and r ∈ R+ to denote the points in a ball of radius r around u . That is B ( u , r ) : = { v ∈ X : d ( u , v ) ≤ r } . Also , for any set of points U ⊆ X , let y ( U ) : = ∑ u∈U yu . Considering ( LP4 ) and ( LP1 ) the following holds . Fact 1 . If y is from a feasible LP solution , then y ( B ( v , r ( v ) ) ) ≥ 1 for all v ∈ X . We now describe a routine Filter due to [ 32 , 21 ] which is used as a subroutine in our main algorithm . Assume all the points are initially “ uncovered ” . The routine takes a function R : X → R+ , sorts the points in order of increasing R ( v ) ’ s . Then it considers the first point in this order , calls it a “ representative ” and “ covers ” all the points v at distance at most 2r ( v ) from it . Call these points D ( v ) . Repeating this procedure until all the points are covered , forms a partition on X such that the representatives are “ far apart ” , while each non-representative is assigned to a “ nearby ” representative , among other useful properties listed in Fact 2 and Fact 3 . Algorithm 1 Filter Input : Metric ( X , d ) , radius function R : X → R+ 1 : U ← X ▷ The set of “ uncovered points ” 2 : S ← ∅ ▷ The set of “ representatives ” 3 : while U ̸= ∅ do 4 : u← argminv∈U R ( v ) ▷ The first point in U in non-decreasing R order 5 : S ← S ∪ u 6 : D ( u ) ← { v ∈ U : d ( u , v ) ≤ 2R ( v ) } ▷ D ( u ) = B ( u , 2R ( v ) ) ∩ U and includes u itself 7 : U ← U\D ( u ) 8 : end while Output : S , { D ( u ) : u ∈ S } Fact 2 . [ 32 , 21 ] The following is true for the output of Filter : ( a ) ∀u , v ∈ S , d ( u , v ) > 2max { R ( u ) , R ( v ) } , ( b ) The balls { B ( u , R ( u ) ) : u ∈ S } are mutually disjoint , ( c ) The set { D ( u ) : u ∈ S } partitions X , ( d ) ∀u ∈ S , ∀v ∈ D ( u ) , R ( u ) ≤ R ( v ) , and ( e ) ∀u ∈ S , ∀v ∈ D ( u ) , d ( u , v ) ≤ 2R ( v ) . Fact 3 . For any u ∈ S and w ∈ B ( u , R ( u ) ) , the unique closest point in S to w is u . Proof . Suppose otherwise . That is , there exists v ∈ S not equal to u s.t . d ( w , v ) ≤ d ( w , u ) . Then d ( u , v ) ≤ d ( u , w ) + d ( w , v ) ≤ 2d ( w , u ) ≤ 2R ( u ) which contradicts Fact 2 as we must have d ( u , v ) > 2max { R ( u ) , R ( v ) } . To elaborate on the importance of Algorithm 1 and build some intuition , we point out the following theorem of [ 32 , 25 ] . Theorem 2 . Take S the output of Algorithm 1 on a FAIR- ( p , k ) -CLUSTERING instance with R : = r. Then if the instance is feasible , |S| ≤ k and d ( v , S ) ≤ 2r ( v ) for all v ∈ X . Proof . By Fact 2 we know that { D ( u ) : u ∈ S } partitions X so for any v ∈ X , there exists a u ∈ S for which v ∈ D ( u ) . Plus , d ( v , u ) ≤ 2R ( v ) = 2r ( v ) . Now it only remains to prove |S| ≤ k. To see this , let S∗ be some feasible solution and observe that two different u , w ∈ S , can not be covered by the same center in S∗ . Since otherwise , if there exists f ∈ S∗ for which d ( u , f ) ≤ r ( u ) and d ( w , f ) ≤ r ( w ) by triangle inequality d ( u , v ) ≤ d ( u , f ) + d ( f , w ) ≤ r ( u ) + r ( v ) and this contradicts d ( u , w ) > 2max { r ( u ) , r ( v ) } from Fact 2 . But of course , the above theorem does not give any guarantees for the clustering cost ( unless p =∞ ) . It might be the case that many points are paying close to 0 towards the clustering cost in the optimal solution , but are made to connect to a point much farther in the above procedure . 3 Algorithm for FAIR- ( p , k ) -CLUSTERING problem Now we are ready to describe our algorithm Fair-Round which establishes Theorem 1 . At a highlevel , we run our Filter routine by defining the input function R in a manner that is conscious of the FAIR- ( p , k ) -CLUSTERING cost : given ( x∗ , y∗ ) which is an optimal solution to ( LP ) , for any v ∈ X let Cv be v ’ s contribution to the LP cost i.e . Cv : = ∑ u∈X d ( v , u ) px∗vu and define R ( v ) : = min { r ( v ) , ( 2Cv ) 1/p } . Let us ponder for a bit to see what changes from Theorem 2 . For the output S , we still have the fairness guarantee d ( v , S ) ≤ 2r ( v ) for all v but since ( 2Cv ) 1/p might be less than r ( v ) for any v , we can not guarantee that |S| ≤ k. Thankfully , in this case , we can still prove |S| ≤ 2k ( Corollary 1 ) . The rest of the algorithm is deciding on a subset of at most k points out of this S to return as the final solution , while ensuring the fairness and cost guarantees are still within constant factor of the optimal . This idea is very similar to existing ideas in [ 3 , 12 ] which look at the problem without fairness considerations . Recall that { D ( u ) : u ∈ S } partitions X and each u ∈ S is responsible for covering all the points in D ( u ) . Here , we could simply move each point in D ( u ) to u and divert the y value of each point to its closest point in S. Note that , y ( S ) = k ( Fact 4 ) and similar to the proof of Theorem 2 we could show yu ≥ 1/2 ( Lemma 1 ) for all u ∈ S hence |S| ≤ 2k ( Corollary 1 ) . If this leads to some y-value reaching 1 , we open those centers . For u ∈ S with yu < 1 , if we do not decide to include it in the final solution , we promise to open Su , its closest point in S other than itself . In this case , all the |D ( u ) | points on u are delegated to Su . Using the fact that yu < 1 , we can prove that fairness guarantee ( Lemma 3 ) approximately holds for the points in D ( u ) even after this delegation . To get the clustering cost guarantee , we need to do more work . Observe that , currently , u is already fractionally assigned to v by 1− yu . So if instead of yu ≥ 1/2 we had yu = 1/2 we could ensure that already u is assigned to v by 1/2 thus integrally assigning u to v only doubles the cost . This is why we need to do more work to get yu ∈ { 1/2 , 1 } for u ∈ S ( see Lemma 2 ) and then bound the clustering cost in Lemma 4 . Fact 4. y ( S ) = k and remains so after Line 11 . Proof . We initialize y to y∗ for which y∗ ( X ) = k by LP2 . The algorithm then transfers y ( X ) to S by Line 11 without changing the total . Afterwards , the modifications in Lines 13 and 19 move the y mass around in S but the total value y ( S ) remains unchanged . Lemma 1 . After Line 14 of Algorithm 2 , 1/2 ≤ yu ≤ 1 for all u ∈ S. Proof . First we argue that yu ≥ 1/2 for all u ∈ S by the end of Line 11 . Fix u ∈ S. Per Fact 3 y ( B ( u , R ( u ) ) ) is entirely moved to yu . By definition of R ( u ) there are two cases : Case I , R ( u ) = r ( u ) thus yu ≥ 1 as y∗ ( B ( v , r ( v ) ) ) ≥ 1 for all v ∈ X according to Fact 1 . Case II , R ( u ) = ( 2Cu ) 1/p then by Markov ’ s inequality y∗ ( B ( u , R ( u ) ) ) ≥ 1/2 and after this point , yu is never decreased to below 1/2 . More precisely , Cu = ∑ v∈X d ( u , v ) px∗uv ≥ ∑ v∈X : d ( u , v ) > R ( u ) d ( u , v ) px∗uv ≥ 2Cu ∑ v∈X : d ( u , v ) > R ( u ) x∗uv . Considering ∑ v∈X x ∗ uv = 1 by LP1 , this implies ∑ v∈X : d ( u , v ) ≤R ( u ) x∗uv ≥ 1/2 thus y∗ ( B ( u , R ( u ) ) ) ≥ 1/2 . As for proving yu ≤ 1 , it might indeed be the case that yu > 1 by the end of Line 11 but the loop ending at Line 14 can guarantee yu ≤ 1 for all u ∈ S. This is because y ( S ) = k ( Fact 4 ) and we already checked in the beginning of Algorithm 2 that |S| > k. Corollary 1 . S in Algorithm 2 has size at most 2k . Algorithm 2 Fair-Round : FAIR- ( p , k ) -CLUSTERING bi-criteria approximation Input : Metric ( X , d ) , radius function r : X → R+ , and ( x∗ , y∗ ) an optimal solution of LP 1 : Cv ← ∑ u∈X d ( v , u ) px∗vu ∀v ∈ X ▷ v ’ s cost share in the LP objective 2 : R ( v ) ← min { r ( v ) , ( 2Cv ) 1/p } ∀v ∈ X 3 : S , { D ( u ) : u ∈ S } ← Filter ( ( X , d ) , R ) 4 : if |S| ≤ k then 5 : return S 6 : end if 7 : ( x , y ) ← ( x∗ , y∗ ) 8 : for all v ∈ X\S do ▷ Direct y mass from outside of S to the closest point in S 9 : u← closest point in S to v 10 : yu ← yu + yv , yv ← 0 ▷ Note : May cause yu to increase above 1 11 : end for 12 : while There are u , v ∈ S with yu > 1 and yv < 1 do ▷ Ensure yu ≤ 1 for all u ∈ S 13 : yu ← yu − δ , yv ← yv + δ , where δ : = min { 1− yv , yu − 1 } 14 : end while ▷ Remark : By now , 1/2 ≤ yu ≤ 1 for all u ∈ S ( see Lemma 1 ) . 15 : Su ← closest point in S\u to u ∀u ∈ S 16 : while There are u , v ∈ S with 17 : 1/2 < yu < 1 , and yv < 1 , 18 : d ( u , Su ) p|D ( u ) | > d ( v , Sv ) p|D ( v ) | do ▷ Move y mass from u to v if u is costlier 19 : yu ← yu − δ , yv ← yv + δ , where δ : = min { 1− yv , yu − 1/2 } 20 : end while ▷ Remark : At this point , yu ∈ { 1/2 , 1 } for all u ∈ S ( see Lemma 2 ) . 21 : T ← u ∈ S with yu = 1 22 : Consider the forest of arbitrary rooted trees on vertices u ∈ S with edges ( u , Su ) . Let O be odd-level vertices in S\T , and E be even-level vertices in S\T . 23 : if |E| ≤ |O| then 24 : T ← T ∪ E 25 : else 26 : T ← T ∪O 27 : end if Output : T Lemma 2 . After Line 20 of Algorithm 2 , yu ∈ { 1/2 , 1 } for all u ∈ S. Proof . Suppose not . Since y ( S ) = k ( Fact 4 ) and k is an integer , by Lemma 1 , there has to be at least two u , v ∈ S with yu , yv ∈ ( 1/2 , 1 ) which is a contradiction , since either one of them has to be changed to 1/2 or 1 in the while loop ending at Line 20 . Lemma 3 . For all v ∈ X , d ( v , T ) ≤ 8r ( v ) . Proof . Fix v ∈ X . Since { D ( u ) : u ∈ S } partitions X there exists u ∈ S such that v ∈ D ( u ) . According to Fact 2 d ( v , u ) ≤ 2R ( v ) ≤ 2r ( v ) by definition of R. If u ends up in T we are done . Else , it has to be that yu < 1 and Su ∈ T . In what follows , we prove that if yu < 1 at Line 20 then d ( u , Su ) ≤ 6r ( v ) . This implies d ( v , Su ) ≤ 8r ( v ) hence the lemma . We know that initially y∗ ( B ( v , r ( v ) ) ) ≥ 1 per Fact 1 but since v /∈ S , the y mass on B ( v , r ( v ) ) has been moved to S by Line 11 . If for all w ∈ B ( v , r ( v ) ) their closest point in S was u , all of y ( B ( v , r ( v ) ) ) would be moved to u then yu = 1 by the end of Line 20 . So there must exist w ∈ B ( v , r ( v ) ) with x∗vw > 0 along with u′ ∈ S , u′ ̸= u , such that d ( w , u′ ) ≤ d ( w , u ) which made yw to be moved to u′ . By definition of Su , d ( u , Su ) ≤ d ( u , u′ ) . Applying the triangle inequality twice gives : d ( u , Su ) ≤ d ( u , u′ ) ≤ d ( u , w ) + d ( w , u′ ) ≤ 2d ( u , w ) ≤ 2 ( d ( u , v ) + d ( v , w ) ) ≤ 2 ( 2r ( v ) + r ( v ) ) = 6r ( v ) where the last inequality comes from Fact 2 stating d ( u , v ) ≤ 2r ( v ) , and from the fact that x∗vw > 0 implies d ( v , w ) ≤ r ( v ) by LP4 . Lemma 4 . ( ∑ v∈X d ( v , T ) p ) 1/p ≤ ( ( 2p+2 ) ∑v , u∈X Cv ) 1/p ≤ 21+2/popt . Proof . For the proof , we compare ∑ v∈X d ( v , T ) p with the optimal LP cost ∑ v∈X Cv which is at most optp . Fix v ∈ X and u ∈ S for which v ∈ D ( u ) . By Fact 2 d ( v , u ) ≤ 2R ( v ) ≤ 2 ( 2Cv ) 1/p per definition of R. So moving D ( u ) to u for all u ∈ S has an additive cost of∑ u∈S ∑ v∈D ( u ) d ( v , u ) p ≤ 2p+1 ∑ v∈X Cv . From now on , assume there are D ( u ) collocated points on a u ∈ S. Moving around the y mass up to Line 11 adds a multiplicative factor of 2p loss in the approximation ratio . The logic is : if u ∈ S was relying on w ∈ X in the LP solution , meaning , x∗uw > 0 and yw was moved to a u′ ∈ S ( due to d ( w , u′ ) ≤ d ( w , u ) ) then the cost u has to pay to connect to u′ is d ( u , u′ ) p ≤ ( d ( u , w ) + d ( w , u′ ) ) p ≤ 2pd ( u , w ) p which is a 2p factor worse than the LP cost d ( u , w ) p it was paying to connect to w earlier . At this point ( Line 14 ) , note that the cost incurred by u is |D ( u ) |d ( u , Su ) p ( 1 − yu ) . To elaborate on this , corresponding to y , we define an x such that ( x , y ) is feasible for LP . Let xuu = yu and xuSu = 1− yu so that LP4 is satisfied for u . Then the cost incurred by u is |D ( u ) | ( d ( u , u ) pxuu + d ( u , Su ) pxuSu ) = |D ( u ) |d ( u , Su ) p ( 1 − yu ) . The while loop ending at Line 20 does not increase the value of the objective function : As we decrease yu and increase yv , only if the cost incurred by u is bigger than that of v. The last multiplicative factor 2 loss comes from when u ∈ S\T . In which case , yu = 1/2 and Su ∈ T . So by assigning u to Su we pay |D ( u ) |d ( u , Su ) p which is twice more than before ( as 1 − yu = 1/2 ) . Observe that this last step is why we needed to do all the work to get yu ∈ { 1/2 , 1 } in Lemma 2 . Putting the three steps together , the overall cost is at most 2× 2p + 2p+1 = 2p+2 times the LP cost or at most 21+2/popt . Proof of Theorem 1 . Using Corollary 1 and the fact that |T | ≤ |S|/2 by construction , we have |T | ≤ k. Lemmas 3 and 4 give the fairness and approximation guarantees . As for runtime , notice that Algorithm 2 runs in time Õ ( n2 ) . But the runtime is dominated by the LP solving time . According to [ 37 ] , finding a ( 1 + ε ) -approximation to LP takes time O ( kn2/ϵ2 ) . Setting ε = 1/n gives the O ( kn4 ) runtime . As evident , the runtime is dominated by the LP solving time . We end this section by descibing the sparsification pre-processing that when applied to the original instance , can tremendously decrease the LP solving time in practice while incurring only a small loss in fairness and clustering cost . One reason why the LP takes a lot of time is because there are many variables ; xvu for every v ∈ X and every u within distance r ( v ) of v. To fix this , we first run the Filter algorithm on the data set with R ( v ) = δr ( v ) , where δ is a tune-able parameter . It is not too hard to quantify the loss in fairness and cost as a function of δ , and we do so in the lemma below . More importantly , note that when δ = 1 , then the number of variables goes down from n to O ( k ) ; this is because of the definition of r ( v ) which guarantees ≈ n/k points in the radius r ( v ) around v. Therefore , running the pre-processing step would make the number of remaining points ≈ k. In our experiments we set δ to be much smaller , and yet observe a great drop in our running times . Algorithm 3 Sparsification + Fair-Round Input : Metric ( X , d ) , radius function r : X → R+ , parameter δ > 0 1 : S , { D ( u ) : u ∈ S } ← Filter ( ( X , d ) , δr ) 2 : ( x′ , y′ ) ← solve LP only on points S with objective function ∑ v , u∈S d ( v , u ) pxvu|D ( v ) | 3 : x∗vw ← x′uw∀v , w ∈ X , u ∈ S : v ∈ D ( u ) ▷ v ’ s assignment is identical to its representative u 4 : y∗u ← y′u if u ∈ S and 0 otherwise Output : Fair-Round ( ( X , d ) , ( 1 + δ ) r , ( x∗ , y∗ ) ) Lemma 5 . Algorithm 3 outputs an ( 8 ( 1 + δ ) , 21+2/p ( 1 + ( δϕ ) p ) 1/p ) -approximation where ϕ = ( ∑ v∈X r ( v ) p ) 1/p/opt . Proof . Observe that ( x∗ , y∗ ) is not a feasible LP solution anymore . Nevertheless , it will be feasible for when r is dilated by a factor of ( 1 + δ ) in the LP4 . Here is how we argue LP4 holds in this case : For any v , w ∈ X for which x∗vw > 0 , recall u ∈ S is chosen so v ∈ D ( u ) and x∗vw : = x′uw > 0 meaning d ( u , w ) ≤ r ( u ) . We have d ( v , w ) ≤ d ( v , u ) +d ( u , w ) ≤ δr ( v ) + r ( u ) ≤ ( 1+ δ ) r ( v ) . The last two inequalities are by Fact 2 as d ( v , u ) ≤ δr ( v ) and r ( u ) ≤ r ( v ) . As for the LP cost of ( x∗ , y∗ ) , it is not longer upper-bounded by optp but rather by optp + ∑ u∈S ∑ v∈D ( u ) d ( v , u ) p. This additive term is at most ∑ v∈X ( δr ( v ) ) p ≤ ( δϕopt ) p. Plugging this into Lemma 4 finishes the proof .

Summary: The paper studies the problem of \ell_p norm and center-based clustering with a fairness consistent. The fairness constraint roughly says that no point should be further from its assigned center than it is from its (n/k)-th furthest neighbor. The authors consider a Linear Programming based (approximation) algorithm for this problem.