Planar p-center problems are solvable in polynomial time when clustering a Pareto Front
aa r X i v : . [ c s . C G ] A ug Planar p-center problems are solvable inpolynomial time when clustering a Pareto Front
Nicolas Dupin, Frank Nielsen, El-Ghazali TalbiAugust 27, 2019
Abstract
This paper is motivated by real-life applications of bi-objective opti-mization. Having many non dominated solutions, one wishes to clusterthe Pareto front using Euclidian distances. The p-center problems, bothin the discrete and continuous versions, are proven solvable in polynomialtime with a common dynamic programming algorithm. Having N pointsto partition in K > O ( KN log N )(resp O ( KN log N )) time and O ( KN ) memory space for the continu-ous (resp discrete) K -center problem. 2-center problems have complexi-ties in O ( N log N ). To speed-up the algorithm, parallelization issues arediscussed. A posteriori, these results allow an application inside multi-objective heuristics to archive partial Pareto Fronts. Keywords : Optimization ; Operational Research ; Computational Geometry; Dynamic programming ; Clustering algorithms ; k-center problems ; p-centerproblems ; complexity ; bi-objective optimization ; Pareto front
This paper is motivated by real-life applications of multi-objective optimization(MOO). Some optimization problems can be driven by more than one objectivefunction, with some conflicts among objectives. For example, one may minimizefinancial costs, while maximizing the robustness to uncertainties [13, 36]. In suchcases, higher levels of robustness are likely to induce financial over-costs. Paretodominance, preferring a solution from another if it is better for all the objectives,is a weak dominance rule. With conflicting objectives, several non-dominatedsolutions, denoted efficient solutions , can be generated. A
Pareto front (PF) isthe projection in the objective space of these efficient solutions [17].For a presentation to decision makers, concise information to summary theshape of solutions are required. Firstly, one can present a view of a PF inclusters and the density of points in the cluster. Secondly, representative pointsalong the PF can be presented to support decision making. Both problems canbe seen as applications of clustering a PF, representative points being computedas the most central points of clusters. We consider the case of two-dimensional12d) PF using the Euclidian distance. Having a 2d PF of size N , the problemis to define K ≪ N clusters, minimizing the dissimilarity between the pointsin the clusters. The K -center problems, both in the discrete and continuousversions, define in this paper the cluster costs, covering the 2d PF with K identical balls while minimizing the radius of the balls to use. The k-centerproblems are NP-complete in the general case [27] but also for the specificcase in R using the Euclidian distance [34]. This paper proves that p-centerclustering in a 2d PF is solvable in polynomial time, using a common DynamicProgramming (DP) algorithm, and discuss properties of the algorithm for anefficient implementation.This paper is organized as following. In Section 2, we describe the consideredproblems with unified notation. In Section 3, we discuss related state-of-the-artelements to appreciate our contributions. In Section 4, intermediate results arepresented. In Section 5, a common DP algorithm is presented with polynomialcomplexity thanks to the results of section 4. In Section 6, the implications andapplications of the results of section 5 are discussed. In Section 7, our contribu-tions are summarized, discussing also future directions of research. To ease thereadability of the paper, some elementary proofs are gathered in Appendix A. We suppose in this paper having a set E = { x , . . . , x N } of N elements of R , such that for all i = j , x i I x j defining the binary relations I , ≺ for all y = ( y , y ) , z = ( z , z ) ∈ R with: y ≺ z ⇐⇒ y < z and y > z (1) y z ⇐⇒ y ≺ z or y = z (2) y I z ⇐⇒ y ≺ z or z ≺ y (3)We note that the definition of E considers minimization of two objective inthe sense of inequalities defining I , ≺ , as illustrated in Figure 1. This is nota loss of generality, one can transform the objectives to maximize f into − f allows to consider the minimization two objectives. xyO = ( x O , y O ) • B = ( x B , y B ) with x O < x B and y O < y B O dominates BC = ( x C , y C ) with x O > x C and y O > y C C dominates O D = ( x D , y D ) with x O < x D and y D < y O O and D not comparable, O I D and O ≺ DA = ( x A , y A ) with x O > x A and y O < y A A and O not comparable, A I O and A ≺ O Figure 1: Pareto dominance and incomparability cadrans minimizing two ob-jectives: zones of A and D are incomparability zones related to O .2e define Π K ( E ), as the set of all the partitions of E in K ∈ N ∗ subsets:Π K ( E ) = n P ⊂ P ( E ) (cid:12)(cid:12)(cid:12) ∀ p, p ′ ∈ P, p ∩ p ′ = ∅ , [ p ∈ P p = E and card( P ) = K o (4)K-center problems are combinatorial optimization problems indexed by Π K ( E ):min π ∈ Π K ( E ) max P ∈ π f ( P ) (5)The cost function f applied for each subset of E measures a dissimilarity ofthe points in the subset. In this paper, we will consider the Euclidian norm todefine distances, defining for all y = ( y , y ) , z = ( z , z ) ∈ R : d ( y, z ) = || y − z || = q ( y − z ) + ( y − z ) (6)The K -center problems covering the PF with K identical balls while minimizingthe radius of the balls to use. The center of the balls for the discrete versionare necessarily points of the 2d PF, so that the cost function for the discrete K -center problem is: ∀ P ⊂ E, f D ctr ( P ) = min y ∈ P max x ∈ P || x − y || (7)The continuous K -center problem is a geometric problem, minimizing the radiusof covering balls without any localization constraint for the center of the coveringballs, the centers can be any subset of points in the plane: ∀ P ⊂ E, f C ctr ( P ) = min y ∈ R max x ∈ P || x − y || (8)For the sake of having unified notations for common results and proofs, we define γ ∈ { , } to indicate which version of the p-center problem is considered. γ = 0(resp 1) indicates that the continuous (resp discrete) p-center problem is used, f γ indicates the cost measure between f C ctr , f D ctr . The continuous and discrete K -center problems in the 2d PF are denoted K - γ -CP2dPF. This section describes related works to appreciate our contributions, regardingthe state of the art of the p-center problems and clustering problems in a PF.For more detailed surveys on the results for the p-center problems, we refer to[7].
We note that some hypotheses may be more general than the ones in Section2. The p-center problem consists in locating p facilities among a set of possiblelocations and assigning N clients, called c , c , . . . , c N , to the facilities in order3o minimize the maximum distance between a client and the facility to whichit is allocated. The continuous p-center problem assumes that any place oflocation can be chosen, whereas the discrete p-center problem considers a subsetof M potential sites denoted f , f , . . . , f M , and distances d i,j for all i ∈ [[1 , n ]]and j ∈ [[1 , M ]]. Discrete p-center problems can be formulated with bipartitegraphs or infinite distances, modeling that some assignments between clientsand facilities are unfeasible. In the discrete p-center problem defined in Section2, as in many applications using k-center for clustering, the points f , f , . . . , f M are exactly c , c , . . . , c N , and the distances are defined using a Euclidian norm.Discrete and continuous p-center problems are NP-hard [27, 33]. Further-more, for all α <
2, any α -approximation for the discrete p-center problemwith triangle inequality is NP-hard [25]. [26, 22] provided 2-approximations forthe discrete p-center problem running respectively in O ( N K log N ) time and in O ( N K ) time. The discrete p-center problem in R with a Euclidian distance isalso NP-hard [33]. Defining binary variables x i,j ∈ { , } and y j ∈ { , } with x i,j = 1 if and only if the customer i is assigned to the depot j and y j = 1if and only if the point f j is chosen as a depot, the following Integer LinearProgramming (ILP) formulation models the discrete p-center problem ([10]):min x,y,z z (9 . s.t : P nj =1 d i,j x i,j z ∀ i ∈ [[1 , N ]] (9 . P Mj =1 y j = p (9 . P Mj =1 x i,j = 1 ∀ i ∈ [[1 , N ]] (9 . x i,j y j ∀ ( i, j ) ∈ [[1 , N ]] × [[1 , M ]] , (9 . x i,j , y j ∈ { , } ∀ i, j ∈ [[1 , N ]] × [[1 , M ]] , (9 . p , constraints (9.4) as-sign each client to exactly one facility, constraints (9.5) enforces facility j tobe opened to use variables x i,j . The objective function is obtained with thelinearization in (9.2). To solve to optimality discrete p-center problems, tighterILP formulations, with efficient exact algorithms, rely on the ILP models [8, 18].Exponential exact algorithms were also provided for the continuous p-centerproblem [9, 11]. An N O ( √ p ) -time algorithm was provided for the continuousEuclidean p-center problem in the plane [28]. An N O ( p − /d ) -time algorithm forsolving the continuous p-center problem in R d under Euclidian and L -metricwas provided in [1]. Meta-heuristics are also efficient for p-center problemswithout furnishing optimality guarantees or dual bounds [21, 19, 35].Some specific cases of p-center problems are also solvable in polynomialtime. The continuous 1-center problem is exactly the minimum covering ballproblem which has a linear complexity in R . Indeed, a ”prune and search”algorithm from [32] finds the optimum bounding sphere and runs in linear timeif the dimension is fixed as a constant. In dimension d , its complexity is in O (( d + 1)( d + 1)! n ) time, which is impractical for high-dimensional applications.The discrete 1-center problem, seeking the smallest disk centered at a point ofP and containing P, can be solved solve in time O ( N log N ) , using the furthest-4eighbor Voronoi diagram of P [4]. The continuous and planar 2-center problemcan be solved in randomized expected O ( N log N ) time, with algorithms pro-vided and improved by [42, 20]. The discrete and planar 2-center problem issolvable in time O ( N / log N ) [2]. In the case of graded distances, the dis-crete p-center problem is polynomial time solvable when the distance matrixis graded up the rows or graded down the rows [24]. The continuous p-centerproblem is solvable in O ( N log N ) time in a tree structure [34]. The continuousand planar k-centers on a line, finding k disks with centers on a given line l , issolvable in polynomial time, in O ( N log N ) time in the first algorithm by [4]and in O ( N K log N ) time and O ( N ) space in the improved version provided by[29]. Selection or clustering points in PF have been studied with applications toMOO algorithms. Firstly, a motivation is to store representative elements ofa large PF (exponential sizes of PF are possible [17]) for exact methods orpopulation meta-heuristics. Maximizing the quality of discrete representationsof Pareto sets was studied with the hypervolume measure in the HypervolumeSubset Selection (HSS) problem [3, 41]. Secondly, a crucial issue in the design ofpopulation meta-heuristics for MOO problems is to select relevant solutions foroperators like cross-over or mutation phases in evolutionary algorithms [44, 46].Selecting knee-points is another known approach for such goals [38].The HSS problem, maximizing the representativity of K solutions among aPF of size N , is known to be NP-hard in dimension 3 (and greater) since [5].An exact algorithm in n O ( √ K ) and a polynomial-time approximation scheme forany constant dimension d are also provided in [5]. The 2d case is solvable inpolynomial time thanks to a DP algorithm with a complexity in O ( KN ) timeand O ( KN ) space provided in [3]. The time complexity of the DP algorithmwas improved in O ( KN + N log N ) by [6] and in O ( K. ( N − K ) + N log N ) by[30].Some similar results were also proven for clustering problems. K-medianand K-medoid problems are known to be NP hard in dimension 2 since [33],the specific case of 2d PF were proven to be solvable in O ( N ) time with DPalgorithms [16, 15]. K-means, one of the most famous unsupervised learningproblem, is also NP-hard for 2d cases [31]. The restriction to 2d PF would bealso solvable in O ( N ) time with a DP algorithm if a conjecture is proven [14].We note that an affine 2d PF is a line in R , clustering is equivalent to 1 di-mensional cases. 1-dimension K-means was proven to be solvable in polynomialtime with a DP algorithm in O ( KN ) time and O ( KN ) space. This complexitywas improved for a DP algorithm in O ( KN ) time and O ( N ) space in [23]. Thisis thus the complexity of K-means in an affine 2d PF. The specific case, alreadymentioned in the previous section, of the continuous p-center problem with cen-ters on a straight line is more general that the case of an affine 2d PF, witha complexity proven in O ( N K log N ) time and O ( N ) space by [29]. 2d casesof clustering problems can also be seen as specific cases of three-dimensional53d) PF, affine 3d PF. Having NP-hard complexities proven for planar cases ofclustering, which is the case for k-means, p-median, k-medoids, p-center prob-lems since [31, 34], it implies that the considered clustering problems are alsoNP-hard for 3d PF. This section gathers the intermediate result necessary for the following devel-opments. Some elementary proofs are gathered in Appendix A.
In this section, we analyze some properties of the relations ≺ and , beforedefining an order relation and a new indexation among the points of E . Thefollowing lemma is extends trivially the properties of and < in R : Lemma 1. is an order relation, and ≺ is a transitive relation: ∀ x, y, z ∈ R , x ≺ y and y ≺ z = ⇒ x ≺ z (10)The following proposition implies an order among the points of E , for areindexation in O ( N log N ) time: Obj Obj x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • Figure 2: Illustration of a 2d PF with 15 points and the indexation implied byProposition 1
Proposition 1 (Total order) . Points ( x i ) can be indexed such that: ∀ ( i , i ) ∈ [[1; N ]] , i < i = ⇒ x i ≺ x i (11) ∀ ( i , i ) ∈ [[1; N ]] , i i = ⇒ x i x i (12) This property is stronger than the property that induces a total order in E .Furthermore, the complexity of the sorting re-indexation is in O ( N log N )6 roof : We index E such that the first coordinate is increasing: ∀ ( i , i ) ∈ [[1; N ]] , i < i = ⇒ x i < x i This sorting procedure has a complexity in O ( N. log N ). Let ( i , i ) ∈ [[1; N ]] ,with i < i . We have thus x i < x i . Having x i I x i implies x i > x i . x i < x i and x i > x i is by definition x i ≺ x i . (cid:3) The re-indexation of Proposition 1 implies also a monotonic structure forthe distances among points of the 2d PF:
Lemma 2.
We suppose that points ( x i ) are sorted following Proposition 1. ∀ ( i , i , i ) ∈ [[1; N ]] , i i < i = ⇒ d ( x i , x i ) < d ( x i , x i ) (13) ∀ ( i , i , i ) ∈ [[1; N ]] , i < i i = ⇒ d ( x i , x i ) < d ( x i , x i ) (14) E in one subset is E . To solve 1-center problems, it is required to compute thecost of the trivial partition, i.e. to compute the radius of minimum enclosingdisk covering all the points of E (and centered in one point of E for the discreteversion). This section provides the complexity result related to these problems. Lemma 3.
Let P ⊂ E such that card ( P ) > . Let i (resp i ′ ) the minimal (respmaximal) index of points of P . f C ctr ( P ) = 12 || x i − x i ′ || (15) Lemma 4.
Let P ⊂ E such that card ( P ) > . Let i (resp i ′ ) the minimal (respmaximal) index of points of P . f D ctr ( P ) = min j ∈ [[ i,i ′ ]] ,x j ∈ P max ( || x j − x i || , || x j − x i ′ || ) (16) Proposition 2.
Let γ ∈ { , } , let E = { x , . . . , x N } a subset of N points of R , such that for all i = j , x i I x j . - γ -CP2dPF has a complexity in O ( N ) time. Proof : Using equations (15) or (16), computations of f γ is in O ( N ) once havingcomputed the extreme elements following the order ≺ . Computing the extremepoints is also in O ( N ), with one traversal of the elements of E . Finally, thecomplexity of 1-center problems are in linear time. (cid:3) In this section, a common characterization of optimal solutions of problems (5)for the p-center problems is given in Proposition 3. The proof is common forthe discrete and continuous cases, using Lemma 5.7 emma 5.
Let γ ∈ { , } . Let P ⊂ P ′ ⊂ E . We have f γ ( P ) f γ ( P ′ ) . Proposition 3.
Let γ ∈ { , } , let K ∈ N , let E = ( x i ) a 2d PF, indexedfollowing Proposition 1. There exists optimal solutions of K - γ -CP2dPF usingonly clusters C i,i ′ = { x j } j ∈ [[ i,i ′ ]] = { x ∈ E | ∃ j ∈ [[ i, i ′ ]] , x = x j } . Proof : We prove the result by induction on K ∈ N . For K = 1, the optimalsolution is unique, the optimal cluster is E = { x j } j ∈ [[1 ,N ]] . Let us suppose K > K - γ -CP2dPF.Let π ∈ Π K ( E ) an optimal solution of K - γ -CP2dPF, let OP T the optimal cost.We denote π = C , . . . , C K the K subsets of the partition π denoting C K the cluster of x N . For all k ∈ [[1 , K ]], f γ ( C k ) OP T . Let i the minimalindex such that x i ∈ C K . We consider the subsets C ′ K = { x j } j ∈ [[ i,N ]] and C ′ k = C k ∩ { x j } j ∈ [[1 ,i − for all k ∈ [[1 , K − C ′ , . . . , C ′ K − is a partition of { x j } j ∈ [[1 ,i − , and C ′ , . . . , C ′ K is a partition of E .For all k ∈ [[1 , K ]], C ′ k ⊂ C k so that f γ ( C ′ k ) ⊂ f γ ( C k ) OP T (Lemma 5). C ′ , . . . , C ′ K is a partition of E , and max k ∈ [[1 ,K ]] f ( C k ) OP T . C ′ , . . . , C ′ K is an optimal solution of K - γ -CP2dPF. C ′ , . . . , C ′ K − is an optimal solution ofthe considered K − γ -CP2dPF applied to points E ′ = ∪ K − k =1 C ′ . Let OP T ′ the optimal cost, we have OP T ′ max k ∈ [[1 ,K − f γ ( C ′ k ) OP T . Applying IHfor K − γ -CP2dPF to points E ′ , we have C ′′ , . . . , C ′′ K − an optimal solution of K − γ -CP2dPF among E ′ on the shape C i,i ′ = { x j } j ∈ [[ i,i ′ ]] = { x ∈ E ′ | ∃ j ∈ [[ i, i ′ ]] , x = x j } . For all k ∈ [[1 , K − f γ ( C ′′ k ) OP T ′ OP T . C ′′ , . . . , C ′′ K − , C ′ K is finally an optimal solution of K - γ -CP2dPF in E using only clusters on theshape C i,i ′ . (cid:3) Algorithm 1: Computation of f D ctr ( C i,i ′ ) input : indexes i < i ′ output : the cost f D ctr ( C i,i ′ ) Initialization: define idInf = i , valInf = || x i − x i ′ || ,define idSup = i ′ , valSup = || x i − x i ′ || , while idSup − idInf > j i + i ′ k , valTemp = f i,i ′ , idMid , valTemp2 = f i,i ′ , idMid +1 if valTemp = valTemp2idInf = idMid , valInf = valTempidSup = 1 + idMid, valSup = valTemp2 if valTemp < valTemp2 // increasing phaseidSup = idMid , valSup = valTemp if valTemp > valTemp2idInf = 1 + idMid , valInf = valTemp2 end whilereturn min(valInf , valSup) 8 .4 Computation of cluster costs Proposition 3 induces that only costs of clusters C i,i ′ shall be computed. Theissue is here to compute the most efficiently such clusters. Once points are sortedfollowing Proposition 1, equation (15) assures that cluster costs f C ctr ( C i,i ′ ) canbe computed in O (1) once E is sorted following Proposition 1.Equation (16) assures that cluster costs f D ctr ( C i,i ′ ) can be computed in O ( i ′ − i ) for all i < i ′ . Actually, Lemma 6 and Proposition 4 allows to have computa-tions in O (log( i ′ − i )) once points are sorted following Proposition 1. It allowsto compute all the f D ctr ( C i,i ′ ) for all i < i ′ in O ( N log N ). Lemma 6.
Let ( i, i ′ ) with i < i ′ . f i,i ′ : j ∈ [[ i, i ′ ]] max ( || x j − x i || , || x j − x i ′ || ) is strictly decreasing before reaching first a minimum f i,i ′ ( l ) , f i,i ′ ( l +1) > f i,i ′ ( l ) ,and then is strictly increasing for j ∈ [[ l + 1 , i ′ ]] Proposition 4. , Let E = { x , . . . , x N } be N points of R , such that for all i = j , x i ≺ x j . Computing cost f D ctr ( C i,i ′ ) for any cluster C i,i ′ has a complexityin O (log( i ′ − i )) time. Proof : Let i < i ′ . Algorithm 1 uses Lemma 6 to have as a loop invariant theexistence of a minimal solution of f i,i ′ ( j ∗ ) with idInf j ∗ idSup. Indeed,Algorithm 2 computes l = idMid = j idInf + idSup k and the values f i,i ′ ( l ) and f i,i ′ ( l + 1), operations in O (1). If f i,i ′ ( l ) < f i,i ′ ( l + 1), the Lemma 6 ensuresthat the center of C i,i ′ is before l ; so that it can be reactualized idSup = l . If f i,i ′ ( l ) > f i,i ′ ( l + 1), the Lemma 6 ensures that the center of C i,i ′ is after l + 1; sothat it can be reactualized idInf = l + 1. Iterating this procedure till M − m < C i,i ′ using at most log( i ′ − i ) operations in O (1). (cid:3) Proposition 3 implies that optimal solutions of the p-center problems (5) canbe designed using only subsets C i,i ′ . Enumerating such partitions with a bruteforce algorithm would lead to Θ( N K ) computations of the cost of the partition.This is not enough to guarantee to have a polynomial algorithm, but it is a firststep for the clustering algorithm of section 5. Proposition 3 allows to derive a common DP algorithm for p-center problems.Defining C i,k as the optimal cost of k - γ -CP2dPF clustering with k cluster amongpoints [[1 , i ]] for all i ∈ [[1 , N ]] and k ∈ [[1 , K ]]. The case k = 1 is given by: ∀ i ∈ [[1 , N ]] , C i, = f γ ( C ,i ) (17)We have following induction relation, with the convention C ,k = 0 for all k > ∀ i ∈ [[1 , N ]] , ∀ k ∈ [[2 , K ]] , C i,k = min j ∈ [[1 ,i ]] max( C j − ,k − , f γ ( C j,i )) (18)9 lgorithm 2: p-center clustering in a 2dPF, general DP algorithmInput: - N points of R , E = { x , . . . , x N } such that for all i = j , x i I x j ;- γ ∈ { , } to specify the clustering measure;- K ∈ N the number of clusters. Output:
OP T the optimal cost and the clustering partition P initialize matrix C with C i,k = 0 for all i ∈ [[1; N ]] , k ∈ [[1; K − E following the order of Proposition 1compute C i, = f γ ( C ,i ) for all i ∈ [[1; N ]] for k = 2 to K − C for i = 2 to N − C i,k = min j ∈ [[2 ,i ]] max( C j − ,k − , f γ ( C j,i )) end forend for set OP T = min j ∈ [[2 ,N ]] max( C j − ,N − , f γ ( C j,N ))set j = argmin j ∈ [[2 ,N ]] max( C j − ,N − , f γ ( C j,N )) i = j //Backtrack phaseinitialize P = { [[ j ; N ]] } , a set of sub-intervals of [[1; N ]]. for k = K to 1 with increment k ← k − j ∈ [[1 , i ]] such that min j ∈ [[1 ,i ]] max( C j − ,k − , f γ ( C j,i ))add [[ j, i ]] in P i = j − end forreturn OP T the optimal cost and the partition P Algorithm 2 uses these relations to compute the optimal values of C i,k . C N,K is the optimal solution of K - γ -CP2dPF, a backtracking algorithm allows tocompute the optimal partitions. In Algorithm 2, the main issue to calculate the complexity is to compute ef-ficiently C i,k = min j ∈ [[2 ,i ]] max( C j − ,k − , f γ ( C j,i )). Lemma 8 and Proposition5 allows to compute each line of the DP matrix with a time complexity in O ( N log N ) and O ( N ) space. Lemma 7.
Let k ∈ [[2 , K ]] . The application g i,k : j ∈ [[1 , N ]] C j,k isincreasing. Lemma 8.
Let i ∈ [[2 , N ]] , k ∈ [[2 , K ]] . Let g i,k : j ∈ [[2 , i ]] max( C j − ,k − , f γ ( C j,i )) .It exists l ∈ [[2 , i ]] such that g i,k is decreasing for j ∈ [[2 , l ]] , and then is increasingfor j ∈ [[ l + 1 , i ]] Proposition 5 (Line computation) . Let i ∈ [[2 , N ]] , k ∈ [[2 , K ]] . Let γ ∈ { , } .Once the values C j,k − in the DP matrix of Algorithm 2 are computed, Algo-rithm 3 computes C i,k = min j ∈ [[2 ,i ]] max( C j − ,k − , f γ ( C j,i )) calling O (log i ) cost lgorithm 3: Dichotomic computation of C i,k = min j ∈ [[2 ,i ]] max( C j − ,k − , f γ ( C j,i )) input : indexes i ∈ [[2 , N ]], k ∈ [[2 , K ]], γ ∈ { , } , a vector v j = C j,k − for all j ∈ [[1 , i − output : min j ∈ [[2 ,i ]] max( C j − ,k − , f ( C j,i )) Initialization: define idInf = 2, valInf = f γ ( C ,i ),define idSup = i , valSup = v i − , while idSup − idInf > j i + i ′ k , valTemp = g i,k ( idMid ) , valTemp2 = g i,k ( idMid + 1) if valTemp < valTemp2 // increasing phaseidSup = idMid , valSup = valTemp else idInf = 1 + idMid , valInf = valTemp2 end whilereturn min(valInf , valSup) computations f γ ( C j,i ) . It induces a time complexity in O (log γ i ) . In other words,once the line of the DP matrix C j,k − is computed for all j ∈ [[1 , N ]] , the line C j,k − can be computed with a complexity in O ( N log γ N ) time and O ( N ) space. Proof : Algorithm 3 is a dichotomic (and thus logarithmic) search based onLemma 8, similarly to Algorithm 1 derived from Lemma 6. The complexity tocall Algorithm 3 is O (log i ) cost computations f γ ( C j,i ). (cid:3) At this stage, the time complexity of Algorithm 1 can be provided with a spacecomplexity in O ( KN ), the size of the DP matrix. This section aims to reducethis complexity into a O ( N ) memory space.Actually, the DP matrix C can be computed line by line, with the index k increasing. The computation of line k + 1 requires only the line k and com-putations of cluster costs requiring O (1) additional memory space. In the DPmatrix, deleting the line k − k is completed allows to have 2 N elements in the memory? IT allows to compute the optimal value C N,K using O ( N ) memory space. The point here is that the backtracking operations, aswritten in Algorithm 2, require stored values of the whole matrix. This sectionaims to provide an alternative backtrack algorithm with a complexity in at most O ( N ) memory space and O ( KN log N ) time.We define Algorithm 4,an algorithm starting from the point z N , computingthe last cluster are the largest one with a size below the optimal cost of p-centeran,d iterating successivelmy by constructing the largest cluster. We prove thatthis procuedure furnish optimal partitions for the p-center problems. Lemma 9.
Let K ∈ N , K > . Let E = { z , . . . , z N } , sorted such that forall i < j , z i ≺ z j . For the discrete and continuous K -center problems, the lgorithm 4: Backtracking algorithm using O ( N ) memory spaceinput : - γ ∈ { , } to specify the clustering measure;- N points of a 2d PF, E = { z , . . . , z N } , sorted such that for all i < j , z i ≺ z j ;- K ∈ N the number of clusters;- OP T , the optimal cost of K - γ -CP2dPF; output : P an optimal partition of K - γ -CP2dPF.initialize maxId = N , minId = N , P = ∅ , a set of sub-intervals of [[1; N ]]. for k = K to 2 with increment k ← k − minId = maxId while f γ ( C minId − ,maxId )) OP T do minId = minId end while add [[ minId, maxId ]] in P set maxId = minId − end for add [[1 , maxId ]] in P return P indexes given by Algorithm 4 are lower bounds of the indexes of any optimalsolution: Denoting [[1 , i ]] , [[ i + 1 , i ]] , . . . , [[ i K − + 1 , N ]] the indexes given byAlgorithm 4, and [[1 , i ′ ]] , [[ i ′ + 1 , i ′ ]] , . . . , [[ i ′ K − + 1 , N ]] the indexes of an optimalsolution, we have for all k ∈ [[1 , K − , i k i ′ k Proposition 6.
Once the optimal cost of p-center problems are computed, Algo-rithm 4 computes an optimal partition in O ( N log N ) time using O (1) additionalmemory space. Proof : Let
OP T , the optimal cost of K -center clustering with f . Let [[1 , i ]] , [[ i +1 , i ]] , . . . , [[ i K − + 1 , N ]] the indexes given by Algorithm 4. By construction,all the clusters C defined by the indexes [[ i k + 1 , i k +1 ]] for all k > f γ ( C ) OP T . Let C the cluster defined by [[1 , i ]], we have to prove that f γ ( C ) OP T to conclude of the optimality of the clustering defined by Algo-rithm 4. Let an optimal solution, let [[1 , i ′ ]] , [[ i ′ + 1 , i ′ ]] , . . . , [[ i ′ K − + 1 , N ]] theindexes defining this solution. Lemma 9 ensures that i i ′ , and thus Lemma5 assures f γ ( C ,i ) f γ ( C ,i ′ ) OP T .Analyzing the complexity, Algorithm 4 calls at most ( K + N ) N timesthe clustering cost function, without requiring stored elements, the complexityis in O ( N log γ N ) time. (cid:3) Remark
Actually, finding the biggest cluster with an extremity given and abounded cost can be proceeded by a dichotomic search. It would induce acomplexity in O ( K log γ N ). To avoid the separate case K = O ( N ) and γ = 1,Algorithm 4 provides a common algorithm running in O ( N log N ) time whichis enough for the following complexity results.12 .4 Complexity results Theorem 1 (Polynomial complexities for K-centers in a 2d PF) . Let E = { x , . . . , x N } a subset of N points of R , such that for all i = j , x i I x j . 1-center problems are solvable in linear time. Applied to the 2d PF E for p > , thep-center problems are solvable to optimality in polynomial time using Algorithm2, and its operators defined in Algorithms 3 and 4. For K > , the continuousK-center problems are solvable in O ( KN log N ) time and O ( N ) space. Thediscrete 2-center is solvable in O ( N log N ) time and O ( N ) space. For K > ,the discrete K-center problems are solvable in O ( KN log N ) time and O ( N ) space. Proof : The induction formula (18) uses only values C i,j with j < k in Algo-rithm 3. C N,k is at the end of each loop in k the optimal value of the k -centerclustering among the N points of E , and the optimal cost. Proposition 6 en-sures that Algorithm 4 gives an optimal partition, which proves the validity ofAlgorithm 1.Let us analyze the complexity. We suppose K >
2, the case K = 1 is givenby Proposition 2. The space complexity is in O ( N ) using section 5.3. Sortingand indexing the elements of E following Proposition 1 has a time complexityin O ( N log N ). Computing the first line of the DP matrix costs has also a timecomplexity in O ( N log N ) With Proposition 5 The construction of the lines ofthe DP matrix C i,k for k ∈ [[2 , K − N × ( K −
2) computations ofmin j ∈ [[1 ,i ]] C j − ,k − + f γ ( C j,i ), which are in O (log γ N ) time, the complexity ofthis phase is in O (( K − N log γ N ) . Proposition 6 ensures that backtrackingoperations are also in O ( N log N ) time. Finally, the final time complexities arein O ( N log N + ( K − N log N ) for the discrete K -center problem, and in O ( N log N + ( K − N log N ) = O ( KN log N ) for the continuous K -centerproblem. For the discrete p-center problem, the case K = 2 induces a specificcomplexity in O ( N log N ), whereas cases K > O ( KN log N ). (cid:3) Planar p-center problems were not studied previously in the case of a PF. Thishypothesis, leading to many applications in bi-objective optimization, is cru-cial for the complexity results. The p-center problems were proven NP-hardin a planar Euclidean space since [33]. Adding the PF hypothesis induces thepolynomial complexity of Theorem 1 and allows an efficient implementation ofthe algorithm. Two properties of 2d PF were crucial for these results. On onehand, the 1-dimensional structure implied by Proposition 1 allows an extensionof DP algorithms [45, 23]. On the other hand, properties (15) and (16) allowfast computations of cluster costs. The specific computations of the minimumenclosing disks allows to improve in the 2d PF case the general results for planar13uclidian spaces. The discrete 1-center problem is proven O ( N ) time for 2d PF,instead of O ( N log N ) ([4]). The continuous 1-center problem has a complexityin O ( N ) time proven, as in in [32], but with a much simpler algorithm. Fur-thermore, when having to compute many cluster costs like in Algorithm 2, theinitial sorting following Proposition 1 is amortized, with marginal computationsto compute cluster costs in O ( N log γ N ) time. The complexity continuous 2-centre case is also improved, in O ( N log N ) time instead of O ( N log N ) timein [20, 42]. Section 5.3 emphasizes that many optimal solution may exist. The backtrack-ing procedure in Algorithm 4 gives the clusters with the minimal indexes. Analternative backtracking procedure, constructing the first clusters starting froma point x i , initially x , and furnishing the maximal clusters represented by [[ i, j ]]of cost bounded by OP T , furnishes also optimal solutions with the maximalindexes. A large funnel of optimal solutions may exist. Many optimal solutionsmay be nested, i.e. non verifying the Proposition 3. For the real-life applica-tion, having well-balanced clusters is more natural. The Algorithm 4 providesbadly balanced solutions. One may will to balance the sizes of covering balls,or the number of points in the clusters. Both types of solutions may be givenusing simple and fast post-processing. For example, one may proceed a steepestdescent local search using 2-center problem types for consecutive clusters in thecurrent solution. For balancing the size of clusters, iterating 2-center computa-tions induces marginal computations in O (log γ N ) time for each iteration. The results can be extended to some variants of the p-center problems. Therectilinear p-center considers the L norm [12]. Considering rectilinear p-centerproblems, Proposition 2 is also true, and similar properties than (15) and (16)can be proven to achieve the same complexity of cluster costs. Lemma 5 hasalso its proof considering the rectilinear p-center, which allows to prove similarlythe Proposition 3. Capacitated p-center problems can also be considered, withbounds on the cardinal of the clusters [7]. Proposition 3 can also be proven inthis case, and Algorithm 2 can be adapted considering the cardinal constraintsin the DP programming, and in the backtracking operations of Algorithm 4. The time complexity in O ( N K log γ N ) can be enough for large scale compu-tations of k - γ -CP2dPF. We mention here that Algorithm 2 with the operatorsspecified in Algorithms 3 and 4 have also good properties for a parallel imple-mentation in multi or many-core environments. As mentioned in section 5.3, theconstruction of the DP matrix can be computed line by line, with N − K ? A crucial point for the real life application of clustering is to select an appropri-ate value of K . A too small value of K can miss that an instance is well-capturedwith K + 1 representative clusters. The real-life application seeks for the bestcompromise between the minimization of K , and the minimization of the dis-similarity among the clusters. Similarly with [15], properties of DP can be usedin such goal. With the DP algorithm, many couples { ( k, C k,N ) } k are computed,the optimal k-center values with k clusters. Having defined a maximal value K ′ , the complexity for computing these points is in O ( N K ′ log γ N ) Searchingfor good values of k , the elbow technique, graph test, gap test as described in[39], may be applied. Algorithm 4 may be applied for many solutions withoutchanging the complexity. The initial motivation of this work was to support the decision makers whena MOO approach without preference furnishes a large set of non dominatedsolutions. In this application, the value of K is small, for human analysesto give some preferences. In this paper, the optimality is not required. Ourwork applies also for partial PF furnished by population meta-heuristics [44].A posteriori, the complexity allows to use Algorithm 3 embedded in MOO ap-proaches, similarly with [3]. Archiving diversified solutions of Pareto sets hasan application for the diversification of genetic algorithms to select diversifiedsolutions for cross-over and mutation phases [46, 40], but also for swarm par-ticle optimization heuristics [37]. In these applications, clustering has to runquickly. The complexity results and the parallelization properties are useful insuch application. This paper examined properties of the p-center problems in the special caseof a discrete set of non-dominated points in a 2d Euclidian space. A commoncharacterization of optimal clusters is proven for the discrete and continuousvariants of the p-center problem. It allows to solve these problems to optimality15ith a unified DP algorithm of a polynomial complexity, whereas both prob-lems are NP-hard without the PF hypothesis in R . A complexity is proven in O ( KN log N ) time for the continuous p-center problem and in O ( KN log N )time for the discrete p-center problem, with a space complexity in O ( N ) forboth cases. Discrete 2-center is also proven solvable in O ( N log N ) time, and1-center problems in O ( N ) time. It improves general results for 1-center, 2-center problems. The DP algorithms can also be parallelized to speed up thecomputational time.Research perspectives are to extend the result of this paper to other cluster-ing algorithms. In these cases, the crucial point is to prove the interval prop-erty of Proposition 3. In the perspectives to extend some results to dimension3, interesting for the MOO application, clustering a 3d PF will be a NP-hardproblem. The perspectives are only to find specific approximation algorithmsfor a 3d PF. References [1] P. Agarwal and C. Procopiuc. Exact and approximation algorithms for clustering.
Algo-rithmica , 33(2):201–226, 2002.[2] P. Agarwal, M. Sharir, and E. Welzl. The discrete 2-center problem.
Discrete & Com-putational Geometry , 20(3):287–305, 1998.[3] A. Auger, J. Bader, D. Brockhoff, and E. Zitzler. Investigating and exploiting the biasof the weighted hypervolume to articulate user preferences. In
Proceedings of GECCO2009 , pages 563–570. ACM, 2009.[4] P. Brass, C. Knauer, H. Na, C. Shin, and A. Vigneron. Computing k-centers on a line. arXiv preprint arXiv:0902.3282 , 2009.[5] K. Bringmann, S. Cabello, and M. Emmerich. Maximum volume subset selection foranchored boxes. arXiv preprint arXiv:1803.00849 , 2018.[6] K. Bringmann, T. Friedrich, and P. Klitzke. Two-dimensional subset selection for hy-pervolume and epsilon-indicator. In
Annual Conference on Genetic and EvolutionaryComputation , pages 589–596. ACM, 2014.[7] H. Calik, M. Labb´e, and H. Yaman. p-center problems. In
Location science , pages 79–92.Springer, 2015.[8] H. Calik and B. Tansel. Double bound method for solving the p-center location problem.
Computers & operations research , 40(12):2991–2999, 2013.[9] B. Callaghan, S. Salhi, and G. Nagy. Speeding up the optimal method of Drezner for thep-centre problem in the plane.
European Journal of Operational Research , 257(3):722–734, 2017.[10] M Daskin.
Network and discrete location: models, algorithms and applications . Wiley,1995.[11] Zvi Drezner. The p-centre problemheuristic and optimal algorithms.
Journal of theOperational Research Society , 35(8):741–748, 1984.[12] Zvi Drezner. On the rectangular p-center problem.
Naval Research Logistics (NRL) ,34(2):229–234, 1987.[13] N. Dupin.
Mod´elisation et r´esolution de grands probl`emes stochastiques combinatoires:application `a la gestion de production d’´electricit´e . PhD thesis, Univ. Lille 1, 2015.[14] N. Dupin, F. Nielsen, and E. Talbi. Dynamic programming heuristic for k-means cluster-ing among a 2-dimensional pareto frontier. , pages 1–8, 2018.
15] N. Dupin, F. Nielsen, and E. Talbi. k-medoids clustering is solvable in polynomial time fora 2d Pareto front. In
World Congress on Global Optimization , pages 790–799. Springer,2019.[16] N. Dupin and E. Talbi. Clustering in a 2-dimensional Pareto Front: p-median and p-center are solvable in polynomial time. arXiv preprint arXiv:1806.02098 , 2018.[17] M. Ehrgott and X. Gandibleux. Multiobjective combinatorial optimization - theory,methodology, and applications. In
Multiple criteria optimization: State of the art anno-tated bibliographic surveys , pages 369–444. Springer, 2003.[18] S. Elloumi, M. Labb´e, and Y. Pochet. A new formulation and resolution method for thep-center problem.
INFORMS Journal on Computing , 16(1):84–94, 2004.[19] A. Elshaikh, S. Salhi, and G. Nagy. The continuous p-centre problem: An investiga-tion into variable neighbourhood search with memory.
European Journal of OperationalResearch , 241(3):606–621, 2015.[20] D. Eppstein. Faster construction of planar two-centers. In
SODA , volume 97, pages131–138, 1997.[21] D. Ferone, P. Festa, A. Napoletano, and M. Resende. A new local search for the p-centerproblem based on the critical vertex concept. In
Internat. Conference on Learning andIntelligent Optimization , pages 79–92. Springer, 2017.[22] T. Gonzalez. Clustering to minimize the maximum intercluster distance.
TheoreticalComputer Science , 38:293 – 306, 1985.[23] A. Grønlund et al. Fast exact k-means, k-medians and Bregman divergence clustering in1d. arXiv preprint arXiv:1701.07204 , 2017.[24] L. Guo-Hui and G. Xue. K-center and k-median problems in graded distances.
Theoreticalcomputer science , 207(1):181–192, 1998.[25] D.S. Hochbaum. When are NP-hard location problems easy?
Annals of OperationsResearch , 1(3):201–214, 1984.[26] D.S. Hochbaum and D.B. Shmoys. A best possible heuristic for the k-center problem.
Mathematics of operations research , 10(2):180–184, 1985.[27] W. Hsu and G. Nemhauser. Easy and hard bottleneck location problems.
DiscreteApplied Mathematics , 1(3):209–215, 1979.[28] R. Hwang, R. Lee, and R. Chang. The slab dividing approach to solve the EuclideanP-Center problem.
Algorithmica , 9(1):1–22, 1993.[29] A. Karmakar, S. Das, S. Nandy, and B. Bhattacharya. Some variations on constrainedminimum enclosing circle problem.
Journal of Combinatorial Optimization , 25(2):176–190, 2013.[30] T. Kuhn, C. Fonseca, L. Paquete, S. Ruzika, M. Duarte, and J. Figueira. Hypervol-ume subset selection in two dimensions: Formulations and algorithms.
EvolutionaryComputation , 24(3):411–425, 2016.[31] M. Mahajan, P. Nimbhorkar, and K. Varadarajan. The planar k-means problem is NP-hard.
Theoretical Computer Science , 442:13–21, 2012.[32] N. Megiddo. Linear-time algorithms for linear programming in R3 and related problems.
SIAM journal on computing , 12(4):759–776, 1983.[33] N. Megiddo and K. Supowit. On the complexity of some common geometric locationproblems.
SIAM journal on computing , 13(1):182–196, 1984.[34] N. Megiddo and A. Tamir. New results on the complexity of p-centre problems.
SIAMJournal on Computing , 12(4):751–758, 1983.[35] N. Mladenovi´c, M. Labb´e, and P. Hansen. Solving the p-center problem with tabu searchand variable neighborhood search.
Networks , 42(1):48–64, 2003.
36] T. Peugeot, N. Dupin, M-J Sembely, and C. Dubecq. MBSE, PLM, MIP and RobustOptimization for System of Systems Management, Application to SCCOA French AirDefense Program. In
Complex Systems Design&Management , pages 29–40. Springer,2017.[37] G. Pulido and C. Coello. Using clustering techniques to improve the performance ofa multi-objective particle swarm optimizer. In
Genetic and Evolutionary ComputationConference , pages 225–237. Springer, 2004.[38] C. Ramirez-Atencia, S. Mostaghim, and D. Camacho. A knee point based evolutionarymulti-objective optimization for mission planning problems. In
Proceedings of the Geneticand Evolutionary Computation Conference , pages 1216–1223. ACM, 2017.[39] J-P Rasson and T. Kubushishi. The gap test: an optimal method for determining thenumber of natural classes in cluster analysis. In
New approaches in classification anddata analysis , pages 186–193. Springer, 1994.[40] M. Samorani, Y. Wang, Z. Lv, and F. Glover. Clustering-driven evolutionary algorithms:an application of path relinking to the quadratic unconstrained binary optimization prob-lem.
Journal of Heuristics , pages 1–14, 2018.[41] S. Sayın. Measuring the quality of discrete representations of efficient sets in multipleobjective mathematical programming.
Mathematical Programming , 87(3):543–560, 2000.[42] M. Sharir. A near-linear algorithm for the planar 2-center problem.
Discrete & Compu-tational Geometry , 18(2):125–134, 1997.[43] E. Sintorn and U. Assarsson. Fast parallel GPU-sorting using a hybrid algorithm.
Journalof Parallel and Distributed Computing , 68(10):1381–1388, 2008.[44] E. Talbi.
Metaheuristics: from design to implementation , volume 74. Wiley, 2009.[45] H. Wang and M. Song. Ckmeans. 1d. dp: optimal k-means clustering in one dimensionby dynamic programming.
The R journal , 3(2):29, 2011.[46] E. Zio and R. Bazzo. A clustering procedure for reducing the number of representativesolutions in the pareto front of multiobjective optimization problems.
European Journalof Operational Research , 210(3):624–634, 2011. ppendix A: Proof of the intermediate lemmas This section gives the elementary proofs of intermediate lemmas.
Proof of Lemma 2 : We note firstly that the equality cases are trivial, so thatwe can suppose i < i < i in the following proof. We prove the propriety (13),the proof of (14) is analogous.Let i < i < i . We note x i = ( x i , x i ), x i = ( x i , x i ) and x i = ( x i , x i ) .Proposition 1 ordering ensures x i < x i < x i and x i > x i > x i . d ( x i , x i ) = ( x i − x i ) + ( x i − x i ) With x i − x i > x i − x i >
0, ( x i − x i ) < ( x i − x i ) With x i − x i < x i − x i <
0, ( x i − x i ) < ( x i − x i ) Thus d ( x i , x i ) < ( x i − x i ) + ( x i − x i ) = d ( x i , x i ) . (cid:3) Proof of Lemma 3 : This Lemma is a reformulated in Lemma 10 below.
Proof of Lemma 4 : Let y ∈ P − { x i , x i ′ } , we denote j ∈ [[ i, i ′ ]] such that y = x j . Applying Lemma 2 to i < j < i ′ : ∀ k ∈ [[ i, i ′ ]] , || x j − x k || max ( || x j − x i || , || x j − x i || ) (19)Finally, Lemma 4 is proven with: f D ctr ( P ) = min y = x j ∈ P max x ∈ P || x − y || = min j ∈ [[ i,i ′ ]] ,x j ∈ P max (cid:18) max ( || x j − x i || , || x j − x i || ) , max k ∈ [[ i,i ′ ]] || x j − x k || (cid:19) = min j ∈ [[ i,i ′ ]] ,x j ∈ P max ( || x j − x i || , || x j − x i ′ || ) (cid:3) Proof of Lemma 5 : Let i (resp i ′ ) the minimal index of points of P (resp P ′ ).Let j (resp j ′ ) the maximal index of points of P (resp P ′ ). f C ctr ( P ) f C ctr ( P ′ ) is trivial using Lemma 2 and Lemma 3.To prove f D ctr ( P ) f D ctr ( P ′ ), we use Lemma 2 and Lemma 4: f D ctr ( P ) = min k ∈ [[ i,j ]] ,x k ∈ P max ( || x k − x i || , || x j − x k || ) min k ∈ [[ i ′ ,j ′ ]] ,x k ∈ P ′ max ( || x k − x i || , || x j − x k || ) min k ∈ [[ i ′ ,j ′ ]] ,x k ∈ P ′ max ( || x k − x i ′ || , || x j ′ − x k || ) = f D ctr ( P ′ ) (cid:3) Proof of Lemma 6 : We define g i,i ′ ,j , h i,i ′ ,j with: g i,i ′ : j ∈ [[ i, i ′ ]] x j − x i || and h i,i ′ : j ∈ [[ i, i ′ ]] x j − x i ′ || i < i ′ . Proposition 2 applied to i and any j, j + 1 with j > i and j < i ′ assures that g is strictly decreasing. Similarly, Proposition 2 applied to i ′ andany j, j + 1 ensures that h is strictly increasing.Let A = { j ∈ [[ i, i ′ ]] |∀ m ∈ [[ i, j ]] g i,i ′ ( m ) < h i,i ′ ( m ) } . g i,i ′ ( i ) = 0 and h i,i ′ ( i ) = || x i ′ − x i || > i ∈ A . A is a non empty and bounded subset of N , so that A has a maximum. We note l = max A . h i,i ′ ( i ′ ) = 0 and g i,i ′ ( i ′ ) = || x i ′ − x i || > i ′ / ∈ A and l < i ′ .Let j ∈ [[ i, l − g i,i ′ ( j ) < g i,i ′ ( j +1) and h i,i ′ ,j ( j +1) < h i,i ′ ( j ) using monotonyof g i,i ′ and h i,i ′ . f i,i ′ ( j + 1) = max ( g i,i ′ ( j + 1) , h i,i ′ ( j + 1)) = h i,i ( j + 1) and f i,i ′ ( j ) = max( g i,i ′ ( j ) , h i,i ′ ( j )) = h i,i ( j ) as j, j + 1 ∈ A .Hence, f i,i ′ ( j + 1) = h i,i ′ ( j + 1) < h i,i ′ ( j ) = f i,i ′ ( j ). It proves that f i,i ′ is strictly decreasing in [[ i, l ]]. l +1 / ∈ A and g i,i ′ ( l +1) > h i,i ′ ( l +1) to be coherent with the fact that l = max A .Let j ∈ [[ l + 1 , i ′ − j + 1 > j > l + 1 so g i,i ′ ( j + 1) > g i,i ′ ( j ) > g i,i ′ ( l + 1) >h i,i ′ ( l + 1) > h i,i ′ ( j ) > h i,i ′ ( j + 1) using monotony of g i,i ′ and h i,i ′ .It proves that f i,i ′ is strictly increasing in [[ l + 1 , i ′ ]].Lastly, the minimum of f can be reached in l or in l + 1, depending on the signof f i,i ′ ( l + 1) − f i,i ′ ( l ). If f i,i ′ ( l + 1) = f i,i ′ ( l ) there are two minimums l, l + 1.Otherwise, there exist a unique minimum l ∈ { l, l + 1 } , f i,i ′ decreasing strictlybefore increasing strictly. (cid:3) Proof of Lemma 7 : We note firstly that the case k = 1 is implied by theLemma 5, so that we can suppose in the following k >
2. Let k ∈ [[2 , K ]] and j ∈ [[2 , N ]]. Let C , . . . , C k an optimal solution of k -center clustering among thepoints ( x l ) l ∈ [[1 ,j ]] , its cost is C j,k . We index the clusters such that x j ∈ C k . ∀ k ′ ∈ [[1 , k ]] , f ( C k ′ ) C j,k We consider the clusters C ′ , . . . , C ′ k = C , . . . , C k − , C k − x k . It is a partitionof ( x l ) l ∈ [[1 ,j − , so that C j − ,k max k ′ ∈ [[1 ,k ]] f ( C ′ k ′ ). Lemma 5 assures that f ( C ′ k ′ ) = f ( C k − x k ) f ( C k ). Hence, C j − ,k max k ′ ∈ [[1 ,k ]] f ( C ′ k ′ ) C j,k , theapplication g i,k is increasing. (cid:3) Proof of Lemma 8 : This lemma is proven noticing that the following appli-cations are monotone: j ∈ [[1 , i ]] f γ ( C j,i ) is decreasing with Lemma 5, j ∈ [[1 , N ]] C j,k is increasing for all k with Lemma 7. (cid:3) Proof of Lemma 9 : This lemma is proven by decreasing induction on k ,starting from k = K −
1. The case k = K − j ∈ [[1 , N ]] f γ ( C j,N ) is decreasing with Lemma5. Having for a given k , i ′ k i k , i k − i ′ k − is implied by lemma 2 and d ( z i k , z i k − − ) > OP T . (cid:3) Lemma 10.
Let P ⊂ E such that card ( P ) > . Let i (resp j ) the minimal resp maximal) index of points of P . We have the following results: ∀ x ∈ R − (cid:26) x i + x j (cid:27) , max p ∈ P || x − p || > || x i − x j || (20)max p ∈ P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x i + x j − p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = || x i − x j || (21) In other word, the application x ∈ R max p ∈ P || x − p || ∈ R as a uniqueminimum reached for x = x i + x j . Proof : We prove (20) and (21) using specific coordinates to ease the analyticcalculations. Indeed, the analytic calculus of (6) is invariant with translationsof the origin and rotations of the axes. We note diam( P ) = || x i − x j || .To prove (20), we use the coordinates defining as origin the point O = x i + x j and rotating the axes such that in the system of coordinates x i = ( − diam( P ) , x j = ( diam( P ) , x a point in R and ( x , x ) its coordinates. Weminimize max p ∈ P d ( x, x p ) distinguishing the cases:if x > d ( x, x i ) = ( x + diam( P )) + x > ( x + diam( P )) > ( diam( P )) max p ∈ P d ( x, x p ) > d ( x, x i ) > diam( P )if x < d ( x, x j ) = ( x − diam( P )) + x > ( x − diam( P )) > ( diam( P )) max p ∈ P d ( x, x p ) > d ( x, x j ) > diam( P )if x = 0 and x = 0, d ( x, x i ) = ( diam( P )) + x > ( x + diam( P )) > ( diam( P )) max p ∈ P d ( x, x p ) > d ( x, x i ) > diam( P ).The three cases allow to reach any point of R except x = x i + x j , proving (20).To prove (21), we use the coordinates translating the axes such that in thesystem of coordinates x i = ( − diam( P ); 0) and x j = ( diam( P ); 0). The originis the ideal point defined by x i and x j . x has coordinates ( √ diam( P ) , √ diam( P )).Let x = ( x , x ) ∈ P , such that x = x i and x = x j . Thus x i ≺ x i ≺ x j . ThePareto dominance imposes that O x , x √ diam( P ). d ( x, x ) = ( x − √ diam( P )) + ( x − √ diam( P )) d ( x, x ) ( √ diam( P )) + ( √ diam( P )) = 2 diam( P )) d ( x, x ) diam( P ), which proves (21) as d ( x , x i ) = d ( x , x j ) = diam( P ). (cid:3)(cid:3)