Minimizing the sum of the k largest functions in linear time - CiteSeerX

Information Processing Letters 85 (2003) 117–122www.elsevier.com/locate/iplMinimizing the sum of the k largest functions in linear timeWlodzimierz Ogryczak a ,ArieTamir b,∗a Institute of Control and Computation Engineering, Warsaw University of Technology, 00-665 Warsaw, Polandb School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, IsraelReceived 28 November 2001; received in revised form 29 July 2002Communicated by M. YamashitaAbstractGiven a collection of n functions defined on R d , and a polyhedral set Q ⊂ R d , we consider the problem of minimizing thesum of the k largest functions of the collection over Q. Specifically we focus on collections of linear functions and severalclasses of convex, piecewise linear functions which are defined by location models. We present simple linear programmingformulations for these optimization models which give rise to linear time algorithms when the dimension d is fixed. Ourresults improve complexity bounds of several problems reported recently by Tamir [Discrete Appl. Math. 109 (2001) 293–307],Tokuyama [Proc. 33rd Annual ACM Symp. on Theory of Computing, 2001, pp. 75–84] and Kalcsics, Nickel, Puerto and Tamir[Oper. Res. Lett. 31 (1984) 114–127].© 2002 Elsevier Science B.V. All rights reserved.Keywords: Algorithms; Computational geometry; k-largest; Location; Rectilinear; k-centrum1. IntroductionGiven a collection {g i (x)} n i=1of n functions definedon R d , and a polyhedral set Q ⊂ R d ,inthispaperwe consider the problem of minimizing the sum ofthe k largest functions of the collection over Q. Specificallywe focus on collections of linear functions andseveral classes of convex, piecewise linear functionswhich are defined by location models. We present simplelinear programming formulations for these optimizationmodels which give rise to linear time algorithmswhen the dimension d is fixed. Tokuyama [22]has recently discussed the case where all the functions* Corresponding author.E-mail addresses: ogryczak@ia.pw.edu.pl (W. Ogryczak),atamir@post.tau.ac.il (A. Tamir).in the collection are linear, and Q is the intersectionof p half spaces. Tokuyama presents a very complex(randomized) algorithm which finds the optimal solutionin O(p + n log n) time when d is fixed. We showhow to use our formulation to improve this bound. Weobtain a deterministic O(p + n) algorithm. In the locationmodels that we consider, each function g i (x)represents a (weighted) distance of x from a givenpoint v i ∈ R d . The distance function is defined eitherby the l 1 or l ∞ norms. These problems are called thek-centrum location models. Again, when d is fixed ourlinear programming formulations lead to O(p + n) algorithmswhich find optimal solutions.In Section 2 we formally define the optimizationmodel and present the linear programming formulation,leading to linear time algorithms. In the last sectionwe discuss extensions to more general problems.0020-0190/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0020-0190(02)00370-8

118 W. Ogryczak, A. Tamir / Information Processing Letters 85 (2003) 117–1222. A linear programming formulationTo facilitate the discussion we first introduce somenotation. For any real number z define (z) + =max(z, 0). Lety = (y 1 ,...,y m ) be a vector in R m .Define θ(y)= (θ 1 (y), θ 2 (y), . . ., θ m (y)) to be the vectorin R m , obtained by sorting the m components ofy in nonincreasing order, i.e., θ 1 (y) θ 2 (y) ···θ m (y). θ i (y) will be referred to as the ith largest componentof y. Finally, for k = 1,...,m,defineΘ k (y) =∑ ki=1θ i (y), the sum of the k largest components of y.Lemma 1. For any vector y ∈ R m and k = 1,...,m,Θ k (y) = 1 (m∑m∑ [k y i + min k(t − yi ) +mt∈Ri=1Moreover, t ∗ = θ k (y) is an optimizer of the aboveminimization problem.Proof. Define h(t) = ∑ mi=1 [k(t −y i ) + +(m−k)(y i −t) + ]. This function is piecewise linear and convex.It is easy to verify that the one sided derivatives att ∗ = θ k (y) are of opposite signs, and therefore t ∗ isa minimum point of h(t). Substituting t ∗ = θ k (y) inthe expression for h(t) we obtainh ( θ k (y) ) = k= mm∑ (θk (y) − θ i (y) )i=k+1− (m − k)k∑θ i (y) − ki=1Hence, h(θ k (y)) = mΘ k (y)−k ∑ mi=1 y i , and thereforeΘ k (y) = 1 ()m∑k y i + minmh(t) . ✷t∈Ri=1It follows from the above lemma that Θ k (y) canbe represented as the solution value of the followinglinear program.Θ k (y) = min 1 m( m ∑i=1subject tosubject toi=1] ) (n∑min kt + di+ + (m − k)(y i − t) + .i=1subject to(n∑k∑ (θk (y) − θ i (y) )min kt + di+i=1i=1subject tom∑θ i (y).i=1[ kd−i+ (m − k)di + ] ) + ky idi + − di − = y i − t, di + ,di − 0, i = 1,...,m.Substituting di − = di + − y i + t, we obtain( )m∑Θ k (y) = min kt +i=1d + i y i − t, d + i 0, i = 1,...,m.d + iGiven the collection of functions, {g i (x)} n i=1 ,andthe polyhedral set Q ⊂ R d , defined in the introduction,let g(x) = (g 1 (x), . . ., g n (x)). The problem of minimizingΘ k (g(x)), the sum of the k largest functions ofthe collection over Q, can now be formulated as)di + g i (x) − t, di + 0, i = 1,...,n,x = (x 1 ,...,x d ) ∈ Q.2.1. Minimizing the sum of the k largest linearfunctionsConsider the linear case. For i = 1,...,n, g i (x) =a i x + b i ,wherea i = (a1 i ,...,ai d ) ∈ Rd ,andb i ∈R 1 . With the above notation the problem can beformulated as the following linear program:di + a i x + b i − t, i = 1,...,n,di + 0, i = 1,...,n,x = (x 1 ,...,x d ) ∈ Q.)Note that this linear program has n + d + 1 variables,d + 1 ,...,d+ n ,x 1,...,x d ,t,and2n+p constraints. Thisformulation constitutes a special case of the classof linear programs defined as the duals of linearmultiple-choice knapsack problems. Therefore, usingthe results in [24], when d is fixed, an optimal solutioncan be obtained in O(p +n) time. (See also [13].) Thisbound improves upon the O(p +n log n) bound in [22]by a factor of O(log n).

W. Ogryczak, A. Tamir / Information Processing Letters 85 (2003) 117–122 1192.2. Solving the rectilinear k-centrum locationproblemFor each pair of points u = (u 1 ,...,u d ), v =(v 1 ,...,v d ) in R d let d(u,v) denote the rectilineardistance between u and v,d(u,v) =d∑|v j − u j |.j=1Given is a set {v 1 ,...,v n } of n points in R d . Supposethat v i ,i= 1,...,n, is associated with a nonnegativereal weight w i . For each point x ∈ R d define the vectorD(x) ∈ R n by D(x) = (w 1 d(x,v 1 ),...,w n d(x,v n )).For a given k = 1,...,n, the single facility rectilineark-centrum problem in R d is to find a point x ∈ R dminimizing the objective f k (x) = ∑ ki=1 θ i (D(x)).(Tothe best of our knowledge the concept of a k-centrumwas first defined by Slater [18] and Andreatta andMason [1,2].) Note that the case k = 1 coincides withthe classical (weighted) rectilinear 1-center problemin R d , while the case k = n defines the classical(weighted) rectilinear 1-median problem, proposedby Hakimi [5,6]. It is well known that the last twoproblems can be formulated as linear programs. Thecenter problem (k = 1) is formulated as,min tsubject tot w i d(x,v i ),x = (x 1 ,...,x d ) ∈ R d .i = 1,...,n,To obtain a linear program we replace each one ofthe n nonlinear constraints t w i d(x,v i ),byasetof 2 d linear constraints. For i = 1,...,n,let∆ i =(δ1 i ,...,δi d) be a vector all of whose components areequal to +1 or−1. Consider the set of 2 d linear constraints,t ∑ dj=1 δj i w i(x j − vj i ), δi j∈{−1, 1}, j=1,...,d. This linear program has d + 1 variables,t,x 1 ,...,x d ,and2 d n constraints. Therefore, when dis fixed, it can be solved in O(n) time by the algorithmof Megiddo [12].Similarly, the median problem (k = n) isformulatedas,n∑minz ii=1subject toz i δ i jd∑δj i w i(xj − vj i ),j=1∈{−1, 1},x = (x 1 ,...,x d ) ∈ R d .j = 1,...,d, i= 1,...,n,This linear program has n + d variables, z 1 ,...,z n ,x 1 ,...,x d ,and2 d n constraints. This formulation isalso a special case of the class of linear programsdiscussed by Zemel [24]. Therefore, when d is fixed,an optimal solution can be obtained in O(n) time.The above formulation of the median problem can bereplaced by another, where the number of variables isnd + d and the number of constraints is only 2dn.n∑ d∑miny i,ji=1 j=1subject toy i,j w i(xj − v i j), j = 1,...,d, i= 1,...,n,y i,j −w i(xj − v i j) , j = 1,...,d, i= 1,...,n,x = (x 1 ,...,x d ) ∈ R d .The latter compact formulation is also solvable inO(n) time by the procedure of Zemel when d is fixed.In fact, it is easy to see from this formulation that thed-dimensional median problem is decomposable intod 1-dimensional problems. Therefore, it can be solvedin O(dn) time.We note in passing that when d is fixed even the1-centdian objective function, defined by Halpern [7–9] and Handler [10], as a convex combination of thecenter objective f 1 (x) and the median objective f n (x)can be solved in linear time (see [20]).To the best of our knowledge for a general valueof k, no linear time algorithms are reported in theliterature even for d = 1. In [19] the case d = 1istreated as a special case of a tree network. In particular,this one dimensional problem is solved in O(n) timewhen k is fixed, and in O(n log n) time when k isvariable. Subquadratic algorithms for any fixed d andvariable k are given in [11]. For example, for d = 2thealgorithm there has O(n log 2 n) complexity.

120 W. Ogryczak, A. Tamir / Information Processing Letters 85 (2003) 117–122Using the above results we can now formulate therectilinear k-centrum problem in R d as the followingoptimization problem:( )n∑min kt +i=1d + isubject todi + w i d(x,v i ) − t, di + 0, i= 1,...,n,x = (x 1 ,...,x d ) ∈ R d .As above, to obtain a linear program we replaceeach one of the n nonlinear constraints di+ w i d(x,v i ) − t, byasetof2 d linear constraints. Therectilinear k-centrum problem is now formulated asthe linear programming problem,( )n∑min kt +subject tod + iδ i j+ t i=1d + id∑δj i w i(xj − vj i ),j=1∈{−1, 1}, j= 1,...,d, i= 1,...,n,di + 0, i = 1,...,n.Note that the linear program has n + d + 1 variables,d1 + ,...,d+ n ,x 1,...,x d ,t, and 2 d n + n constraints.This formulation is again a special case of the classof linear programs defined as the duals of linearmultiple-choice knapsack problems. Therefore, usingthe results of Zemel [24], when d is fixed, an optimalsolution can be obtained in O(n) time.We note in passing that the results for the rectilinearproblem can be extended to other polyhedralnorms. For example, if we use the l ∞ norm, and let thedistance between u, v ∈ R d be defined by d(u,v) =max j=1,...,d |v j − u j |, we get the following formulationfor the respective k-centrum problem:( )n∑min kt +i=1d + isubject todi + (+ t w i xj − vj i ), j = 1,...,d, i= 1,...,n,di + (+ t −w i xj − vj i ) ,j = 1,...,d, i= 1,...,n.3. Related problems and extensionsRecently, a new type of objective function inlocation modeling, called ordered median function,has been introduced and analyzed. See, for example,[15,17,14,4]. (This criterion was introduced alreadyin [23] in the context of multi-criteria decision making.)In our context this objective function generalizesthe k-centrum objective. We extend the formulation inSection 2 to the rectilinear ordered median problem.To define this general model we first need to introducesome notation. Given is a nonnegative vector λ =(λ 1 ,...,λ m ) ∈ R m , satisfying λ 1 ··· λ m . For conveniencedefine λ m+1 = 0. For each y = (y 1 ,...,y m )definem∑Λ(y) = λ i θ i (y).i=1Although Λ(y) may also be considered for arbitrarysequences of (λ 1 ,...,λ m ) [23], the monotonicityassumption λ 1 ··· λ m is important whitin thelocation analysis context since it guarantees the socalledequitable properties of solutions [16]. (Note thatif λ is the vector whose first k components are equal to1 and the others are equal to 0, then Λ(y) = Θ k (y).)For each vector λ satisfying the above we have thefollowing expression,m∑Λ(y) = (λ k − λ k+1 )Θ k (y).k=1Using the results in Section 2, due to the monotonicityassumption, λ 1 ··· λ m 0, we can representΛ(y) as the solution value of the following linear programming:()m∑m∑Λ(y) = min (λ k − λ k+1 ) kt k +k=1di,k+i=1subject todi,k + y i − t k , di,k + 0, i= 1,...,m, k= 1,...,m.The latter formulation can be used to generalize theresults in Sections 2.1 and 2.2. For example, considera generalization of the rectilinear k-centrum problem,called the rectilinear ordered median problem,definedas follows:Find a point x ∈ R d minimizing the objective functionΛ(D(x)). (To use the above model set m = n.)

W. Ogryczak, A. Tamir / Information Processing Letters 85 (2003) 117–122 121This problem can now be formulated as a linear program.()n∑n∑min (λ k − λ k+1 ) kt k +k=1subject tod + i,k + t k δ i jd + i,k 0,d∑δj i w i(xj − vj i ) ,j=1∈{−1, 1},di,k+i=1j = 1,...,d, i= 1,...,n,k = 1,...,n,i = 1,...,n, k= 1,...,n.Note that this linear program has n 2 + d + n variables,{d + i,k }, i, k = 1,...,n, x 1,...,x d , t 1 ,...,t n ,and2 d n 2 + n 2 constraints.The best known algorithm to solve the aboverectilinear ordered median in a fixed dimension d isthe O(n log 2d n) procedure described in [11]. Thisprocedure is actually a direct application of the generalalgorithm of Cohen and Megiddo [3]. At this pointin time we do not yet know whether the abovelinear programming formulation, which involves aquadratic number of variables, can be used to improvethe bounds reported in [11]. Nevertheless, for somespecial cases of the vector λ = (λ 1 ,...,λ n ), the aboveproblem can still be solved in linear time. Specifically,if the components of λ can take on a constant, say q,number of distinct values, the above problem reducesto a linear program with qn + d + n variables and2 d qn + qn constraints. Therefore, it can be solvedin O(n) time when both, d and q are fixed. As anexample, consider the centdian problem mentioned inSection 2.2. In this case λ = (1,µ,...,µ),whereµ isa positive number bounded above by 1.Finally, we address the solvability of the rectilinearordered median problem when the dimension d isvariable. (Note that the formulation given above has2 d n 2 + n 2 constraints.) Consider the alternative formulation:()n∑n∑min (λ k − λ k+1 ) kt k +k=1di,k+i=1subject tod + i,k + t k d∑y i,j ,j=1i = 1,...,n, k= 1,...,n,y i,j w i(xj − v i j), j = 1,...,d, i= 1,...,n,y i,j −w i(xj − v i j), j = 1,...,d, i= 1,...,n,d + i,k 0,i = 1,...,n, k= 1,...,n.This linear program has n 2 + (n + 1)d + n variablesand 2n 2 + 2nd constraints. Therefore, the problemis polynomially solvable. Moreover, if w i = w, i =1,...,n, for some positive constant w, it followsfrom [21] that the problem can be solved in stronglypolynomial time.References[1] G. Andreatta, F.M. Mason, k-eccentricity and absolutek-centrum of a tree, European J. Oper. Res. 19 (1985) 114–117.[2] G. Andreatta, F.M. Mason, Properties of the k-centrum in anetwork, Networks 15 (1985) 21–25.[3] E. Cohen, N. Megiddo, Maximizing concave functions in fixeddimension, in: P.M. Pardalos (Ed.), Complexity in NumericalOptimization, World Scientific, Singapore, 1993, pp. 74–87.[4] R.L. Francis, T.J. Lowe, A. Tamir, Aggregation error boundsfor a class of location models, Oper. Res. 48 (2000) 294–307.[5] S.L. Hakimi, Optimum locations of switching centers and theabsolute centers and medians of a graph, Oper. Res. 12 (1964)450–469.[6] S.L. Hakimi, Optimum distribution of switching centers ina communication network and some related graph-theoreticproblems, Oper. Res. 13 (1965) 462–475.[7] J. Halpern, The location of a cent-dian convex combination onan undirected tree, J. Regional Sci. 16 (1976) 237–245.[8] J. Halpern, Finding minimal center-median convex combination(cent-dian) of a graph, Management Sci. 24 (1978) 535–544.[9] J. Halpern, Duality in the cent-dian of a graph, Oper. Res. 28(1980) 722–735.[10] G.Y. Handler, Medi-centers of a tree, Transportation Sci. 19(1985) 246–260.[11] J. Kalcsics, S. Nickel, J. Puerto, A. Tamir, Algorithmic resultsfor ordered median problems, Oper. Res. Lett. 30 (2002) 149–158.[12] N. Megiddo, Linear programming in linear time when thedimension is fixed, J. ACM 31 (1984) 114–127.[13] N. Megiddo, A. Tamir, Linear time algorithms for some separablequadratic programming problems, Oper. Res. Lett. 13(1993) 203–211.

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