Improved LP-based Approximation Algorithms for Facility Location with Hard Capacities
IImproved LP-based Approximation Algorithms forFacility Location with Hard Capacities
Mong-Jen KaoDepartment of Computer Science and Information Engineering,National Chung-Cheng University, Taiwan. [email protected]
Abstract
We present LP-based approximation algorithms for the capacitated facility location problem(CFL), a long-standing problem with intriguing unsettled approximability in its literature datedback to the 90s. We present an elegant iterative rounding scheme for the MFN relaxation thatyields an approximation guarantee of (cid:0)
10 + √ (cid:1) / ≈ . We consider the facility location problem with hard capacities (CFL), a long-standing problem withintriguing unsettled complexity and literature dated back to the 90s. In this problem, we are givena set F of facilities, a set D of clients, and a distance metric c defined over F ∪ D . Each i ∈ F isassociated with an open cost o i and a capacity u i , which is the number of clients it can serve whenopened up. The cost of serving a client j using a facility i equals the distance between them. Thegoal is to determine a set of facilities to open up and an assignment of the clients to the openedfacilities which respects their capacity limits so that the total cost is minimized.The CFL problem was first considered by Shmoys, Tardos, and Aardal [13]. First, for facilitieswith uniform capacities, Koropolu et al. [8] showed that the local search heuristic proposed byKuehn and Hamburger [9] yields a constant factor approximation. Chudak and Williamson [6]improved the analysis of Korupolu et al. [8] and obtained a (6 , , , , a r X i v : . [ c s . D S ] F e b n contrast to the rich LP-based toolboxes developed for the uncapacitated facility locationproblem (UFL), the fact that no LP-based algorithms with constant approximation guarantee wereknown for CFL was intriguing and surprising. In fact, devising an LP-based approximation withconstant guarantee for CFL was listed as one of ten open problems in the textbook on approximationalgorithm due to Williamson and Shmoys [14]. This problem was resolved by the notable work ofAn, Singh, and Svensson [3], in which a strong multi-commodity flow network (MFN) relaxationwith constant integrality gap is presented. The approximation guarantee obtained in this work,however, is in the order of couple hundreds, and it remains an open problem to devise a betterLP-based guarantee for CFL or a better integrality gap for the MFN relaxation.In the pursuit of settling down the approximability of CFL, an important variation betweenthe general problem and the case with uniform capacities is when we have cardinality-type facilitycosts, i.e., uniform facility cost, This was studied by Levi, Shmoys, and Swamy [10], in whichan LP-based 5-approximation was presented. Interestingly, the approximation ratio of 5 not onlyremains the best known ratio for a long time since 2004, but also coincide with the best knownapproximation result obtained later for the general CFL problem [4] using local search heuristic. In this paper we present improved LP-based approximation algorithms for the general CFL problemand the case with cardinality facility costs. Our first result is the following theorem.
Theorem 1.
There is an LP-based algorithm for CFL that produces a (cid:0)
10 + √ (cid:1) / ≈ . Theorem 2.
There is an LP-based algorithm for CFL-CFC that produces a 4-approximation inpolynomial-time.This surpasses the decades-old ratio of 5 due to Levi et al. which ages up since 2004 and untiesthe match to the best approximation for CFL that is obtained via local search in 2012 [4]. Ourresult for CFL-CFC is built on a two-staged rounding scheme that combines the ideas developedin the past for both facility location and capacitated covering problems [1, 5, 7, 10].Our results considerably deepen the current understanding for the CFL problem and indicatethat an LP-based ratio strictly better than 5 for the general problem may still be possible to pursue.
Overview of the Algorithms
The core part of our results can be seen as procedures for rounding small instances incurred in theLP relaxations, i.e., the rounding decisions for the small facilities along with the assignments madeto them. Our rounding procedures aim at fractionally serving the clients while making sure thatthe rounded facilities of interests (if not all) are reasonably sparsely-loaded by the assignments.Our algorithm for CFL inherits the “round-or-separate“ framework from An et al. [3]. Insideeach iteration incurred in this framework, we are given as input a candidate solution and the task2s either to round this candidate solution integrally, or to produce a separating hyperplane so thatEllipsoid method can proceed. Our result for this part is the following theorem.
Theorem (Sketch) . Given a candidate solution ( x , y ) for the MFN relaxation and a target param-eter α with 0 < α ≤ /
3, we can compute in polynomial-time either (i) a hyperplane separating( x , y ) and the feasible region, or (ii) an integral solution ( x ∗ , y ∗ ) rounded from ( x , y ) withcost( x ∗ , y ∗ ) ≤ max (cid:26) α , − α (1 − α ) (cid:27) × cost( x, y ) . In our algorithm, instead of guaranteeing a firm fraction of demand be sent to the small in-stances, as was did in the previous work by An et al. [3], we emphasize that, with proper construc-tion, the large facilities are sparsely-loaded by the flow sent to them and are ready for a reasonablerounding blow-up of 1 / (1 − α ) when necessary.Our rounding procedure for the small facilities is built on the idea due to Abraham et al. [1] anda couple of prior works developed for uncapacitated facility location. In each iteration, the facilitywith the least “per-unit-flow-rerouting-cost“, i.e., the cost incurred if we relocate flow simultaneously and proportionally from all facilities in the vicinity to the facility, is selected to be rounded. Afterthe rounding process is done, we guarantees that at least (1 − α )-fraction of each client is assigned.To bound the extra rerouting cost incurred during the process, in the previous works [1, 3]a filtering technique is applied to ensure a low assignment radius for each client throughout theprocess. However, this technique inevitably causes a tremendous blowup in the resulting guarantee.In our result, we apply an implicit primal-dual schema for an exact pricing on the cost incurred.This bounds the assignment radius tightly for each client that gets reassigned. To ensure thatcorrect pricing is obtained for the need of our rounding process in each iteration, we apply thistechnique in an iterative manner. Together this yields our approximation for CFL.Our approximation algorithm for CFL-CFC is built on a two-staged rounding scheme thatcombines the ideas developed in the past for both facility location and covering problems [1,5,7,10].As is done in our first result, we apply primal-dual schema to ensure a tightly-bounded assignmentradius for each client throughout the rounding process. However, the large facilities are no longersparsely-loaded by the assignment given by natural LPs.To overcome this issue, when the residue demand of a client drops below the target threshold,we discard the residue client and redistribute part of it to the large facilities in the vicinity to formthe so-called “ outlier clients .“ The outlier clients participate in the rounding process and act asnormal clients except for that, there is no threshold for the outlier clients to be discarded, and weguarantee that the outlier clients will be fully-assigned for the final feasibility.To achieve this guarantee, in the second stage of the rounding process, we formulate the assign-ment problem of the remaining outlier clients as a carefully designed assignment LP. We apply atechnique, which is originally developed for the capacitated covering problems [5, 7], to show that,basic feasible solutions of this simple LP corresponds naturally to an implicit matching betweenthe large facilities at which the outlier clients reside and the non-integral facilities given by this LP.Together this yields a bound for our final rounding. Organization of this paper.
The rest of this paper is organized as follows. In the rest ofthis section we describe the preliminaries necessary to present our approximation algorithms. Wepresent our approximation algorithm for CFL in Section 2, page 5, and our approximation algorithmfor CFL-CFC in Section 3, page 8.The additional content is organized as follows. We establish our approximation guarantee forCFL in Section 4, page 11, and our guarantee for CFL-CFC in Section 5, page 25.3 .2 Preliminaries
In the CFL problem, we are given a set F of facilities, a set D of clients, and a distance metric c defined over F ∪ D . Each i ∈ F is associated with an open cost o i and a capacity u i , which is thenumber of clients it can serve when opened up. A facility multiplicity function y : F → [0 ,
1] denotesthe decision whether each facility is selected to be opened up. A client assignment x : F ×D → [0 , j is assigned to facility i (served by facility i ). A solution forCFL consists of a multiplicity function y and an assignment function x such that the followingconditions are met: (a) (cid:80) i ∈F x i,j ≥
1, for each j ∈ D . (b) (cid:80) j ∈D x i,j ≤ u i · y i , for each i ∈ F . (c) x i,j ≤ y i , for each i ∈ F , j ∈ D . The cost of a solution ( x, y ) is defined to be ψ ( x, y ) := (cid:88) i ∈F o i · y i + (cid:88) i ∈F , j ∈D c i,j · x i,j . Given an instance Ψ = ( F , D , c , o , u ) of CFL, the goal of this problem is to compute an integralsolution ( x, y ) such that ψ ( x, y ) is minimized. The MFN Relaxation.
As the natural LP formulation for CFL is known to have an unboundedintegrality gap even for simple settings, An, Singh, and Svensson [3] designed a strong LP relax-ation based on multicommodity flow networks (MFN). The idea is to impose Knapsack-cover typeconstraints, formulated as reassignable partial assignments given as free in each qualifying teston both the facility values and the assignments of the clients. In the following we introduce theframework and the construction of this relaxation.
Definition 1 (Partial Assignments) . A partial assignment g is a function g : F × D → [0 , g is said to be valid if (i) (cid:80) i ∈F g i,j ≤
1, for each j ∈ D , and (ii) (cid:80) j ∈D g i,j ≤ u i ,for each i ∈ F .Given a candidate fractional solution ( x, y ) and a valid partial assignment g , An, Singh, andSvensson [3] designed the following multi-commodity flow network, denoted MFN Ψ ( x, y, g ), as aqualifying test for the candidate solution ( x, y ). Definition 2 (Multi-commodity Flow Network) . For a valid partial assignment g and a candidatesolution ( x, y ), MFN Ψ ( x, y, g ) is a multicommodity flow network defined as follows. • Each client j ∈ D corresponds to two nodes j s and j t in the network and is associated witha commodity j with source-sink pair ( j s , j t ) and demand r ( g ) j := 1 − (cid:80) i ∈F g i,j . • Each facility i ∈ F corresponds to two nodes i and i t that are connected by an arc ( i, i t ) ofcapacity u ( g ) i := y i · (cid:16) u i − (cid:80) j ∈D g i,j (cid:17) . • For each j ∈ D and each i ∈ F , there is an arc ( j s , i ) of capacity x i,j , an arc ( i, j s ) of capacity g i,j , and an arc ( i t , j t ) of capacity r ( g ) j · y i .For any i ∈ F , j ∈ D , let P ( g )( x,y ) ( i, j ) to denote the set of paths in MFN Ψ ( x, y, g ) for commodity j to sink via i t . The superscript ( g ) and the subscript ( x, y ) is omitted when there is no confusionin the context. Let P := ∪ i ∈F ,j ∈D P ( i, j ) to denote the set of possible paths. See also Figure 1 foran illustration on the construction of MFN Ψ ( x, y, g ) and the corresponding constraints. Lemma 3 (An, Singh, Svensson [3]) . Given an instance Ψ = ( F , D , c , o , u ) of CFL, the constraintsdefined by MFN Ψ ( x, y ) := { MFN Ψ ( x, y, g ) feasible : ∀ valid g } is a valid relaxation for integralsolutions on Ψ. Furthermore, the separation problem for the feasibility of MFN Ψ ( x, y, g ) for anyvalid g can be answered in weakly polynomial-time, and a basic feasible flow can be obtained.4 s j s j sn i i i m i t i t i tm · · ·· · ·· · · j t j t j tn · · · x i,j g i,j u ( g ) i r ( g ) j y i demand r ( g ) j sink of j (cid:88) i ∈F , p ∈ P ( i,j ) f p ≥ r ( g ) j , ∀ j ∈ D , (1.a) (cid:88) p ∈ P : ( j s ,i ) ∈ p f p ≤ x i,j , ∀ i ∈ F , j ∈ D , (1.b) (cid:88) p ∈ P : ( i,j s ) ∈ p f p ≤ g i,j , ∀ i ∈ F , j ∈ D , (1.c) (cid:88) j ∈D , p ∈ P ( i,j ) f p ≤ u ( g ) i · y i , ∀ i ∈ F , (1.d) (cid:88) p ∈ P ( i,j ) f p ≤ r ( g ) j · y i , ∀ i ∈ F , j ∈ D , (1.e) f p ≥ , ∀ p ∈ P. (1.f)Figure 1: The construction of MFN Ψ ( x, y, g ) and the corresponding LP constraints. In this section we describe our approximation algorithm for CFL and prove Theorem 1. Ouralgorithm inherits the round-or-separate framework used in [3] for producing candidate solutionsthat can either be rounded or separated. For completeness we first describe the details of theframework in the following. Then we present our modified construction for obtaining sparsely-loaded flow and our iterative rounding scheme.
The Round-or-Separate Framework.
Let Ψ = ( F , D , o , c , u ) be an instance of CFL. Thealgorithm starts by guessing the cost of the optimal solution using a standard binary search. Foreach guess, say, γ , the algorithm applies the Ellipsoid algorithm on LP-(MFN) in Figure 2 withcost γ . For each separation problem incurred in the Ellipsoid algorithm, say, for ( x (cid:48) , y (cid:48) ), weapply Theorem 4, which is stated below, for either a separating hyperplane or an integral solution (cid:0) x ( γ ) , y ( γ ) (cid:1) with the claimed approximation guarantee. MFN Ψ ( x, y ) ,ψ ( x , y ) ≤ γ, x ∈ [0 , F×D , y ∈ [0 , F . Figure 2: LP-(MFN).When an integral solution is successfully rounded or whenthe Ellipsoid algorithm concludes the infeasibility of ( x (cid:48) , y (cid:48) ),the algorithm proceeds to the next iteration of the binary searchprocess until the desired precision is attained. When the binarysearch process terminates, the algorithm outputs (cid:0) x ( γ ) , y ( γ ) (cid:1) for the smallest γ for which an integral solution is successfullyrounded when γ is considered.Our result for CFL is the following theorem. Theorem 4.
Given a candidate solution ( x (cid:48) , y (cid:48) ) for LP-(MFN) and a target parameter α with0 < α ≤ /
3, we can compute in polynomial-time either (i) a separating hyperplane for ( x (cid:48) , y (cid:48) )and LP-(MFN), or (ii) an integral solution ( x ∗ , y ∗ ) rounded from ( x (cid:48) , y (cid:48) ) for Ψ with ψ ( x ∗ , y ∗ ) ≤ max (cid:26) α , − α (1 − α ) (cid:27) × ψ ( x (cid:48) , y (cid:48) ) .
5o prove Theorem 1, it suffices to prove the statement of Theorem 4 and observe that the specificchoice of α := (cid:0) − √ (cid:1) /
11 gives the claimed result. We also note that, it suffices to considerthe MFN-type constraints as the remaining constraints in LP-(MFN) can be verified directly.In the rest of this section, we describe our iterative rounding process that establishes Theorem 4.We complete the proof in Section 4, page 11.
Obtaining a Sparsely-Loaded Flow or a Separating Hyperplane.
Let ( x (cid:48) , y (cid:48) ) be a candi-date solution and 0 < α ≤ / I := { i ∈ F : 0 < y (cid:48) i < α } and U := { i ∈ F : y (cid:48) i ≥ α } . The facilities in U are further classifiedinto two categories. Let U ( > ) := i ∈ U : (cid:88) j ∈D x (cid:48) i,j > (1 − α ) · u i and U ( ≤ ) := U \ U ( > ) . Provided that the facilities in U are to be rounded up in the approximatesolution, we know that the assignments to U ( ≤ ) are ready for rounded up as well.Consider the maximum b-matching problem in the bipartite graph G = (cid:0) D , U ( > ) , E (cid:1) , wherefor each j ∈ D and i ∈ U ( > ) , there exists an edge ( j, i ) in E with capacity x (cid:48) i,j / (1 − α ). Solve themaximum b-matching problem on G for an optimal assignment h .We say that a client j ∈ D is partially-assigned in h if (cid:80) i ∈ U ( > ) h i,j < P between D and U ( > ) that starts at a partially-assigned client j ∈ D is an augmenting path if the following holds. • For each ( j (cid:48) , i (cid:48) ) ∈ P , where j (cid:48) ∈ D , i (cid:48) ∈ U ( > ) , we have h i (cid:48) ,j (cid:48) < x (cid:48) i,j / (1 − α ). • For each ( i (cid:48) , j (cid:48) ) ∈ P , where i (cid:48) ∈ U ( > ) , j (cid:48) ∈ D , we have h i (cid:48) ,j (cid:48) > j is a way of increasing itsassignment without altering the optimality of maximum b-matching between D and U ( > ) .We say that a facility i ∈ U ( > ) is tightly-occupied if there exists an augmenting path betweena partially-assigned client and i . For each i ∈ F , j ∈ D , define the partial assignment g i,j as g i,j := (cid:40) h i,j , if i ∈ U ( > ) is tightly-occupied,0 , otherwise.Apply Lemma 3 for either a basic feasible flow f or a separating hyperplane for MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ),where y (cid:48)(cid:48) i := 1 for all i ∈ U and y (cid:48)(cid:48) i := y (cid:48) i otherwise. If a violated constraint is found, report it tothe Ellipsoid algorithm. Otherwise, we proceed to round the flow f .Since feasible solutions of MFN Ψ ( x (cid:48) , y (cid:48) , g ) are also feasible for MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ), hyperplanesseparating MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ) also separate MFN Ψ ( x (cid:48) , y (cid:48) , g ). Furthermore, since f is a basic feasiblesolution, it follows that the number of paths on which nonzero flow are sent in f is at most thenumber of constraints in MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ), which is polynomial in F and D . Rounding the Small Instance via Sparsely-Loaded Flow.
In the following we describe ouriterative rounding process for facilities in I using the flow f obtained in the previous step.During the rounding process, the algorithm maintains a parameter tuple Ψ (cid:48) = ( F (cid:48) , D (cid:48) , r (cid:48) ) whichdenotes the remaining instance to be processed, where r (cid:48) is the residue demand of each j ∈ D (cid:48) .The algorithm updates the parameter tuple gradually during the process until D (cid:48) becomes empty.6nitially, for each j ∈ D , the residue demand of j is defined to be r (cid:48) j := (cid:80) i ∈ I, p ∈ P (cid:48) ( i,j ) f p , where P (cid:48) ( i, j ) is the set of paths for which nonzero flow for commodity j is sent to sink via i in f . Thealgorithm sets F (cid:48) := I and D (cid:48) := { j ∈ D : r (cid:48) j > α · r j } , where r j := 1 − (cid:80) i ∈ U ( > ) g i,j is thedemand to be sent in MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ) for commodity j ∈ D .For any i ∈ F (cid:48) with (cid:80) j ∈ D (cid:48) x i,j > x , y ) of interests, define the per-unit-assignmentrerouting cost to i as θ ( i, D (cid:48) , x , y ) := 1 (cid:80) j ∈ D (cid:48) x i,j · · o i · y i + 2 · (cid:88) j ∈ D (cid:48) c i,j · x i,j . Provided that ( x , y ) is optimal for LP-(M) given in Figure 3, standard complementary slacknessguarantees that c i,j ≤ λ j for all x i,j >
0, where λ comes from an optimal solution for LP-(DM).Furthermore, it follows that 3 · o i · y i + 2 · (cid:80) j ∈ D (cid:48) c i,j · x i,j = o i · y i + 2 · (cid:80) j ∈ D (cid:48) λ j · x i,j , and θ ( i, D (cid:48) , x, y )corresponds implicitly and naturally to the cost incurred if we reroute the assignments which areoriginally assigned to facilities in the vicinity of i simultaneously and proportionally to i .Minimize ψ ( x, y ) LP-(M) (cid:88) i ∈ F (cid:48) x i,j = r (cid:48) j , ∀ j ∈ D (cid:48) , (cid:88) j ∈ D (cid:48) x i,j ≤ u i · y i , ∀ i ∈ F (cid:48) ,x i,j ≤ α − α · r j · y i , ∀ i ∈ F (cid:48) , j ∈ D (cid:48) ,y i ≤ − α , ∀ i ∈ F (cid:48) ,x i,j ≥ , y i ≥ , ∀ i ∈ F (cid:48) , j ∈ D (cid:48) . LP-(DM)Maximize (cid:88) j ∈ D (cid:48) r (cid:48) j · λ j − (cid:88) i ∈ F (cid:48) · (1 − α ) · η i λ j ≤ β i + Γ i,j + c i,j , ∀ i ∈ F (cid:48) , j ∈ D (cid:48) ,u i · β i + 2 α − α · (cid:88) j ∈ D (cid:48) r j · Γ i,j ≤ o i + η i , ∀ i ∈ F (cid:48) ,λ j ∈ R , β i , Γ i,j , η i ≥ , ∀ i ∈ F (cid:48) , j ∈ D (cid:48) . Figure 3: The natural LP formulations for our iterative rounding process.In the following, we describe the rounding process in details. In each iteration, the algorithmsolves LP-(M) on the current parameter tuple Ψ (cid:48) for optimal ( x † , y † ) and selects a facility i ∈ F (cid:48) to form a cluster. Depending on the status of the facilities in F (cid:48) , we have two cases. • If y † i = (1 − α ) / i ∈ F (cid:48) , then the algorithm picks an arbitrary i with y † i = (1 − α ) / σ ( i ) k = 0 for all k ∈ F (cid:48) \ { i } . • If y † i < (1 − α ) / i ∈ F (cid:48) , then the algorithm selects among the facilities i ∈ F (cid:48) with (cid:80) j ∈ D (cid:48) x † i,j >
0, the facility i with the minimum θ ( i, D (cid:48) , x † , y † ). Intuitively, the facility i has the lowest per-unit-assignment rerouting cost, and any other facility in F (cid:48) can afford thererouting cost within the cluster centered at i .For each j ∈ D (cid:48) , let δ ( i ) j := (cid:16) (1 − α ) / (2 y † i ) − (cid:17) · x † i,j denote the amount of assignment to begathered from the vicinity of facility i to i via client j .7or each k ∈ F (cid:48) \ { i } , define the contribution (fraction) of k towards i to be σ ( i ) k := 1 (cid:80) (cid:96) ∈ D (cid:48) x † k,(cid:96) · (cid:88) j ∈ D (cid:48) σ ( i ) k,j , where σ ( i ) k,j := x † k,j (cid:80) (cid:96) ∈ F (cid:48) \{ i } x † (cid:96),j · δ ( i ) j isthe contribution of k to be sent via client j . Note that, from the definition it follows that (cid:80) k ∈ F (cid:48) \{ i } σ ( i ) k,j = δ ( i ) j , and the amount of flow to be gathered via j can be fulfilled.The algorithm updates the parameter tuple Ψ (cid:48) as follows. The algorithm removes i from F (cid:48) andsets r (cid:48) j := (cid:80) k ∈ F (cid:48) (1 − σ ( i ) k ) · x † k,j for all j ∈ D (cid:48) . For each j ∈ D (cid:48) with r (cid:48) j ≤ α · r j , the algorithmremoves j from D (cid:48) . Then the algorithm proceeds to the next iteration until D (cid:48) becomes empty. Final Output.
When D (cid:48) becomes empty, the algorithm forms the integral multiplicity function y ∗ by setting y ∗ i := 1 for all i ∈ U ∪ ( I \ F (cid:48) ) and y ∗ i := 0 otherwise. The algorithm solves the min-cost assignment problem on D and F ∗ := (cid:8) i ∈ F : y ∗ i = 1 (cid:9) for an optimal integral assignment x ∗ and reports ( x ∗ , y ∗ ) as the claimed integral solution. -Approximation for CFL-CFC In the following, we describe our approximation algorithm A for CFL-CFC and prove Theorem 2.Let Ψ = ( F , D , c , u ) be an instance of CFL-CFC.For the ease of presentation, for any A ⊆ F , any assignment x , and any j ∈ D , we use N ( A,x ) ( j ) := (cid:8) i ∈ A : x i,j > (cid:9) to denote the set of facilities in A to which j is assigned to in x .Similarly, for any B ⊆ D and any i ∈ F , we use N ( B,x ) ( i ) := (cid:8) j ∈ B : x i,j > (cid:9) to denote the setof clients in B that is assigned to i in x .Solve LP-(N) and its dual LP, given below in Figure 4, on Ψ for optimal primal and dualsolutions ( x (cid:48) , y (cid:48) ) and ( α , β , Γ , η ). It follows that, α j ≥ c i,j for each i ∈ F , j ∈ D with x (cid:48) i,j >
0. Wewill use the fact that α j is a valid estimation on the assignment radius for each j ∈ D in x (cid:48) .min (cid:88) i ∈F y i + (cid:88) i ∈F ,j ∈D c i,j · x i,j LP-(N) (cid:88) i ∈F x i,j ≥ , ∀ j ∈ D , (cid:88) j ∈D x i,j ≤ u i · y i , ∀ i ∈ F , ≤ x i,j ≤ y i , ∀ i ∈ F , j ∈ D , ≤ y i ≤ , ∀ i ∈ F , max (cid:88) j ∈D α j − (cid:88) i ∈F η i LP-(DN) α j ≤ β i + Γ i,j + c i,j , ∀ i ∈ F , j ∈ D ,u i · β i + (cid:88) j ∈D Γ i,j ≤ η i , ∀ i ∈ F ,α j , β i , Γ i,j , η i ≥ , ∀ i ∈ F , j ∈ D . Figure 4: The natural LP formulations for CFL-CFC.
Initial Classification.
Let I := (cid:8) i ∈ F : 0 < y (cid:48) i < (cid:9) and U := (cid:8) i ∈ F : y (cid:48) i ≥ (cid:9) . Theclients in D are divided into three categories, namely, those that are served merely by I , those8hat are served jointly by I and U , and those that are served merely by U . Let J ( I ) := (cid:110) j ∈ D : x (cid:48) i,j = 0 for all i ∈ U (cid:111) , J ( ↔ ) := (cid:26) j ∈ D : min (cid:18) max i ∈ I x (cid:48) i,j , max i ∈ U x (cid:48) i,j (cid:19) > (cid:27) , and J ( U ) := D \ (cid:0) J ( I ) ∪ J ( ↔ ) (cid:1) denote the clients in the three categories, respectively. Note that, inaddition to the rounding problem for facilities in I , we need to resolve the rounding problem forassignments of the clients in the first two categories as well. Our Rounding Process.
Let F (cid:48) and D (cid:48) be the set of facilities and the set of clients yet to beprocessed, and x ∗ be an intermediate assignment function our algorithm will maintain during theprocess. Initially, F (cid:48) := I , D (cid:48) := J ( I ) ∪ J ( ↔ ) , and x ∗ := 0.Our rounding algorithm consists of two phases. In the first phase, it proceeds in iterations toform clusters. In each of such iterations, the algorithm checks if (cid:80) i ∈ F (cid:48) x (cid:48) i,j ≥ / j ∈ D (cid:48) . If not, the algorithm makes it so by repeatedly removing small clients from D (cid:48) ∩ J ( ↔ ) andredistributing their demand to facilities in U to form a set of outlier clients . We use H to denotethe set of outlier clients created in this step and H (cid:48) ⊆ H to denote those that are created but notyet processed by the rounding algorithm. Initially H := ∅ and H (cid:48) := ∅ .When (cid:80) i ∈ F (cid:48) x (cid:48) i,j ≥ / j ∈ D (cid:48) , a cluster centered at a client is formed andpossibly rounded, depending on whether or not the client forming the cluster is outlier, and thecorresponding parts of the cluster are removed from F (cid:48) , D (cid:48) , and H (cid:48) , respectively. The clusterforming process repeats until D (cid:48) ∪ H (cid:48) becomes empty.In the second phase the algorithm rounds the remaining clusters centered at the outlier clientsusing an assignment LP to form an integral multiplicity function. In the following we describe thethree components of our rounding algorithm in details. Creating the Outlier Clients.
When (cid:80) i ∈ F (cid:48) x (cid:48) i,j < / j ∈ J ( ↔ ) ∩ D (cid:48) , the algorithmdiscards j and relocates some of the remaining demand to facilities in N ( U,x (cid:48) ) ( j ) to form outlierclients in a way as if the demand were originated from these facilities. U b j ∈ J ( ↔ ) wj w I Figure 5: Construction ofthe outlier clients.Let r (cid:48) j := min { (cid:80) i ∈ F (cid:48) x (cid:48) i,j , (cid:80) i ∈ U x (cid:48) i,j } be the amount of residuedemand of j to be redistributed. For each w ∈ N ( U,x (cid:48) ) ( j ), we createa client j w at the facility w with demand d j w := r (cid:48) j · x (cid:48) w,j / (cid:80) i ∈ U x (cid:48) i,j and add j w to both H and H (cid:48) . We set α j w := α j + c w,j . For each i ∈ N ( F (cid:48) ,x (cid:48) ) ( j ), we further set x (cid:48) i,j w := d j w · x (cid:48) i,j / (cid:80) i ∈ F (cid:48) x (cid:48) i,j . See also Figure 5 for an illustration on the construction of j w .It follows by definition that d j w ≤ x (cid:48) w,j , and the residue demand of j is fully redistributed since (cid:80) w ∈ N ( U,x (cid:48) ) ( j ) d j w = r (cid:48) j . We note that (cid:80) i ∈ N ( F (cid:48) ,x (cid:48) ) ( j ) x (cid:48) i,j w = d j w , and the demand of j w is fully-assigned.After the outlier client j w is created for each w ∈ N ( U,x (cid:48) ) ( j ), thealgorithm removes j from D (cid:48) and set x (cid:48) i,j to be zero for all i ∈ F (cid:48) . Forming Clusters and Rounding.
When (cid:80) i ∈ F (cid:48) x (cid:48) i,j ≥ / j ∈ D (cid:48) , the algorithmselects a client j ∈ D (cid:48) ∪ H (cid:48) that minimizes α j to form a cluster. Depending on the set to which j belongs, the algorithm proceeds differently. 9 If j ∈ H (cid:48) , then a cluster centered at j with satellite facilities N ( F (cid:48) ,x (cid:48) ) ( j ) is formed. We use B ( j ) := N ( F (cid:48) ,x (cid:48) ) ( j ) to denote the set of satellite facilities at this moment. The algorithm thenremoves j from H (cid:48) and B ( j ) from F (cid:48) . The rounding problem for this cluster is handled laterin the second phase of the algorithm. • If j ∈ D (cid:48) , the algorithm further selects a facility i ∈ N ( F (cid:48) ,x (cid:48) ) ( j ) with the maximum u i . Thealgorithm reroutes the assignment and facility values from the facilities in N ( F (cid:48) ,x (cid:48) ) ( j ) to i asfollows. Let δ i := (cid:0) − y (cid:48) i (cid:1) / (cid:80) k ∈ N ( F (cid:48) ,x (cid:48) ) ( j ) \{ i } y (cid:48) k be the factor to reroute from N ( F (cid:48) ,x (cid:48) ) ( j ) \{ i } .For each (cid:96) ∈ N ( F (cid:48) ,x (cid:48) ) ( j ) \ { i } , the algorithm scales y (cid:48) (cid:96) down by (1 − δ i ). For each k ∈ N ( D (cid:48) ∪ H (cid:48) ,x (cid:48) ) ( (cid:96) ), the algorithm further scales x (cid:48) (cid:96),k down by (1 − δ i ) and increases x ∗ i,k by thesame amount x (cid:48) (cid:96),k has decreased in this step.The algorithm increases x ∗ i,k by x (cid:48) i,k for each k ∈ D (cid:48) and then removes i from F (cid:48) .When the above updates are done, for each k ∈ J ( I ) ∩ D (cid:48) with (cid:80) i ∈ F (cid:48) x (cid:48) i,k < /
2, the algorithmremoves k from D (cid:48) and sets x (cid:48) i,k to be zero for all i ∈ F (cid:48) . Then the algorithm proceeds to the nextiteration until D (cid:48) ∪ H (cid:48) becomes empty. min (cid:88) i ∈ G y i + (cid:88) i ∈ G, j ∈ U c i,j · x i,j (cid:88) i ∈ G x i,j ≥ d j , ∀ j ∈ U, (cid:88) j ∈ U x i,j ≤ u i · y i , ∀ i ∈ G,y i ≤ , ∀ i ∈ G,x i,j , y i ≥ , ∀ i ∈ G, j ∈ U. Figure 6: LP-(O).
Rounding the Outlier Clusters.
When D (cid:48) ∪ H (cid:48) becomes empty, the algorithm proceeds to handle therounding problem for clusters centered at clients in H ,which is formulated as an instance of CFL-CFC withfacility set G := (cid:83) j ∈ H B ( j ) and client set U as follows.Each w ∈ U is associated with a demand d w definedas d w := (cid:88) k ∈ H,k located at w (cid:88) i ∈ B ( k ) , (cid:96) ∈D∪ H t (cid:48) (cid:96) · x (cid:48) i,(cid:96) , where the scaling factor t (cid:48) (cid:96) is defined as t (cid:48) (cid:96) := 1 (cid:80) k ∈ I x ∗ k,(cid:96) + (cid:80) k ∈ G x (cid:48) k,(cid:96) · (cid:32) − (cid:88) i ∈ U x (cid:48) i,(cid:96) − r (cid:48) (cid:96) (cid:33) if (cid:96) ∈ D and (cid:80) k ∈ I x ∗ k,(cid:96) + (cid:80) k ∈ G x (cid:48) k,(cid:96) >
0, and t (cid:48) (cid:96) := 1 otherwise. Intuitively, t (cid:48) (cid:96) is the factor for whichthe assignments made for (cid:96) should be scaled up, and the demand d w of each w ∈ U is the totaldemand of the clusters centered at the outlier clients located at w .The algorithm solves the assignment LP for the formulated instance, denoted LP-(O) and givenin Figure 6, for a basic optimal solution ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ). Final Output.
Define the multiplicity function y ∗ as follows. Let y ∗ i := 1 for i ∈ I \ F (cid:48) , y ∗ i := (cid:100) y (cid:48)(cid:48) i (cid:101) for i ∈ G , and y ∗ i := 0 otherwise. The algorithm solves the min-cost assignment problemon D and F ∗ := { i ∈ F : y ∗ i = 1 } for an optimal integral assignment x † , and reports ( x † , y ∗ ) asthe approximation solution.The following theorem, which is proved in Section 5, page 25, bounds the cost incurred by( x † , y ∗ ) and completes the proof for Theorem 2. Theorem 5.
Let Ψ be the input instance of CFL-CFC and ( x (cid:48) , y (cid:48) ) be an optimal solution for LP-(N) on Ψ. Algorithm A computes in polynomial time a feasible integral solution ( x † , y ∗ ) with ψ ( x † , y ∗ ) ≤ · ψ ( x (cid:48) , y (cid:48) ) . Proof of Theorem 4.
It is clear that the first phase of the algorithm runs in polynomial time. Since y (cid:48) ≤ y (cid:48)(cid:48) , the feasibleregion of MFN Ψ ( x (cid:48) , y (cid:48) , g ) is contained in that of MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ). Hence, any hyperplane sep-arating ( x (cid:48) , y (cid:48) ) from MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ) also separates ( x (cid:48) , y (cid:48) ) from MFN Ψ ( x (cid:48) , y (cid:48) , g ). To completethe proof of Theorem 4, it suffices to prove the following lemma. Lemma 6.
Provided that
MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ) is feasible, our iterative rounding process in the secondphase is well-defined and terminates in polynomial time. Furthermore, the feasible region of themin-cost assignment problem on D and F ∗ is non-empty, and ψ ( x ∗ , y ∗ ) ≤ max (cid:26) α , − α (1 − α ) (cid:27) × ψ ( x (cid:48) , y (cid:48) ) . The proof for Lemma 6 is outlined as follows. First, we define in Section 4.1 the intermediate flow f (cid:48) which corresponds to the assignments made to facilities in U and in Section 4.2 the intermediateassignment x (cid:48)(cid:48) which corresponds to the assignments made by our iterative rounding process tofacilities in I \ F (cid:48) , respectively. We prove in the same section that the iterative rounding process iswell-defined and runs in polynomial time.In Section 4.3, page 18, we show that, the assignment x (cid:48)(cid:48)(cid:48) defined jointly from f (cid:48) and x (cid:48)(cid:48) ,together with y ∗ forms a feasible solution for Ψ. This shows that the feasible region of the min-costassignment problem on D and F ∗ is nonempty.For the approximation guarantee, in Section 4.4, page 20, we analyze the cost incurred by x (cid:48)(cid:48)(cid:48) and y ∗ , and show that it satisfies the claimed approximation guarantee. Note that, this completesthe proof of this lemma since x ∗ is the optimal min-cost assignment between D and F ∗ . j s j s j sn i i i m i t i t i tm · · ·· · ·· · · j t j t j tn · · · x i,j g i,j u ( g ) i r ( g ) j y i demand r ( g ) j sink of j (cid:88) i ∈F , p ∈ P ( i,j ) f p ≥ r ( g ) j , ∀ j ∈ D , (2.a) (cid:88) p ∈ P : ( j s ,i ) ∈ p f p ≤ x i,j , ∀ i ∈ F , j ∈ D , (2.b) (cid:88) p ∈ P : ( i,j s ) ∈ p f p ≤ g i,j , ∀ i ∈ F , j ∈ D , (2.c) (cid:88) j ∈D , p ∈ P ( i,j ) f p ≤ u ( g ) i · y i , ∀ i ∈ F , (2.d) (cid:88) p ∈ P ( i,j ) f p ≤ r ( g ) j · y i , ∀ i ∈ F , j ∈ D , (2.e) f p ≥ , ∀ p ∈ P. (2.f)Figure 7: (Restate for further reference) The construction of MFN Ψ ( x, y, g ) and the correspondingLP constraints. 11 .1 The Intermediate Flow f (cid:48) for Assignments to U In this section we define the intermediate flow f (cid:48) which corresponds to the assignments made tofacilities in U . Consider the first phase of the algorithm for obtaining the partial assignment g andthe corresponding feasible flow f for MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ).Let D ( (cid:96) ) denote the set of clients that are fully-assigned in h and unreachable from any partially-assigned clients using augmenting paths. Let D ( p ) := D \ D ( (cid:96) ) denote the set of clients that areeither partially-assigned or reachable by partially-assigned clients via augmenting paths.The following proposition shows that, in the flow f , none of j (cid:48) ∈ D ( p ) has sent nonzero flowthat passes through j s for any j ∈ D ( (cid:96) ) , which implies that for any i ∈ F and j ∈ D ( (cid:96) ) , the arc( j s , i ) carries only flow for commodity j . Proposition 7.
For any j ∈ D ( (cid:96) ) , i ∈ F , and any p ∈ P with ( j s , i ) ∈ p , we have f p > p ∈ (cid:83) i (cid:48) ∈F P ( i (cid:48) , j ), that is, f p > p is a path for commodity j . Proof.
Since j ∈ D ( (cid:96) ) , it follows that g k,j = 0 for all k ∈ F by the definition of g . Hence, p muststart from j s and is thereby a path for commodity j .Define the intermediate flow function f (cid:48) as follows. • For any i ∈ U ( > ) that is tightly-occupied and any j ∈ D , we define f (cid:48) p := ( 1 − α ) · g i,j , where p = j s → i → i t → j t . • For any i ∈ U ( > ) that is not tightly-occupied, we further consider two cases. – For any j ∈ D ( (cid:96) ) , we define f (cid:48) p := ( 1 − α ) · h i,j , where p = j s → i → i t → j t . – For any j ∈ D ( p ) and any p ∈ P ( i, j ), we define f (cid:48) p := f p . • Finally, for any i ∈ U ( ≤ ) , any j ∈ D , and any p ∈ P ( i, j ), we define f (cid:48) p := f p . For any i ∈ U , j ∈ D , let Φ (cid:48) ( i, j ) denote the set of paths with nonzero flow for commodity j tosink via i t in the flow functions f (cid:48) . The following lemma, which is valid from the construction of f (cid:48) , shows that the facilities in U are sparsely-loaded. Lemma 8.
For any i ∈ U , we have (cid:88) j ∈D , p ∈ Φ (cid:48) ( i,j ) f (cid:48) p ≤ ( 1 − α ) · u i . Proof of Lemma 8.
Depending on the category to which each i belongs, we consider the followingthree cases separately. • i ∈ U ( > ) is tightly-occupied. By the construction rules of f (cid:48) and the definition of g , we have (cid:88) j ∈D , p ∈ Φ (cid:48) ( i,j ) f (cid:48) p = (cid:88) j ∈D f (cid:48) j s → i → i t → j t = (1 − α ) · (cid:88) j ∈D g i,j = (1 − α ) · (cid:88) j ∈D h i,j = (1 − α ) · u i , where the last equality follows from the fact that there exists an augmenting path connectinga partially-assigned client to i , which implies that (cid:80) j ∈D h i,j = u i by the optimality of h .12 i ∈ U ( > ) is not tightly-occupied. Since i is not tightly-occupied, by the optimality of h itfollows that, h i,j = x (cid:48) i,j / (1 − α ) holds for any j ∈ D ( p ) . By the construction rules of f (cid:48) andProposition 7, we have (cid:88) j ∈ D ( p ) ,p ∈ Φ (cid:48) ( i,j ) f (cid:48) p = (cid:88) j ∈ D ( p ) , p ∈ P ( i,j ) f p ≤ (cid:88) j ∈ D ( p ) ,p ∈ P ( i,j ) with ( j s ,i ) ∈ p f p ≤ (cid:88) j ∈D ( p ) x (cid:48) i,j = (1 − α ) · (cid:88) j ∈D ( p ) h i,j , (3)where in the second last inequality we apply Constraint (2.b) due to the fact that f is afeasible flow for MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ). From the second construction rule of f (cid:48) , we have (cid:88) j ∈D ( (cid:96) ) f (cid:48) j s → i → i t → j t = (1 − α ) · (cid:88) j ∈D ( (cid:96) ) h i,j . (4)Combining (3) and (4) with the definition of Φ (cid:48) ( i, j ), we have (cid:88) j ∈D , p ∈ Φ (cid:48) ( i,j ) f (cid:48) p = (cid:88) j ∈ D ( p ) , p ∈ Φ (cid:48) ( i,j ) f (cid:48) p + (cid:88) j ∈ D ( (cid:96) ) , p ∈ Φ (cid:48) ( i,j ) f (cid:48) p ≤ (1 − α ) · u i , where the last inequality follows from the feasibility of h , i.e., (cid:80) j ∈D h i,j ≤ u i . • i ∈ U ( ≤ ) . By the third construction rule of f (cid:48) and the feasibility of f , we have (cid:88) j ∈D , p ∈ Φ (cid:48) ( i,j ) f (cid:48) p = (cid:88) j ∈D , p ∈ P ( i,j ) f p ≤ (cid:88) j ∈D x (cid:48) i,j ≤ (1 − α ) · u i , where in the last inequality we apply the definition of U ( ≤ ) .This completes the proof of this lemma. x (cid:48)(cid:48) for Facilities in I \ F (cid:48) Consider the rounding process in the second phase of the algorithm. For each i ∈ I \ F (cid:48) that isselected to be rounded by the algorithm, let Ψ ( i ) = ( F (cid:48) ( i ) , D (cid:48) ( i ) , r (cid:48) ( i ) ) denote the parameter tuplethe algorithm maintains in the beginning of the iteration when facility i is selected.Let Ψ (0) = (cid:0) F (cid:48) (0) , D (cid:48) (0) , r (cid:48) (0) (cid:1) denote the initial parameter tuple the algorithm constructs inthe beginning of the second phase. The following lemma, which shows that the feasible regionof LP-(M) on Ψ (0) is nonempty, is proved by verifying the corresponding constraints of LP-(M)combined with the fact that 0 < α ≤ / Lemma 9.
The solution (cid:0) x † (0) , y † (0) (cid:1) defined by x † (0) i,j := (cid:88) p ∈ P ( i,j ) f p for all i ∈ I, j ∈ D and y † (0) i := 1 − α α · y (cid:48) i for all i ∈ I is feasible for LP-(M) on the initial parameter tuple Ψ (0) .13in (cid:88) i ∈ F (cid:48) o i · y i + (cid:88) i ∈ F (cid:48) , j ∈ D (cid:48) c i,j · x i,j LP-(M)s.t. (cid:88) i ∈ F (cid:48) x i,j = r (cid:48) j , ∀ j ∈ D (cid:48) (M-1) (cid:88) j ∈ D (cid:48) x i,j ≤ u i · y i , ∀ i ∈ F (cid:48) (M-2)0 ≤ x i,j ≤ α − α · r j · y i , ∀ i ∈ F (cid:48) , j ∈ D (cid:48) , (M-3)0 ≤ y i ≤ − α , ∀ i ∈ F (cid:48) , (M-4)max (cid:88) j ∈ D (cid:48) r (cid:48) j · λ j − (cid:88) i ∈ F (cid:48) · (1 − α ) · η i LP-(DM)s.t. λ j ≤ β i + Γ i,j + c i,j , ∀ i ∈ F (cid:48) , j ∈ D (cid:48) , (DM-1) u i · β i + 2 α − α · (cid:88) j ∈ D (cid:48) r j · Γ i,j ≤ o i + η i , ∀ i ∈ F (cid:48) , (DM-2) λ j ∈ R , β i , Γ i,j , η i ≥ , ∀ i ∈ F (cid:48) , j ∈ D (cid:48) . (DM-3)Figure 8: (Restate for further reference) The natural LP formulation and its dual LP for ouriterative rounding process. Proof of Lemma 9.
We prove this lemma by verifying the constraints of LP-(M) separately. • For Constraint (M-1), consider any j ∈ D (cid:48) (0) . Since F (0) = I by definition, further applyingthe definition of r (cid:48) (0) j , we have (cid:88) i ∈ F (cid:48) (0) x † (0) i,j = (cid:88) i ∈ I, p ∈ P ( i,j ) f p = r (cid:48) (0) j . • For Constraint (M-2), consider any i ∈ F (cid:48) (0) . By the definition of D (cid:48) (0) and Constraint (2.d)from the fact that f is a feasible flow for MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ), it follows that (cid:88) j ∈ D (cid:48) (0) x † (0) i,j = (cid:88) j ∈ D (cid:48) (0) , p ∈ P ( i,j ) f p ≤ (cid:88) j ∈D , p ∈ P ( i,j ) f p ≤ u ( g ) i · y (cid:48) i . Since F (cid:48) (0) := I by definition and i ∈ F (cid:48) (0) , it follows that (cid:80) j ∈D g i,j = 0 by the way g isdefined and therefore u ( g ) i = u i . Moreover, observe that (1 − α ) / (2 α ) is strictly decreasing for α >
0. Since 0 < α ≤ /
3, it follows that y † (0) i := 1 − α α · y (cid:48) i ≥ − / / · y (cid:48) i = y (cid:48) i . (cid:88) j ∈ D (cid:48) (0) x † (0) i,j ≤ u i · y † (0) i . • For Constraint (M-3), consider any i ∈ F (cid:48) (0) and any j ∈ D (cid:48) (0) . Apply Constraint (2.e) bythe fact that f is a feasible flow for MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ) and the definition of y † (0) i and r j , wehave x † (0) i,j = (cid:88) p ∈ P ( i,j ) f p ≤ r ( g ) j · y (cid:48) i = 2 α − α · r j · y † (0) i . • For Constraint (M-4), consider any i ∈ F (cid:48) (0) . By the definition of y † (0) i and the fact that F (cid:48) (0) = I , which implies that y (cid:48) i < α , it follows that y † (0) i = 1 − α α · y (cid:48) i ≤ − α . This proves the lemma.Consider any i ∈ I \ F (cid:48) and the iteration when i is processed. Suppose for the moment thatthe feasible region of LP-(M) on Ψ ( i ) is nonempty. Let (cid:0) x † ( i ) , y † ( i ) (cid:1) denote the optimal solutionthe algorithm computes for LP-(M) on Ψ ( i ) . The following lemma shows that the scaled-downoperation in each iteration is well-defined. Lemma 10.
For any i ∈ I \ F (cid:48) with y † ( i ) i < (1 − α ) /
2, the following holds. • For any j ∈ D (cid:48) ( i ) , we have 0 ≤ δ ( i ) j ≤ (cid:88) (cid:96) ∈ F (cid:48) ( i ) \{ i } x † ( i ) (cid:96),j . • For any k ∈ F (cid:48) ( i ) \ { i } , we have σ ( i ) k ≤ Proof of Lemma 10.
Consider any j ∈ D (cid:48) ( i ) . Since (cid:0) x † ( i ) , y † ( i ) (cid:1) is feasible for LP-(M) on Ψ ( i ) , wehave y † ( i ) i ≤ (1 − α ) /
2, which implies that (1 − α ) / (2 · y † ( i ) i ) ≥
1. Hence δ ( i ) j := (cid:32) − α · y † ( i ) i − (cid:33) · x † ( i ) i,j ≥ . On the other hand, applying Constraint (M-3) of LP-(M), it follows that δ ( i ) j := 1 − α · y † ( i ) i · x † ( i ) i,j − x † ( i ) i,j ≤ α · r j − x † ( i ) i,j < r (cid:48) ( i ) j − x † ( i ) i,j = (cid:88) (cid:96) ∈ F (cid:48) ( i ) \{ i } x † ( i ) (cid:96),j , where in the second last inequality we apply the fact that r (cid:48) ( i ) j > α · r j holds for all j ∈ D (cid:48) ( i ) bythe design of the algorithm, and in the last equality we apply Constraint (M-1) of LP-(M). Thisproves the first part of this lemma. 15onsider any k ∈ F (cid:48) ( i ) \ { i } . By the conclusion of the first part, for any j ∈ D (cid:48) ( i ) , we have σ ( i ) k,j := δ ( i ) j (cid:80) (cid:96) ∈ F (cid:48) ( i ) \{ i } x † ( i ) (cid:96),j · x † ( i ) k,j ≤ x † ( i ) k,j , which implies that σ ( i ) k := 1 (cid:80) (cid:96) ∈ D (cid:48) ( i ) x † ( i ) k,(cid:96) · (cid:88) j ∈ D (cid:48) ( i ) σ ( i ) k,j ≤ . Let Ψ (cid:48)(cid:48) ( i ) = ( F (cid:48)(cid:48) ( i ) , D (cid:48)(cid:48) ( i ) , r (cid:48)(cid:48) ( i ) ) denote the updated parameter tuple the algorithm maintains atthe end of the iteration i . Note that, by the algorithm design, F (cid:48)(cid:48) ( i ) := F (cid:48) ( i ) \ { i } . The followinglemma shows that the feasible region of LP-(M) on the updated tuple Ψ (cid:48)(cid:48) ( i ) is nonempty andestablishes the feasibility of our iterative rounding process. Lemma 11.
For any i ∈ I \ F (cid:48) , the solution (cid:0) x (cid:48) † ( i ) , y (cid:48) † ( i ) (cid:1) defined by x (cid:48)† ( i ) k,j := (cid:16) − σ ( i ) k (cid:17) · x † ( i ) k,j for any k ∈ F (cid:48)(cid:48) ( i ) , j ∈ D (cid:48)(cid:48) ( i ) ,y (cid:48)† ( i ) k := (cid:16) − σ ( i ) k (cid:17) · y † ( i ) k for any k ∈ F (cid:48)(cid:48) ( i ) , is feasible for LP-(M) on the updated parameter tuple Ψ (cid:48)(cid:48) ( i ) . Proof of Lemma 11.
We prove this lemma by verifying the corresponding constraints in LP-(M).First, by Lemma 10, 1 − σ ( i ) k ≥ k ∈ F (cid:48)(cid:48) ( i ) , and (cid:0) x (cid:48) † ( i ) , y (cid:48) † ( i ) (cid:1) is thereby nonnegative. • For Constraint (M-1), consider any j ∈ D (cid:48)(cid:48) ( i ) . By the definition of x (cid:48)† ( i ) k,j for any k ∈ F (cid:48)(cid:48) ( i ) andthe definition of r (cid:48)(cid:48) ( i ) j , we have (cid:88) k ∈ F (cid:48)(cid:48) ( i ) x (cid:48)† ( i ) k,j = (cid:88) k ∈ F (cid:48)(cid:48) ( i ) (cid:16) − σ ( i ) k (cid:17) · x † ( i ) k,j = r (cid:48)(cid:48) ( i ) j . • For Constraint (M-2) and (M-3), consider any k ∈ F (cid:48)(cid:48) ( i ) and any j ∈ D (cid:48)(cid:48) ( i ) . Since (cid:0) x † ( i ) , y † ( i ) (cid:1) is feasible for LP-(M) on Ψ ( i ) and since F (cid:48) ( i ) ⊇ F (cid:48)(cid:48) ( i ) , D (cid:48) ( i ) ⊇ D (cid:48)(cid:48) ( i ) , we have (cid:88) j ∈ D (cid:48)(cid:48) ( i ) x † ( i ) k,j ≤ (cid:88) j ∈ D (cid:48) ( i ) x † ( i ) k,j ≤ u k · y † ( i ) k and x † ( i ) k,j ≤ α − α · r j · y † ( i ) k . Multiplying both inequalities by (cid:16) − σ ( i ) k (cid:17) and applying the definition of (cid:0) x (cid:48) † ( i ) , y (cid:48) † ( i ) (cid:1) , weobtain (cid:88) j ∈ D (cid:48)(cid:48) ( i ) x (cid:48)† ( i ) k,j ≤ u k · y (cid:48)† ( i ) k and x (cid:48)† ( i ) k,j ≤ α − α · r j · y (cid:48)† ( i ) k . • For Constraint (M-4), consider any k ∈ F (cid:48)(cid:48) ( i ) . Applying the definition of y (cid:48)† ( i ) k and the factthat (cid:0) x † ( i ) , y † ( i ) (cid:1) is feasible for LP-(M), we have y (cid:48)† ( i ) k = (cid:16) − σ ( i ) k (cid:17) · y † ( i ) k ≤ y † ( i ) k ≤ − α . F (cid:48) (0) := I and since exactly one facility is selected to be removed from F (cid:48) ineach iteration, it follows that the process repeats for at most | I | iterations before the set D (cid:48) becomesempty. Therefore, our iterative rounding process terminates in polynomial time.For any i ∈ I \ F (cid:48) and j ∈ D , define the assignment x (cid:48)(cid:48) as x (cid:48)(cid:48) i,j := (cid:88) k ∈ F (cid:48) ( i ) σ ( i ) k · x † ( i ) k,j , where in the above we extend the original definition of σ ( i ) and define σ ( i ) i := 1 for notationalbrevity. Recall that, for any j ∈ D , the initial residue demand of j is defined as r (cid:48) (0) j = (cid:88) i ∈ I, p ∈ P (cid:48) ( i,j ) f p = (cid:88) i ∈ I, p ∈ P ( i,j ) f p . The following lemma, which is valid from the algorithm design, shows that the residue demand ofeach j is mostly assigned by our rounding process to facilities in I \ F (cid:48) . Lemma 12.
For any j ∈ D , we have (cid:88) i ∈ I \ F (cid:48) x (cid:48)(cid:48) i,j ≥ r (cid:48) (0) j − α · r j . Proof of Lemma 12.
By Lemma 9, Lemma 11, the definition of r (cid:48) ( i ) j and the definition of x (cid:48)(cid:48) i,j foreach i ∈ I \ F (cid:48) , it follows that, each unit of assignment of any j ∈ D that is scaled down during theiteration for i ∈ I \ F (cid:48) is included in (cid:80) i ∈ I \ F (cid:48) x (cid:48)(cid:48) i,j . Since the algorithm removes a client j from D (cid:48) only when its residue demand r (cid:48) j is no greater than α · r j and since the initial residue demand of j is r (cid:48) (0) j , it follows that (cid:80) i ∈ I \ F (cid:48) x (cid:48)(cid:48) i,j ≥ r (cid:48) (0) j − α · r j .The following lemma shows that the facilities in I \ F (cid:48) is also sparsely-loaded under the assign-ments given in x (cid:48)(cid:48) . Lemma 13.
For any i ∈ I \ F (cid:48) , we have (cid:88) j ∈D x (cid:48)(cid:48) i,j = 1 − α · y † ( i ) i · (cid:88) (cid:96) ∈ D (cid:48) ( i ) x † ( i ) i,(cid:96) ≤ − α · u i . Proof of Lemma 13.
By the definition of x (cid:48)(cid:48) and the definition of σ ( i ) k for all k ∈ F (cid:48) ( i ) , we have (cid:88) j ∈D x (cid:48)(cid:48) i,j = (cid:88) j ∈ D (cid:48) ( i ) , k ∈ F (cid:48) ( i ) σ ( i ) k · x † ( i ) k,j = (cid:88) j ∈ D (cid:48) ( i ) x † ( i ) i,j + (cid:88) k ∈ F (cid:48) ( i ) \{ i } (cid:80) (cid:96) ∈ D (cid:48) ( i ) x † ( i ) k,(cid:96) (cid:88) (cid:96) ∈ D (cid:48) ( i ) σ ( i ) k,(cid:96) · (cid:88) j ∈ D (cid:48) ( i ) x † ( i ) k,j = (cid:88) j ∈ D (cid:48) ( i ) x † ( i ) i,j + (cid:88) k ∈ F (cid:48) ( i ) \{ i } , (cid:96) ∈ D (cid:48) ( i ) σ ( i ) k,(cid:96) . (5)17onsider the second item in (5). Further applying the definition of σ ( i ) k,(cid:96) for all k ∈ F (cid:48) ( i ) \ { i } , (cid:96) ∈ D (cid:48) ( i ) and the definition of δ ( i ) (cid:96) , we obtain (cid:88) k ∈ F (cid:48) ( i ) \{ i } , (cid:96) ∈ D (cid:48) ( i ) σ ( i ) k,(cid:96) = (cid:88) k ∈ F (cid:48) ( i ) \{ i } ,(cid:96) ∈ D (cid:48) ( i ) x † ( i ) k,(cid:96) (cid:80) p ∈ F (cid:48) ( i ) \{ i } x † ( i ) p,(cid:96) · δ ( i ) (cid:96) = (cid:88) (cid:96) ∈ D (cid:48) ( i ) δ ( i ) (cid:96) = (cid:88) (cid:96) ∈ D (cid:48) ( i ) (cid:32) − α · y † ( i ) i − (cid:33) · x † ( i ) i,(cid:96) . (6)Combining (5) and (6), and applying applying (M-2) in LP-(M) by the fact that (cid:0) x † ( i ) , y † ( i ) (cid:1) isa feasible solution for LP-(M) on Ψ ( i ) , we have (cid:88) j ∈D x (cid:48)(cid:48) i,j = 1 − α · y † ( i ) i · (cid:88) (cid:96) ∈ D (cid:48) ( i ) x † ( i ) i,(cid:96) ≤ − α · u i . This completes the proof of this lemma. x (cid:48)(cid:48)(cid:48) and the Feasibility Define the assignment x (cid:48)(cid:48)(cid:48) as follows. For any i ∈ F ∗ and j ∈ D , let x (cid:48)(cid:48)(cid:48) i,j := (cid:80) p ∈ Φ (cid:48) ( i,j ) f (cid:48) p , if i ∈ U , x (cid:48)(cid:48) i,j , if i ∈ I \ F (cid:48) .The following lemma establishes the feasibility of our rounding algorithm. Lemma 14.
For any j ∈ D , we have (cid:88) i ∈F ∗ x (cid:48)(cid:48)(cid:48) i,j ≥ − α. Proof of Lemma 14.
By the definition of x (cid:48)(cid:48)(cid:48) , to prove this lemma, it suffices to prove that (cid:88) i ∈F ∗ x (cid:48)(cid:48)(cid:48) i,j = (cid:88) i ∈ U, p ∈ Φ (cid:48) ( i,j ) f (cid:48) p + (cid:88) i ∈ I \ F (cid:48) x (cid:48)(cid:48) i,j ≥ − α. Depending on the category to which j belongs in the first phase of the algorithm, we consider thefollowing two cases. • j ∈ D ( (cid:96) ) , i.e., j is fully-assigned by h to facilities in U ( > ) and unreachable from any partially-assigned client via augmenting paths.Since j is fully-assigned by h to U ( > ) , we have (cid:88) i ∈ U ( > ) h i,j = 1 . Furthermore, since j is unreachable from any partially-assigned client via augmenting paths,it follows that for any i ∈ U ( > ) , h i,j > i is not tightly-occupied.Hence, by the construction rules of f (cid:48) , it follows that (cid:88) i ∈ U, p ∈ Φ (cid:48) ( i,j ) f (cid:48) p + (cid:88) i ∈ I \ F (cid:48) x (cid:48)(cid:48) i,j ≥ (cid:88) i ∈ U ( > ) f (cid:48) j s → i → i t → j t = (1 − α ) (cid:88) i ∈ U ( > ) h i,j = 1 − α. For the other case, j ∈ D ( p ) , i.e., j is either partially-assigned or reachable from partially-assigned clients via augmenting paths,since f is feasible for MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ), it follows from constraint (2.a) and the definition of MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ) that (cid:88) i ∈ U, p ∈ P ( i,j ) f p + (cid:88) i ∈ I, p ∈ P ( i,j ) f p = r j := 1 − (cid:88) i ∈ U g i,j . (7)Consider the two items in the LHS of (7) separately. By the construction of MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ),no flow is sent in f to sink via tightly-occupied facilities in U ( > ) . Hence, by the second andthird construction rules of f (cid:48) , we know that the flow sent in the first item, (cid:80) i ∈ U, p ∈ P ( i,j ) f p ,is included entirely in f (cid:48) , i.e., (cid:88) i ∈ U ( > ) ,i not tightly-occupied (cid:88) p ∈ Φ (cid:48) ( i,j ) f (cid:48) p + (cid:88) i ∈ U ( ≤ ) , p ∈ Φ (cid:48) ( i,j ) f (cid:48) p = (cid:88) i ∈ U, p ∈ P ( i,j ) f p . (8)For the second item in the LHS of (7), (cid:80) i ∈ I, p ∈ P ( i,j ) f p , consider the rounding process for theclusters in I \ F (cid:48) . Applying Lemma 12 and the definition of r (cid:48) (0) j , we have (cid:88) i ∈ I \ F (cid:48) x (cid:48)(cid:48) i,j ≥ r (cid:48) (0) j − α · r j = (cid:88) i ∈ I, p ∈ P ( i,j ) f p − α · r j . (9)By the first construction rule of f (cid:48) , we know that exactly (1 − α ) · (cid:80) i ∈ U g i,j units of flow isincluded in f (cid:48) . Therefore, (cid:88) i ∈ U ( > ) ,i tightly-occupied (cid:88) p ∈ Φ (cid:48) ( i,j ) f (cid:48) p = (1 − α ) · (cid:88) i ∈ U ( > ) ,i tightly-occupied g i,j = (1 − α ) · (cid:88) i ∈ U g i,j . (10)Combining the definition of U with (8), we obtain (cid:88) i ∈ U, p ∈ Φ (cid:48) ( i,j ) f (cid:48) p + (cid:88) i ∈ I \ F (cid:48) x (cid:48)(cid:48) i,j = (cid:88) i ∈ U ( > ) ,i tightly-occupied (cid:88) p ∈ Φ (cid:48) ( i,j ) f (cid:48) p + (cid:88) i ∈ U, p ∈ P ( i,j ) f p + (cid:88) i ∈ I \ F (cid:48) x (cid:48)(cid:48) i,j ≥ (1 − α ) · (cid:88) i ∈ U g i,j + (1 − α ) · r j = 1 − α, where in the second inequality we apply (10), (9), and then (7), respectively, and in the lastequality we apply the definition of r j .This completes the proof of this lemma.It follows directly from Lemma 14, Lemma 8, and Lemma 13 that ( x (cid:48)(cid:48)(cid:48) / (1 − α ) , y ∗ ) is a feasiblesolution for Ψ, and the feasible region of the min-cost assignment problem on D and F ∗ is therebynonempty. We have the following corollary. Corollary 15. (cid:16) − α · x (cid:48)(cid:48)(cid:48) , y ∗ (cid:17) is a feasible solution for Ψ.19 .4 Approximation Guarantee In the following we bound the cost incurred by facilities in F ∗ and the assignment cost incurred by x (cid:48)(cid:48)(cid:48) / (1 − α ). Note that this establishes an upper-bound on the cost incurred by ( x ∗ , y ∗ ) since x ∗ isthe optimal solution of the min-cost problem between F ∗ and D .For any i ∈ U, j ∈ D and any p ∈ Φ (cid:48) ( i, j ) ∪ P ( i, j ), define the absolute length of path p as | p | := (cid:88) i (cid:48) ∈F , j (cid:48) ∈D , ( j (cid:48) s ,i (cid:48) ) ∈ p c i (cid:48) ,j (cid:48) + (cid:88) i (cid:48) ∈F , j (cid:48) ∈D , ( i (cid:48) ,j (cid:48) s ) ∈ p c i (cid:48) ,j (cid:48) . By the definition of x (cid:48)(cid:48)(cid:48) and the triangle inequality c i,j ≤ | p | which holds for any i ∈ U, j ∈ D , andany p ∈ Φ (cid:48) ( i, j ), we have (cid:88) i ∈F ∗ , j ∈D c i,j · x (cid:48)(cid:48)(cid:48) i,j = (cid:88) i ∈ U, j ∈D (cid:88) p ∈ Φ (cid:48) ( i,j ) c i,j · f (cid:48) p + (cid:88) i ∈ I \ F (cid:48) , j ∈D c i,j · x (cid:48)(cid:48) i,j ≤ (cid:88) i ∈ U, j ∈D (cid:88) p ∈ Φ (cid:48) ( i,j ) | p | · f (cid:48) p + (cid:88) i ∈ I \ F (cid:48) , j ∈D c i,j · x (cid:48)(cid:48) i,j . (11)In (11), we decompose the assignment cost of x (cid:48)(cid:48)(cid:48) into two parts, namely, (i) the assignment costmade by f , g , and h to facilities in U , and (ii) the assignment cost incurred by our iterativerounding process to facilities in I \ F (cid:48) . In the following we consider the two parts separately.We begin with the cost incurred by assignments to facilities in I \ F (cid:48) . For any i ∈ I \ F (cid:48) , let( λ † ( i ) , β † ( i ) , Γ † ( i ) , η † ( i ) ) be an optimal dual solution for LP-(DM) on Ψ ( i ) . The following lemma,which is proven by standard complementary slackness conditions, establishes the equivalence be-tween the cost incurred around a non-extremal facility and the dual values it receives. Lemma 16.
For any i ∈ I \ F (cid:48) and any k ∈ F (cid:48) ( i ) with 0 < y † ( i ) k < (1 − α ) /
2, we have (cid:88) j ∈ D (cid:48) ( i ) λ † ( i ) j · x † ( i ) k,j = o k · y † ( i ) k + (cid:88) j ∈ D (cid:48) ( i ) c k,j · x † ( i ) k,j . Proof of Lemma 16.
Consider the complementary slackness conditions which follow from the factthat ( x † ( i ) , y † ( i ) ) and ( λ † ( i ) , β † ( i ) , Γ † ( i ) , η † ( i ) ) are optimal primal and dual solutions for LP-(M)and LP-(DM), respectively. Since x † ( i ) k,j > λ j ≤ β k + Γ k,j + c k,j , must be tight, we have (cid:88) j ∈ D (cid:48) ( i ) λ † ( i ) j · x † ( i ) k,j = β † ( i ) k · (cid:88) j ∈ D (cid:48) ( i ) x † ( i ) k,j + (cid:88) j ∈ D (cid:48) ( i ) Γ † ( i ) k,j · x † ( i ) k,j + (cid:88) j ∈ D (cid:48) ( i ) c k,j · x † ( i ) k,j .β † ( i ) k > † ( i ) k,j > β † ( i ) k · u k · y † ( i ) k + 2 α − α (cid:88) j ∈ D (cid:48) ( i ) r j · y † ( i ) k · Γ † ( i ) k,j + (cid:88) j ∈ D (cid:48) ( i ) c k,j · x † ( i ) k,j . The assumption that y † ( i ) k > y † ( i ) k < (1 − α ) / η † ( i ) k must be zero. The above equality becomes y † ( i ) k · u k · β † ( i ) k + 2 α − α (cid:88) j ∈ D (cid:48) ( i ) r j · Γ † ( i ) k,j + (cid:88) j ∈ D (cid:48) ( i ) c k,j · x † ( i ) k,j = o k · y † ( i ) k + (cid:88) j ∈ D (cid:48) ( i ) c k,j · x † ( i ) k,j , i ∈ I \ F (cid:48) selected inour iterative rounding process. Lemma 17.
For any i ∈ I \ F (cid:48) , we have(1 − α ) · o i + (cid:88) j ∈ D (cid:48) ( i ) c i,j · x (cid:48)(cid:48) i,j ≤ · (cid:88) k ∈ F (cid:48) ( i ) σ ( i ) k · o k · y † ( i ) k + (cid:88) j ∈ D (cid:48) ( i ) c k,j · x † ( i ) k,j , where we extend the definition of σ ( i ) and define σ ( i ) i := 1 for notational brevity. Proof of Lemma 17. If y † ( i ) i = (1 − α ) / i is selectedfor rounding, then σ ( i ) k = 0 for all k ∈ F (cid:48) ( i ) \ { i } and this lemma holds trivially.In the following we assume that y † ( i ) i < (1 − α ) /
2. Note that this implies that y † ( i ) k < (1 − α ) / k ∈ F (cid:48) ( i ) by the design of the algorithm. By the algorithm design, we have (cid:80) j ∈ D (cid:48) ( i ) x † ( i ) i,j > y † ( i ) i >
0. Apply the definition of x (cid:48)(cid:48) and the triangle inequality, we have (cid:88) j ∈ D (cid:48) ( i ) c i,j · x (cid:48)(cid:48) i,j = (cid:88) j ∈ D (cid:48) ( i ) c i,j · (cid:88) k ∈ F (cid:48) ( i ) σ ( i ) k · x † ( i ) k,j ≤ (cid:88) j ∈ D (cid:48) ( i ) , k ∈ F (cid:48) ( i ) σ ( i ) k · ( c i,k + c k,j ) · x † ( i ) k,j = (cid:88) j ∈ D (cid:48) ( i ) , k ∈ F (cid:48) ( i ) σ ( i ) k · c i,k · x † ( i ) k,j + (cid:88) k ∈ F (cid:48) ( i ) , j ∈ D (cid:48) ( i ) σ ( i ) k · c k,j · x † ( i ) k,j . (12)By (12), to prove this lemma, it suffices to show that(1 − α ) · o i + (cid:88) j ∈ D (cid:48) ( i ) ,k ∈ F (cid:48) ( i ) σ ( i ) k · c i,k · x † ( i ) k,j ≤ (cid:88) k ∈ F (cid:48) ( i ) σ ( i ) k · · o k · y † ( i ) k + 2 (cid:88) j ∈ D (cid:48) ( i ) c k,j · x † ( i ) k,j . (13)In the following we prove Inequality (13). First, since (cid:0) x † ( i ) , y † ( i ) (cid:1) and ( λ † ( i ) , β † ( i ) , Γ † ( i ) , η † ( i ) )are optimal primal and dual solutions for LP-(M) and LP-(DM), it follows that, λ ( i ) j = β ( i ) k + Γ ( i ) k,j + c k,j ≥ c k,j for all k ∈ F (cid:48) ( i ) , j ∈ D (cid:48) ( i ) with x † ( i ) k,j > . (14)Therefore, applying the definition of σ ( i ) k , the fact that c i,i = 0, and the triangle inequality, we have (cid:88) j ∈ D (cid:48) ( i ) , k ∈ F (cid:48) ( i ) σ ( i ) k · c i,k · x † ( i ) k,j = (cid:88) j ∈ D (cid:48) ( i ) , k ∈ F (cid:48) ( i ) \{ i } (cid:80) (cid:96) ∈ D (cid:48) ( i ) x † ( i ) k,(cid:96) · (cid:88) (cid:96) ∈ D (cid:48) ( i ) σ ( i ) k,(cid:96) · c i,k · x † ( i ) k,j = (cid:88) k ∈ F (cid:48) ( i ) \{ i } , (cid:96) ∈ D (cid:48) ( i ) c i,k · σ ( i ) k,(cid:96) ≤ (cid:88) k ∈ F (cid:48) ( i ) \{ i } , (cid:96) ∈ D (cid:48) ( i ) ( c i,(cid:96) + c k,(cid:96) ) · σ ( i ) k,(cid:96) ≤ (cid:88) k ∈ F (cid:48) ( i ) \{ i } , (cid:96) ∈ D (cid:48) ( i ) (cid:16) c i,(cid:96) + λ ( i ) (cid:96) (cid:17) · σ ( i ) k,(cid:96) , (15)where in the last inequality we apply the bound obtained in (14).21urther applying the definition of σ ( i ) k,(cid:96) for all k ∈ F (cid:48) ( i ) \ { i } and (cid:96) ∈ D (cid:48) ( i ) , we obtain (cid:88) k ∈ F (cid:48) ( i ) \{ i } , (cid:96) ∈ D (cid:48) ( i ) (cid:16) c i,(cid:96) + λ ( i ) (cid:96) (cid:17) · σ ( i ) k,(cid:96) = (cid:88) k ∈ F (cid:48) ( i ) \{ i } , (cid:96) ∈ D (cid:48) ( i ) (cid:16) c i,(cid:96) + λ ( i ) (cid:96) (cid:17) · x † ( i ) k,(cid:96) (cid:80) p ∈ F (cid:48) ( i ) \{ i } x † ( i ) p,(cid:96) · δ ( i ) (cid:96) = (cid:88) (cid:96) ∈ D (cid:48) ( i ) (cid:16) c i,(cid:96) + λ ( i ) (cid:96) (cid:17) · δ ( i ) (cid:96) ≤ − α · y † ( i ) i · (cid:88) j ∈ D (cid:48) ( i ) (cid:16) c i,j + λ ( i ) j (cid:17) · x † ( i ) i,j , (16)where in the last inequality we apply the definition of δ ( i ) (cid:96) for all (cid:96) ∈ D (cid:48) ( i ) . Combining (15) and (16),we obtain(1 − α ) · o i + (cid:88) j ∈ D (cid:48) ( i ) ,k ∈ F (cid:48) ( i ) σ ( i ) k · c i,k · x † ( i ) k,j ≤ − α · y † ( i ) i · · o i · y † ( i ) i + (cid:88) j ∈ D (cid:48) ( i ) (cid:16) c i,j + λ ( i ) j (cid:17) · x † ( i ) i,j = 1 − α · y † ( i ) i · o i · y † ( i ) i + 2 · (cid:88) j ∈ D (cid:48) ( i ) λ ( i ) j · x † ( i ) i,j = 1 − α · y † ( i ) i · θ (cid:16) i, D (cid:48) ( i ) , x † ( i ) , y † ( i ) (cid:17) · (cid:88) j ∈ D (cid:48) ( i ) x † ( i ) i,j , (17)where the second last equality follows from Lemma 16 and the fact that 0 < y † ( i ) i < (1 − α ) /
2, andthe last equality follows from Lemma 16 and the definition of θ (cid:0) i, D (cid:48) ( i ) , x † ( i ) , y † ( i ) (cid:1) .By Lemma 13 and the definition of x (cid:48)(cid:48) , we have1 − α · y † ( i ) i · (cid:88) j ∈ D (cid:48) ( i ) x † ( i ) i,j = (cid:88) j ∈ D (cid:48) ( i ) x (cid:48)(cid:48) i,j = (cid:88) j ∈ D (cid:48) ( i ) , k ∈ F (cid:48) ( i ) σ ( i ) k · x † ( i ) k,j . (18)Moreover, by the design of the rounding algorithm, for any k ∈ F (cid:48) ( i ) , we have θ (cid:16) i, D (cid:48) ( i ) , x † ( i ) , y † ( i ) (cid:17) ≤ θ (cid:16) k, D (cid:48) ( i ) , x † ( i ) , y † ( i ) (cid:17) . Combining this property with (18), we have1 − α · y † ( i ) i · θ (cid:16) i, D (cid:48) ( i ) , x † ( i ) , y † ( i ) (cid:17) · (cid:88) j ∈ D (cid:48) ( i ) x † ( i ) i,j ≤ (cid:88) k ∈ F (cid:48) ( i ) σ ( i ) k · θ (cid:16) k, D (cid:48) ( i ) , x † ( i ) , y † ( i ) (cid:17) · (cid:88) j ∈ D (cid:48) ( i ) x † ( i ) k,j = (cid:88) k ∈ F (cid:48) ( i ) σ ( i ) k · · o k · y † ( i ) k + 2 · (cid:88) j ∈ D (cid:48) ( i ) c k,j · x † ( i ) k,j , (19)where in the last inequality we apply the definition of θ (cid:0) k, D (cid:48) ( i ) , x † ( i ) , y † ( i ) (cid:1) for all k ∈ F (cid:48) ( i ) with (cid:80) j ∈ D (cid:48) ( i ) x † ( i ) k,j >
0. Combining (17) and (19) proves Inequality (13) and completes the proof of thislemma. 22pplying Lemma 9, Lemma 11, and Lemma 17, we have the following lemma which establishesthe guarantee for our iterative rounding process.
Lemma 18. (cid:88) i ∈ I \ F (cid:48) o i · y ∗ i + 11 − α · (cid:88) j ∈ D (cid:48) ( i ) c i,j · x (cid:48)(cid:48) i,j (20) ≤ · α · (cid:88) i ∈ I o i · y (cid:48) i + 31 − α · (cid:88) i ∈ I, j ∈D , p ∈ P ( i,j ) | p | · f p . Proof of Lemma 18.
Let ( x ∗† (0) , y ∗† (0) ) be an optimal solution for LP LP-(M) on the initial param-eter tuple Ψ (0) . By Lemma 9 and triangle inequality, it follows that (cid:88) i ∈ I o i · y ∗† (0) i + (cid:88) i ∈ I, j ∈D c i,j · x ∗† (0) i,j ≤ (cid:88) i ∈ I o i · y † (0) i + (cid:88) i ∈ I, j ∈D c i,j · x † (0) i,j ≤ − α · α · (cid:88) i ∈ I o i · y (cid:48) i + (cid:88) i ∈ I, j ∈D (cid:88) p ∈ P ( i,j ) | p | · f p . (21)For any i ∈ I \ F (cid:48) , let Ψ (cid:48)(cid:48) ( i ) = ( F (cid:48)(cid:48) ( i ) , D (cid:48)(cid:48) ( i ) , r (cid:48)(cid:48) ( i ) ) be the updated parameter tuple the algorithmmaintains at the end of the iteration for rounding i , and ( x (cid:48)∗† ( i ) , y (cid:48)∗† ( i ) ) be an optimal solutionfor LP-(M) on Ψ (cid:48)(cid:48) ( i ) . By Lemma 11, it follows that (cid:88) k ∈ F (cid:48)(cid:48) ( i ) o k · y (cid:48)∗† ( i ) k + (cid:88) k ∈ F (cid:48)(cid:48) ( i ) , j ∈ D (cid:48)(cid:48) ( i ) c k,j · x (cid:48)∗† ( i ) k,j ≤ (cid:88) k ∈ F (cid:48)(cid:48) ( i ) (cid:16) − σ ( i ) k (cid:17) · o k · y † ( i ) k + (cid:88) j ∈ D (cid:48)(cid:48) ( i ) c k,j · x † ( i ) k,j . Combining the above with Lemma 17 and apply the fact that F (cid:48)(cid:48) ( i ) = F (cid:48) ( i ) \ { i } , we have (1 − α ) · o i + (cid:88) j ∈ D (cid:48) ( i ) c i,j · x (cid:48)(cid:48) i,j + 3 · (cid:88) i ∈ F (cid:48)(cid:48) ( i ) o i · y (cid:48)∗† ( i ) i + (cid:88) i ∈ F (cid:48)(cid:48) ( i ) , j ∈ D (cid:48)(cid:48) ( i ) c i,j · x (cid:48)∗† ( i ) i,j ≤ · (cid:88) k ∈ F (cid:48) ( i ) o k · y † ( i ) k + (cid:88) j ∈ D (cid:48) ( i ) c k,j · x † ( i ) k,j . (22)Inequality (22) shows that, the total cost incurred by i can be bounded within three times thedifference between the optimal values of the successive iterations. Taking the summation over i ∈ I \ F (cid:48) and applying Inequality (21), we obtain (cid:88) i ∈ I \ F (cid:48) (1 − α ) · o i + (cid:88) j ∈ D (cid:48) ( i ) c i,j · x (cid:48)(cid:48) i,j ≤ · (cid:88) i ∈ I o i · y ∗† (0) i + (cid:88) i ∈ I,j ∈D c i,j · x ∗† (0) i,j ≤ · − α · α · (cid:88) i ∈ I o i · y (cid:48) i + 3 · (cid:88) i ∈ I, j ∈D , p ∈ P ( i,j ) | p | · f p . Multiplying the above by 1 / (1 − α ) completes the proof of this lemma.23ombining Lemma 18 and Inequality (11), it follows thatΨ (cid:18) − α · x (cid:48)(cid:48) , y ∗ (cid:19) = (cid:88) i ∈ U ∪ ( I \ F (cid:48) ) o i · y ∗ i + 11 − α · (cid:88) i ∈F ∗ , j ∈D c i,j · x (cid:48)(cid:48)(cid:48) i,j ≤ α · (cid:88) i ∈ U o i · y (cid:48) i + 11 − α · (cid:88) i ∈ U, j ∈D (cid:88) p ∈ Φ (cid:48) ( i,j ) | p | · f (cid:48) p (23)+ 32 α · (cid:88) i ∈ I o i · y (cid:48) i + 31 − α · (cid:88) i ∈ I, j ∈D , p ∈ P ( i,j ) | p | · f p . (24)Recall that, in the definition of absolute length | p | of any path p ∈ Φ (cid:48) ( i, j ) ∪ P ( i, j ), we take intoaccount the length of the edges between j (cid:48) s and i (cid:48) for any j (cid:48) ∈ D and any i (cid:48) ∈ F . Therefore, tobound the assignment cost incurred in (23) and (24), it suffices to consider the contribution of theedges between each possible pair of j s and i for j ∈ D and i ∈ F .Let i ∈ F and j ∈ D be an arbitrary pair of interests. Consider the contribution of the edges( j s , i ) and ( i, j s ) in the last items of (23) and (24). For notational brevity, let a ( ⇒ ) i,j := (cid:88) p ∈ P, ( j s ,i ) ∈ p f p and b ( ⇐ ) i,j := (cid:88) p ∈ P, ( i,j s ) ∈ p f p denote the total flow that is sent on the edge ( j s , i ) and ( i, j s ) by f , respectively.Among a ( ⇒ ) i,j and b ( ⇐ ) i,j , part of them is sent to sink via facilities in U while the remaining ofthem is sent to sink via facilities in I . In the former case, each unit of it contributes − α · c i,j tothe last item of (23). In the latter case, each unit of it contributes − α · c i,j to the last item of (24).Hence, the contribution is maximized when the flow is sent to sink via facilities in I .By the construction of MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ) and the fact that f is feasible for MFN Ψ ( x (cid:48) , y (cid:48)(cid:48) , g ),we have a ( ⇒ ) i,j ≤ x (cid:48) i,j and b ( ⇐ ) i,j ≤ g i,j ≤ − α · x (cid:48) i,j . Therefore, the contribution of ( j s , i ) and ( i, j s ) is bounded by (cid:18) − α · (cid:16) a ( ⇒ ) i,j + b ( ⇐ ) i,j (cid:17) + g i,j (cid:19) · c i,j ≤ − α (1 − α ) · c i,j · x (cid:48) i,j . Combining the above with (23) and (24), we obtainΨ (cid:18) − α · x (cid:48)(cid:48) , y ∗ (cid:19) ≤ α · (cid:88) i ∈F o i · y (cid:48) i + 7 − α (1 − α ) · (cid:88) i ∈F , j ∈D c i,j · x (cid:48) i,j ≤ max (cid:26) α , − α (1 − α ) (cid:27) · Ψ( x (cid:48) , y (cid:48) ) . This completes the proof of Lemma 6 and proves Theorem 4.24in (cid:88) i ∈F y i + (cid:88) i ∈F , j ∈D c i,j · x i,j LP-(N)s.t. (cid:88) i ∈F x i,j ≥ , ∀ j ∈ D (N-1) (cid:88) j ∈D x i,j ≤ u i · y i , ∀ i ∈ F (N-2)0 ≤ x i,j ≤ y i , ∀ i ∈ F , j ∈ D (N-3)0 ≤ y i ≤ , ∀ i ∈ F . (N-4)max (cid:88) j ∈D α j − (cid:88) i ∈F η i LP-(DN)s.t. α j ≤ β i + Γ i,j + c i,j , ∀ i ∈ F , j ∈ D , (D-1) u i · β i + (cid:88) j ∈D Γ i,j ≤ η i , ∀ i ∈ F , (D-2) α j , β i , Γ i,j , η i ≥ , ∀ i ∈ F , j ∈ D . (D-3)Figure 9: (Restate for further reference) The natural LP formulations for CFL-CFC. The proof of this theorem is outlined as follows. First we define the notations and notions thatare used throughout this section. In Section 5.1, page 26, we show that our rounding algorithmis well-defined and terminates in polynomial time. We define in the same section an intermediateassignment x ◦ and establish the feasibility of ( x ◦ , y ∗ ) for Ψ. This shows that the feasible region ofthe min-cost assignment problem between D and F ∗ is nonempty. We establish the approximationguarantee for ( x ◦ , y ∗ ) in Section 5.2, page 32. This completes the proof since x † is the optimalsolution for the min-cost assignment problem between D and F ∗ . Notations.
Consider the cluster-forming process. Let C D (cid:48) and C H (cid:48) denote the sets of clusterscentered at the non-outlier clients and outlier clients, respectively. For each q ∈ C D (cid:48) , we use j ( q )to denote the center client of q and i ( q ) denote the facility that is selected to be rounded up in theiteration when q is formed. Let F ∗ D (cid:48) := (cid:8) i ( q ) : q ∈ C D (cid:48) (cid:9) denote the set of facilities rounded up forthe clusters in C D (cid:48) . Note that, by the setting of the algorithm, F ∗ D (cid:48) and G are mutually exclusive,and the set of satellite facilities B ( j ) for each j ∈ H jointly forms a partition of G .To prevent notational ambiguity, we will use ( x (cid:48) , y (cid:48) ) to denote the initial solution the algorithmhas for Ψ. For each q ∈ C D (cid:48) , we use D (cid:48) ( q ) , F (cid:48) ( q ) , H ( q ) , H (cid:48) ( q ) , x (cid:48) ( q ) , and y (cid:48) ( q ) to denote the set D (cid:48) , theset F (cid:48) , the sets H , the set H (cid:48) , the assignment x (cid:48) , and the multiplicity y (cid:48) the algorithm maintains atthe moment when the cluster q is formed. We use B ( q ) to denote the set N ( F (cid:48) ,x (cid:48) ) ( j ( q )) of satellitefacilities at that moment. Similarly, we use x (cid:48) (II) and y (cid:48) (II) to denote the assignment x (cid:48) and the25ultiplicity y (cid:48) the algorithm maintains when it enters the second phase.For each outlier client j ∈ H , we use w ( j ) to denote the facility in U at which j is located. Weuse p ( j ) to denote the specific parent client in J ( ↔ ) from which j is created. On the contrary, forany w ∈ U , we use H ( w ) to denote the set of outlier clients located at w . For any j ∈ J ( ↔ ) , we use H ( j ) to denote the set of outlier clients that are created from j .For notational brevity, for any assignment function x of interest, we will use x | A,B to denotethe assignments made in x between A ⊆ F and B ⊆ D . Similarly, for any multiplicity function y of interest, we will use y | A to denote the multiplicity of facilities in A ⊆ F in y . The value of zerois assumed for pairs and facilities not in the defining domain of x and y . We first consider our rounding process for clusters in C D (cid:48) in the first phase and show that therounding algorithm is well-defined and runs in polynomial time. We consider in page 28 therounding process for clusters in C H (cid:48) in the second phase. In page 30 we define the assignment x ◦ and show that ( x ◦ , y ∗ ) forms a feasible solution for Ψ. The rounding process for C D (cid:48) . In the following we consider the first phase of our roundingprocess. To make the presentation precise, we will further use ( ˜ x (cid:48) , ˜ y (cid:48) ), ˜ F (cid:48) , ˜ D (cid:48) , and ˜ H to denotethe solution ( x (cid:48) , y (cid:48) ), the sets F (cid:48) , D (cid:48) , and H the algorithm maintains at any particular moment,referred to by the context, during the rounding process in the first phase.In order to characterize the behavior of the rounding process, we consider two types of con-straints for ( ˜ x (cid:48) , ˜ y (cid:48) ) throughout the process, namely, the capacity constraints for facilities in ˜ F (cid:48) andconstraint (N-3) from LP-(N) for facilities in ˜ F (cid:48) and clients in ˜ D (cid:48) , listed as follows. (cid:88) j ∈D∪ ˜ H ˜ x (cid:48) i,j ≤ u i · ˜ y (cid:48) i , ∀ i ∈ ˜ F (cid:48) . (MN-2)˜ x (cid:48) i,j ≤ ˜ y (cid:48) i , ∀ i ∈ ˜ F (cid:48) , j ∈ ˜ D (cid:48) . (N-3)It is clear that (MN-2) and (N-3) hold in the beginning of the rounding process since initially F (cid:48) := I , D (cid:48) := J ( I ) ∪ J ( ↔ ) , H := ∅ , and ( ˜ x (cid:48) , ˜ y (cid:48) ) is a feasible solution for LP-(N).The following two lemmas establish that, throughout the first phase of the rounding process,the constraints (MN-2) and (N-3) remain valid with respect to ( ˜ x (cid:48) , ˜ y (cid:48) ), ˜ F (cid:48) , ˜ D (cid:48) , and ˜ H . Lemma 19.
The process of creating outlier clients does not render (MN-2) invalid.
Proof of Lemma 19.
Consider the process of creating outlier clients from a client, say, j ∈ J ( ↔ ) ∩ D (cid:48) .By the algorithm design, this happens when (cid:80) i ∈ ˜ F (cid:48) ˜ x (cid:48) i,j < / i ∈ N ( ˜ F (cid:48) , ˜ x (cid:48) ) ( j ), consider the assignments the algorithm has made from H ( j ) to i after H ( j ) is created. By the algorithm setting and the definition of d (cid:96) for any (cid:96) ∈ H ( j ), we have (cid:88) (cid:96) ∈ H ( j ) x (cid:48) i,(cid:96) := (cid:88) (cid:96) ∈ H ( j ) d (cid:96) · ˜ x (cid:48) i,j (cid:80) k ∈ ˜ F (cid:48) ˜ x (cid:48) k,j = ˜ x (cid:48) i,j (cid:80) k ∈ ˜ F (cid:48) ˜ x (cid:48) k,j · (cid:88) (cid:96) ∈ H ( j ) r (cid:48) j · ˜ x (cid:48) w ( (cid:96) ) ,j (cid:80) w ∈ U ˜ x (cid:48) w,j .
26y the definition of r (cid:48) j , we have r (cid:48) j ≤ (cid:80) k ∈ ˜ F (cid:48) ˜ x (cid:48) k,j . Hence, the above becomes (cid:88) (cid:96) ∈ H ( j ) x (cid:48) i,(cid:96) ≤ ˜ x (cid:48) i,j · (cid:88) (cid:96) ∈ H ( j ) ˜ x (cid:48) w ( (cid:96) ) ,j (cid:80) w ∈ U ˜ x (cid:48) w,j = ˜ x (cid:48) i,j , (25)where in the last equality we apply the fact that (cid:80) (cid:96) ∈ H ( j ) ˜ x (cid:48) w ( (cid:96) ) ,j = (cid:80) w ∈ U ˜ x (cid:48) w,j . Since the algorithmresets ˜ x i,j to be zero after H ( j ) is created, it follows from (25) that the LHS of (MN-2) for i doesnot increase after H ( j ) is created and the new assignments to i are made. Lemma 20.
We have 0 < δ i ( q ) ≤ q ∈ C D (cid:48) . Furthermore, the scaled-down operation thealgorithm performs when rounding i ( q ) does not render (MN-2) and (N-3) invalid. Proof of Lemma 20.
Consider any q ∈ C D (cid:48) . We will show that, provided that constraints (MN-2)and (N-3) are valid in the beginning of the iteration for which q is formed, we have 0 < δ i ( q ) ≤ i ( q ) does not render (MN-2) and (N-3) invalid. Note that this proves thelemma together with Lemma 19 since (MN-2) and (N-3) are valid in the beginning of the process.Since i ( q ) ∈ F (cid:48) ( q ) , we know that y (cid:48) ( q ) i ( q ) < /
2. Furthermore, since j ( q ) is selected as the centerclient and since j ( q ) ∈ D (cid:48) ( q ) by assumption, it follows that (cid:80) i ∈ F (cid:48) ( q ) x (cid:48) ( q ) i,j ( q ) ≥ /
2. This implies that (cid:88) i ∈ B ( q ) \{ i ( q ) } y (cid:48) ( q ) i ≥ (cid:88) i ∈ B ( q ) \{ i ( q ) } x (cid:48) ( q ) i,j ( q ) ≥ − x (cid:48) ( q ) i ( q ) ,j ( q ) ≥ − y (cid:48) ( q ) i ( q ) > , where in the above inequalities we apply constraint (N-3) for i ∈ B ( q ) and j ( q ). This shows that δ i ( q ) := (cid:18) − y (cid:48) ( q ) i ( q ) (cid:19) · (cid:80) i ∈ B ( q ) \{ i ( q ) } y (cid:48) ( q ) i > . On the contrary, by (N-3) we have (cid:80) i ∈ B ( q ) y (cid:48) ( q ) i ≥ (cid:80) i ∈ B ( q ) x (cid:48) ( q ) i,j ( q ) ≥ /
2. This implies that1 / − y (cid:48) ( q ) i ( q ) ≤ (cid:80) i ∈ B ( q ) \{ i } y (cid:48) ( q ) i and δ i ( q ) ≤ i ∈ B ( q ) \ { i ( q ) } and any j ∈ D (cid:48) ( q ) , both x (cid:48) ( q ) i,j and y (cid:48) ( q ) i are scaled down simultaneously bythe constant (cid:0) − δ i ( q ) (cid:1) . This completes the proof of this lemma.By Lemma 19 and Lemma 20, we have the following corollary. Corollary 21.
Throughout the first phase of the rounding process, the constraints (MN-2) and (N-3) remain valid with respect to ( ˜ x (cid:48) , ˜ y (cid:48) ), ˜ F (cid:48) , ˜ D (cid:48) , and ˜ H .Lemma 20 shows that the scaled-down operation in our rounding process is well-defined. Sinceconstraint (N-3) holds throughout the process, it follows that, for any particular moment, (cid:88) i ∈ ˜ F (cid:48) ˜ x (cid:48) i,j ≤ (cid:88) i ∈ ˜ F (cid:48) ˜ y (cid:48) i holds for any j ∈ ˜ D (cid:48) . (26)By the algorithm design, we know that at least one facility is removed from F (cid:48) after each iterationin the first phase. Therefore, the rounding process repeats for at most | F (cid:48) | := | I | iterations before F (cid:48) becomes empty, which in turn implies that D (cid:48) ∪ H (cid:48) is empty by (26). This shows that therounding algorithm terminates in polynomial time.The following lemma, which shows that the rounded facility is sparsely loaded by the reroutedassignments, is straightforward to verify. 27 emma 22. We have (cid:80) j ∈D∪ H x ∗ i ( q ) ,j ≤ u i ( q ) / q ∈ C D (cid:48) . Proof of Lemma 22.
Consider any q ∈ C D (cid:48) . By the design of the algorithm and the fact thatconstraint (N-3) holds throughout the process, we have (cid:88) j ∈D∪ H x ∗ i ( q ) ,j = (cid:88) j ∈D∪ H x (cid:48) ( q ) i ( q ) ,j + (cid:88) i ∈ B ( q ) \{ i ( q ) } (cid:88) j ∈D∪ H δ i ( q ) · x (cid:48) ( q ) i,j ≤ u i ( q ) · y (cid:48) ( q ) i ( q ) + (cid:88) i ∈ B ( q ) \{ i ( q ) } δ i ( q ) · u i · y (cid:48) ( q ) i ≤ u i ( q ) · y (cid:48) ( q ) i ( q ) + u i ( q ) · (cid:18) − y (cid:48) ( q ) i ( q ) (cid:19) = 12 · u i ( q ) , where in the second last inequality we use apply fact that u i ( q ) ≥ u i for all i ∈ B ( q ) by the way i ( q ) is selected and the definition of δ i ( q ) . The rounding process for C H (cid:48) . In the following we consider the rounding process for clustersin C H (cid:48) in the second phase. The following lemma summarizes the status of the facilities and clientswhen the algorithm enters the this phase. Lemma 23.
When the algorithm enters the second phase, the following holds. • For any i ∈ G , (cid:80) j ∈D∪ H x (cid:48) (II) i,j ≤ u i · y (cid:48) (II) i . • For any j ∈ J ( I ) , (cid:80) i ∈ I x ∗ i,j + (cid:80) i ∈ G x (cid:48) (II) i,j > / • For any j ∈ J ( ↔ ) , (cid:80) i ∈ I x ∗ i,j + (cid:80) i ∈ G x (cid:48) (II) i,j + (cid:80) i ∈ U x (cid:48) i,j > / • For any j ∈ H , (cid:80) i ∈ F ∗ D (cid:48) x ∗ i,j + (cid:80) i ∈ G x (cid:48) (II) i,j = (cid:80) i ∈ I x (cid:48) i,j . Proof of Lemma 23.
The first statement of this lemma follows directly from Corollary 21 and thedefinition of ( x (cid:48) (II) , y (cid:48) (II) ). The remaining of this lemma follows from the way how the algorithmhandles the residue demand of each client. Consider the moment for which each j ∈ J ( I ) ∪ J ( ↔ ) ∪ H is removed from consideration in the first phase, and the fact that G := (cid:83) j ∈ H B ( j ).For j ∈ J ( I ) ∪ J ( ↔ ) , it is removed when (cid:80) i ∈ F (cid:48) x (cid:48) i,j < /
2. It follows that the assignmentsrerouted to clusters in C D (cid:48) , the assignments taken into clusters in C H (cid:48) , and possibly the originalassignments to facilities in U , account for at least 1 / j ∈ H , it is removed when selected as the center of a cluster that is possibly empty. Whenthis happens, all of the remaining assignments for j are taken into this cluster.Lemma 23 leads to the following corollary. Corollary 24. ≤ t (cid:48) j ≤ j ∈ D . Proof of Corollary 24.
It suffices to prove the statement for j ∈ D with (cid:80) i ∈ I x ∗ i,j + (cid:80) i ∈ G x (cid:48) (II) i,j > (cid:80) i ∈ I x ∗ i,j + (cid:80) i ∈ G x (cid:48) (II) i,j > j ∈ J ( I ) ∪ J ( ↔ ) , by the definition of r (cid:48) j , we have1 − (cid:88) i ∈ U x (cid:48) i,j − r (cid:48) j ≥ − (cid:88) i ∈ U x (cid:48) i,j − (cid:88) i ∈ I x (cid:48) i,j ≥ , which implies that t (cid:48) j > . In the following we show that t (cid:48) j ≤
2. Since j ∈ J ( I ) ∪ J ( ↔ ) , it suffices to prove the statementfor the following cases. 28 If j ∈ J ( I ) , then t (cid:48) j < (cid:80) i ∈ I x ∗ i,j + (cid:80) i ∈ G x (cid:48) (II) i,j > / − (cid:80) i ∈ U x (cid:48) i,j − r (cid:48) j ) ≤ • If j ∈ J ( ↔ ) and r (cid:48) j (cid:54) = (cid:80) i ∈ U x (cid:48) i,j , then all of the residue demand of j has been redistributed asoutlier clients when j is to be removed from D (cid:48) . It follows that1 − (cid:88) i ∈ U x (cid:48) i,j − r (cid:48) j = (cid:88) i ∈ I x ∗ i,j + (cid:88) i ∈ G x (II) i,j and t (cid:48) j = 1 . • If j ∈ J ( ↔ ) and r (cid:48) j := (cid:80) i ∈ U x (cid:48) i,j , then by the conclusion of Lemma 23 we have (cid:80) i ∈ I x ∗ i,j + (cid:80) i ∈ G x (cid:48) (II) i,j + (cid:80) i ∈ U x (cid:48) i,j > /
2, which implies that1 − (cid:88) i ∈ U x (cid:48) i,j − r (cid:48) j = 1 − · (cid:88) i ∈ U x (cid:48) i,j < · (cid:32) (cid:88) i ∈ I x ∗ i,j + (cid:88) i ∈ G x (cid:48) (II) i,j (cid:33) and t (cid:48) j < t (cid:48) (cid:96) ≤ (cid:88) i ∈ G y i + (cid:88) i ∈ G,j ∈ U c i,j · x i,j LP-(O)s.t. (cid:88) i ∈ G x i,j ≥ d j , ∀ j ∈ U, (O-1) (cid:88) j ∈ U x i,j ≤ u i · y i , ∀ i ∈ G, (O-2) y i ≤ , ∀ i ∈ G, (O-3) x i,j ≥ , y i ≥ , ∀ i ∈ G, j ∈ U. (O-4)Figure 10: (Restate for further reference) The assignment LP for clusters centered at outlier clients. The bundled assignment g for LP-(O). In the following we consider the assignment LP forrounding the clusters in C H (cid:48) . For any w ∈ U and i ∈ G such that i ∈ B ( k ) for some k ∈ H ( w ), i.e., i belongs to the clusters centered at some k ∈ H ( w ), define the bundled assignment g i,w as g i,w := (cid:88) (cid:96) ∈D∪ H t (cid:48) (cid:96) · x (cid:48) (II) i,(cid:96) . The following lemma shows that the feasible region of LP-(O) is nonempty, and the basic optimalsolution ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) exists. Lemma 25. (cid:16) g | G,U , y (cid:48) (II) (cid:12)(cid:12) G (cid:17) is a feasible solution for LP-(O). Proof of Lemma 25.
We prove by verifying the constraints of LP-(O).29
Consider (O-1) in LP-(O).For any w ∈ U , apply the definition of g and the definition of d w , we have (cid:88) i ∈ G g i,w = (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) (cid:88) (cid:96) ∈D∪ H t (cid:48) (cid:96) · x (cid:48) (II) i (cid:48) ,(cid:96) = d w ≥ d w . • Consider (O-3) in LP-(O).For any i ∈ G , since G ⊆ I , it follows that y (cid:48) (II) i ≤ y (cid:48) i ≤ /
2, and we have 2 · y (cid:48) (II) i ≤ • Consider (O-2) in LP-(O).For any i ∈ G , let k ∈ H be the outlier client such that i ∈ B ( k ). Applying the definition of g and Corollary 24, we have (cid:88) w ∈ U g i,w = g i,w ( k ) = (cid:88) (cid:96) ∈D∪ H t (cid:48) (cid:96) · x (cid:48) (II) i,(cid:96) ≤ (cid:88) (cid:96) ∈D∪ H · x (cid:48) (II) i,(cid:96) ≤ · u i · y (cid:48) (II) i , where in the last inequality we apply the conclusion of Corollary 21 which states that con-straint (MN-2) holds for i when i is removed from F (cid:48) in the first phase.This proves the lemma. The unbundled assignment h from x (cid:48)(cid:48) . For each i ∈ G and j ∈ D ∪ H , define the unbundledassignment h i,j from x (cid:48)(cid:48) as h i,j := (cid:88) w ∈ U x (cid:48)(cid:48) i,w · d w · (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) t (cid:48) j · x (cid:48) (II) i (cid:48) ,j . Intuitively, in h we redistribute the bundled assignment x (cid:48)(cid:48) back for the original clients, and itfollows that for any j ∈ D ∪ H , (cid:88) i ∈ G h i,j = (cid:88) i ∈ G, w ∈ U x (cid:48)(cid:48) i,w · d w · (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) t (cid:48) j · x (cid:48) (II) i (cid:48) ,j = (cid:88) w ∈ U (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) t (cid:48) j · x (cid:48) (II) i (cid:48) ,j = (cid:88) i ∈ G t (cid:48) j · x (cid:48) (II) i,j , (27)where in the second equality we apply (O-1) from LP-(O) for the fact that (cid:80) i ∈ G x (cid:48)(cid:48) i,w = d w for all w ∈ U since ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) is a feasible solution for LP-(O) and in the last equality we use the fact thatthe set of satellite facilities for each j ∈ H forms a partition of G . The Assignment x ◦ . Provided above, the assignment x ◦ made for each j ∈ D is defined as x ◦ | j := x (cid:48) (cid:12)(cid:12) U, { j } + t (cid:48) j · x ∗ (cid:12)(cid:12)(cid:12) F ∗ D (cid:48) , { j } + x ∗ (cid:12)(cid:12)(cid:12) F ∗ D (cid:48) ,H ( j ) + h (cid:12)(cid:12)(cid:12) G, { j }∪ H ( j ) . More precisely, for any i ∈ F , j ∈ D , x ◦ i,j := x (cid:48) i,j , if i ∈ U , t (cid:48) j · x ∗ i,j + (cid:80) k ∈ H ( j ) x ∗ i,k , if i ∈ F ∗ D (cid:48) , h i,j + (cid:80) k ∈ H ( j ) h i,k , if i ∈ G ,0 , otherwise . j ∈ D in x ◦ consists of its original assignments to U and therounded assignments for clients in { j } ∪ H ( j ) to facilities in F ∗ D (cid:48) ∪ G .The following lemma, which asserts the feasibility of x ◦ , is straightforward to verify. Lemma 26. ( x ◦ , y ∗ ) is feasible for LP-(N) on the input instance Ψ. Proof of Lemma 26.
In the following we show that ( x ◦ , y ∗ ) is feasible for LP-(N) on Ψ. Since y ∗ is already integral and takes values only from { , } , it suffices to argue that x ◦ fully-assigns each j ∈ D and respects the capacity constraints given by y ∗ .For the latter part, since x ◦ keeps the assignments of D to U unchanged, it suffices to argue forthe assignments to F ∗ D (cid:48) ∪ G . By the definition of x ◦ , Corollary 24, and Lemma 22, for any i ∈ F ∗ D (cid:48) ,we have (cid:88) j ∈D x ◦ i,j = (cid:88) j ∈D t (cid:48) j · x ∗ i,j + (cid:88) j ∈ H x ∗ i,j ≤ (cid:88) j ∈D∪ H · x ∗ i,j ≤ u i = u i · y ∗ i . Similarly, for any i ∈ G , applying the definition of x ◦ and h , and the fact that ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) is feasiblefor LP-(O), we have (cid:88) j ∈D x ◦ i,j = (cid:88) j ∈D∪ H h i,j = (cid:88) j ∈D∪ H (cid:88) w ∈ U x (cid:48)(cid:48) i,w · d w · (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) t (cid:48) j · x (cid:48) (II) i (cid:48) ,j = (cid:88) w ∈ U x (cid:48)(cid:48) i,w · d w · (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) (cid:88) j ∈D∪ H t (cid:48) j · x (cid:48) (II) i (cid:48) ,j = (cid:88) w ∈ U x (cid:48)(cid:48) i,w ≤ u i · y (cid:48)(cid:48) i , where in the second last equality we apply the definition of d w for each w ∈ U .In the following, we show that x ◦ fully-assigns each j ∈ D . It suffices to argue for the clients in J ( I ) ∪ J ( ↔ ) . For the former case, for any j ∈ J ( I ) , we have (cid:80) i ∈ U x (cid:48) i,j = 0 and H ( j ) = ∅ . Hence, (cid:88) i ∈F x ◦ i,j = (cid:88) i ∈ F ∗ D (cid:48) t (cid:48) j · x ∗ i,j + (cid:88) i ∈ G h i,j = (cid:88) i ∈ F ∗ D (cid:48) t (cid:48) j · x ∗ i,j + (cid:88) i ∈ G t (cid:48) j · x (cid:48) (II) i,j = t (cid:48) j · (cid:88) i ∈ F ∗ D (cid:48) x ∗ i,j + (cid:88) i ∈ G x (cid:48) (II) i,j = 1 , where in the second equality we apply Equality (27), and in the last equality we apply the definitionof t (cid:48) j with the fact that (cid:80) i ∈ U x (cid:48) i,j = r (cid:48) j = 0.For j ∈ J ( ↔ ) , we have (cid:88) i ∈F x ◦ i,j = (cid:88) i ∈ U x (cid:48) i,j + (cid:88) i ∈ F ∗ D (cid:48) t (cid:48) j · x ∗ i,j + (cid:88) k ∈ H ( j ) x ∗ i,k + (cid:88) i ∈ G h i,j + (cid:88) k ∈ H ( j ) h i,k (28)Applying Equality (27) and the definition of t (cid:48) k for k ∈ H ( j ), we have (cid:88) i ∈ G h i,j + (cid:88) k ∈ H ( j ) h i,k = (cid:88) i ∈ G t (cid:48) j · x (cid:48) (II) i,j + (cid:88) k ∈ H ( j ) , i ∈ G x (cid:48) (II) i,k (29)31ombining (28) and (29), we have (cid:88) i ∈F x ◦ i,j = (cid:88) i ∈ U x (cid:48) i,j + (cid:88) i ∈ F ∗ D (cid:48) t (cid:48) j · x ∗ i,j + (cid:88) i ∈ G t (cid:48) j · x (cid:48) (II) i,j + (cid:88) k ∈ H ( j ) (cid:88) i ∈ F ∗ D (cid:48) x ∗ i,k + (cid:88) i ∈ G x (cid:48) (II) i,k = (cid:88) i ∈ U x (cid:48) i,j + (cid:88) i ∈ F ∗ D (cid:48) t (cid:48) j · x ∗ i,j + (cid:88) i ∈ G t (cid:48) j · x (cid:48) (II) i,j + (cid:88) k ∈ H ( j ) (cid:88) i ∈ I x (cid:48) i,k , (30)where in the last equality we apply the conclusion of Lemma 23. By the construction of outlierclients, each k ∈ H ( j ) is fully-assigned to facilities in I by x (cid:48) . Hence, we have (cid:80) i ∈ I x (cid:48) i,k = d k foreach k ∈ H ( j ). Further applying the definition of d k , we have (cid:88) k ∈ H ( j ) (cid:88) i ∈ I x (cid:48) i,k = (cid:88) k ∈ H ( j ) d k = (cid:88) k ∈ H ( j ) r (cid:48) j · x (cid:48) w ( k ) ,j (cid:80) i ∈ U x (cid:48) i,j = r (cid:48) j , (31)where the last equality follows from the fact that exactly one outlier client is created for each i ∈ U with x (cid:48) i,j >
0. Combining (30) and (31) and applying the definition of t (cid:48) j , we have (cid:88) i ∈F x ◦ i,j = (cid:88) i ∈ U x (cid:48) i,j + (cid:88) i ∈ F ∗ D (cid:48) t (cid:48) j · x ∗ i,j + (cid:88) i ∈ G t (cid:48) j · x (cid:48) (II) i,j + r (cid:48) j = 1 . This completes the proof of this lemma.
In the following we establish our approximation guarantee for CFL-CFC. We will bound the costincurred by clusters in C H (cid:48) and C D (cid:48) separately. First we bound below the cost incurred by C H (cid:48) . Webound the cost incurred by C D (cid:48) in page 36 and establish the overall guarantee in page 39.Recall that we use p ( j ) for each j ∈ H to denote the client in D from which j is created. In thefollowing, we extend this definition and define p ( k ) := k for each k ∈ D for notational brevity. The Clusters in C H (cid:48) . Consider the cost incurred by clusters in C H (cid:48) . We begin with the followinglemma regarding the assignment radius of the outlier clients in H . Lemma 27.
For any j ∈ H and i ∈ G such that x (cid:48) (II) i,j >
0, we have c i,j ≤ α j . Proof of Lemma 27.
By the algorithm design, the outlier client j is created and assigned to i in x (cid:48) only when x (cid:48) i,p ( j ) > c i,p ( j ) ≤ α p ( j ) by complementary slackness condition. By triangle inequality and the definition of α j , it follows that c i,j = c i,w ( j ) ≤ c w ( j ) ,p ( j ) + c i,p ( j ) ≤ c w ( j ) ,p ( j ) + α p ( j ) = α j .The following lemma, which bounds the assignment cost of x ◦ | G, D in terms of that in x (cid:48)(cid:48) and x (cid:48) (II) (cid:12)(cid:12) G, D∪ H , follows from the way the clusters in C H (cid:48) are formed and the assignments are bundled. Lemma 28. (cid:88) i ∈ G, j ∈D c i,j · x ◦ i,j ≤ (cid:88) i ∈ G, j ∈ U c i,j · x (cid:48)(cid:48) i,j + (cid:88) i ∈ G (cid:88) j ∈D∪ H t (cid:48) j · (cid:0) c i,p ( j ) + α j (cid:1) · x (cid:48) (II) i,j . k ∈ H ( w ) ip ( k ) i ′ p ( j ) j Figure 11: An illustration on the bundled assignment from w ∈ U to i ∈ G and unbundledassignments for k ∈ H ( w ), i (cid:48) ∈ B ( k ) such that x (cid:48) (II) i (cid:48) ,j > Proof of Lemma 28.
By the definition of x ◦ and h , we have (cid:88) i ∈ G, j ∈D c i,j · x ◦ i,j = (cid:88) i ∈ G, j ∈D (cid:88) k ∈{ j }∪ H ( j ) c i,j · h i,k = (cid:88) i ∈ G (cid:88) j ∈D∪ H c i,p ( j ) · h i,j = (cid:88) i ∈ G (cid:88) j ∈D∪ H c i,p ( j ) · (cid:88) w ∈ U x (cid:48)(cid:48) i,w · d w · (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) t (cid:48) j · x (cid:48) (II) i (cid:48) ,j . (32)By triangle inequality, for any i ∈ G , j ∈ D ∪ H , w ∈ U , k ∈ H ( w ), and i (cid:48) ∈ B ( k ) such that x (cid:48) (II) i (cid:48) ,j >
0, we have c i,p ( j ) ≤ c i,w + c i (cid:48) ,w + c i (cid:48) ,p ( j ) ≤ c i,w + α k + c i (cid:48) ,p ( j ) , where the last inequality follows from Lemma 27 and the fact that i (cid:48) ∈ B ( k ) implies that x (cid:48) (II) i (cid:48) ,k > (cid:88) i ∈ G, w ∈ U c i,w · x (cid:48)(cid:48) i,w · d w · (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) (cid:88) j ∈D∪ H t (cid:48) j · x (cid:48) (II) i (cid:48) ,j + (cid:88) i ∈ G, w ∈ U x (cid:48)(cid:48) i,w · d w · (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) (cid:88) j ∈D∪ H (cid:0) c i (cid:48) ,p ( j ) + α k (cid:1) · t (cid:48) j · x (cid:48) (II) i (cid:48) ,j . Applying the definition of d w on the former term and the fact that (cid:80) i ∈ G x (cid:48)(cid:48) i,w = d w by the fact that( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) is a feasible solution for LP-(O) on the latter term, the above becomes (cid:88) i ∈ G, w ∈ U c i,w · x (cid:48)(cid:48) i,w + (cid:88) w ∈ U (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) (cid:88) j ∈D∪ H (cid:0) c i (cid:48) ,p ( j ) + α k (cid:1) · t (cid:48) j · x (cid:48) (II) i (cid:48) ,j . By the design of the algorithm, for any w ∈ U, k ∈ H ( w ) , i (cid:48) ∈ B ( k ), and for any j ∈ D ∪ H with x (cid:48) (II) i (cid:48) ,j >
0, we have α k ≤ α j , since k is selected as cluster center because of having the lowest α value. Therefore, the above is further bounded by (cid:88) i ∈ G, w ∈ U c i,w · x (cid:48)(cid:48) i,w + (cid:88) w ∈ U (cid:88) k ∈ H ( w ) , i (cid:48) ∈ B ( k ) (cid:88) j ∈D∪ H (cid:0) c i (cid:48) ,p ( j ) + α j (cid:1) · t (cid:48) j · x (cid:48) (II) i (cid:48) ,j . C H (cid:48) forms a partition of G , the above isexactly (cid:88) i ∈ G, j ∈ U c i,j · x (cid:48)(cid:48) i,j + (cid:88) i ∈ G (cid:88) j ∈D∪ H t (cid:48) j · (cid:0) c i,p ( j ) + α j (cid:1) · x (cid:48) (II) i,j . The following lemma bounds the cost incurred by ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ). Lemma 29.
We have (cid:88) i ∈ G (cid:6) y (cid:48)(cid:48) i (cid:7) + (cid:88) i ∈ G, j ∈ U c i,j · x (cid:48)(cid:48) i,j ≤ · (cid:88) i ∈ G y (cid:48) (II) i + | L | + (cid:88) i ∈ G (cid:88) j ∈D∪ H t (cid:48) j · α j · x (cid:48) (II) i,j , where L := (cid:8) i ∈ G : 0 < y (cid:48)(cid:48) i < (cid:9) . Proof of Lemma 29.
Since ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) is optimal for LP-(O), by Lemma 25, the cost of ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) is nomore than that of (cid:16) g | G,U , y (cid:48) | G (cid:17) . Hence, (cid:88) i ∈ G y (cid:48)(cid:48) i + (cid:88) i ∈ G, j ∈ U c i,j · x (cid:48)(cid:48) i,j ≤ · (cid:88) i ∈ G y (cid:48) (II) i + (cid:88) i ∈ G, j ∈ U c i,j · g i,j = 2 · (cid:88) i ∈ G y (cid:48) (II) i + (cid:88) i ∈ G, w ∈ U,i ∈ B ( k ) for some k ∈ H ( w ) c i,w · (cid:88) (cid:96) ∈D∪ H t (cid:48) (cid:96) · x (cid:48) (II) i,(cid:96) . By the algorithm setting and Lemma 27, for any i ∈ G and w ∈ U such that i ∈ B ( k ) for some k ∈ H ( w ), i.e., facility i belongs to some cluster centered at some k ∈ H ( w ), we have c i,w ≤ α k ≤ α (cid:96) ,for any (cid:96) ∈ D ∪ H with x (cid:48) (II) i,(cid:96) >
0. Therefore, (cid:88) i ∈ G y (cid:48)(cid:48) i + (cid:88) i ∈ G, j ∈ U c i,j · x (cid:48)(cid:48) i,j ≤ · (cid:88) i ∈ G y (cid:48) (II) i + (cid:88) i ∈ G, w ∈ U,i ∈ B ( k ) for some k ∈ H ( w ) (cid:88) (cid:96) ∈D∪ H t (cid:48) (cid:96) · α (cid:96) · x (cid:48) (II) i,(cid:96) = 2 · (cid:88) i ∈ G y (cid:48) (II) i + (cid:88) i ∈ G, j ∈D∪ H t (cid:48) j · α j · x (cid:48) (II) i,j , where the last equality follows from the fact that the set of satellite facilities of clusters in C H (cid:48) forms a partition of G . Applying the definition of L competes the proof of this lemma.The following lemma follows from the fact that ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) is a basic solution for LP-(O). Lemma 30. | L | ≤ | U | , where L := (cid:110) i ∈ G : 0 < y (cid:48)(cid:48) i < (cid:111) . Proof of Lemma 30.
Consider the set of constraints in LP-(O) that hold with equality at ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ),for which we denote by E (=) in the following. Let M and M denote the set of constraints in E (=) of the types (O-3) and (O-4), respectively. Formally, M := (cid:26) i : y i ≤ ∈ E (=) (cid:27) M := (cid:26) ( i, j ) : x i,j ≥ ∈ E (=) (cid:27) ∪ (cid:26) i : y i ≥ ∈ E (=) (cid:27) . Let X be the number of variables in LP-(O). Since ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) is a basic solution for LP-(O), itfollows that, the coefficient matrix of E (=) is of full-rank, i.e., has rank X . Since M and M arelinearly independent, there exists a subset E (cid:48) ⊆ E (=) of linearly independent constraints such that M ∪ M ⊆ E (cid:48) and |E (cid:48) | = X .Let E (cid:48)(cid:48) := E (cid:48) \ ( M ∪ M ) and modify the constraints in E (cid:48)(cid:48) by setting the variable y i to be 1for all i ∈ M . Similarly, modify E (cid:48)(cid:48) by setting x i,j to be zero for all ( i, j ) ∈ M and y i to be zerofor all i ∈ M . By doing so, we removed equally many constraints and variables from E (cid:48) . Since M and M are linearly independent, it follows thatrank( E (cid:48)(cid:48) ) = rank( E (cid:48) ) − | M ∪ M | , and the coefficient matrix of E (cid:48)(cid:48) is still of full-rank.Let M := (cid:26) j : (cid:80) i ∈ G x i,j ≥ d j ∈ E (cid:48)(cid:48) (cid:27) and M := (cid:26) i : (cid:80) j ∈ U x i,j ≤ u i · y i ∈ E (cid:48)(cid:48) (cid:27) . Also let H := (cid:110) ( i, j ) : x (cid:48)(cid:48) i,j (cid:54) = 0 (cid:111) . It follows by the above setting that, L ∪ H corresponds exactlyto the set of variables in E (cid:48)(cid:48) . Since the coefficient matrix of E (cid:48)(cid:48) has full rank, the pivot in each rowof the matrix defines a one-to-one mapping φ : L ∪ H → M ∪ M between the variables and theconstraints.Consider each i ∈ L . Since the variable y i appears exactly in one constraint in M , the mappingmust map y i to the constraint it corresponds to, i.e., φ ( i ) = i . Since the constraint i corresponds toin M contributes one rank, it is non-degenerated and contains at least one variable in H . Let x i,j be one such variable. Since x i,j appears in exactly two constraints, i.e., in the one i corresponds toin M and the one j corresponds to in M , and since φ ( i ) = i , it follows that φ (( i, j )) = j . Sincethe mapping φ is one-to-one, j cannot be mapped to by other pairs.Applying the above argument for each i ∈ L results in a set consisting of distinct clients j from U with the same cardinality. This shows that | L | ≤ | U | .Applying Lemma 28, Lemma 29, Lemma 30, and the fact that y (cid:48) i ≥ / i ∈ U , we obtainthe following bound for the cost incurred by (cid:16) x ◦ | G, D , y ∗ | G (cid:17) . (cid:88) i ∈ G y ∗ i + (cid:88) i ∈ G, j ∈D c i,j · x ◦ i,j ≤ · (cid:88) i ∈ G y (cid:48) (II) i + 2 · (cid:88) i ∈ U y (cid:48) i + (cid:88) i ∈ G, j ∈ H t (cid:48) j · (cid:0) c i,p ( j ) + 2 · α j (cid:1) · x (cid:48) (II) i,j + (cid:88) i ∈ G, j ∈D t (cid:48) j · ( c i,j + 2 · α j ) · x (cid:48) (II) i,j ≤ · (cid:88) i ∈ G y (cid:48) (II) i + 2 · (cid:88) i ∈ U y (cid:48) i + (cid:88) i ∈ G, j ∈ H (cid:0) c i,p ( j ) + 2 · α j (cid:1) · x (cid:48) (II) i,j + (cid:88) i ∈ G, j ∈D (cid:0) · c i,j + 2 · t (cid:48) j · α j (cid:1) · x (cid:48) (II) i,j , (33)where in the last inequality we apply Corollary 24 the fact that t (cid:48) j ≤ j ∈ D and thedefinition that t (cid:48) j = 1 for all j ∈ H . 35 he Clusters in C D (cid:48) . In the following we consider the the clusters in C D (cid:48) . The following lemmabounds the cost incurred by each individual cluster q in C D (cid:48) . Lemma 31.
For any q ∈ C D (cid:48) , we have(i) y ∗ i ( q ) ≤ y (cid:48) ( q ) i ( q ) + 2 δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } y (cid:48) ( q ) k , and(ii) (cid:88) j ∈D c i ( q ) ,j · x ◦ i ( q ) ,j ≤ (cid:88) j ∈ D · c i ( q ) ,j · x (cid:48) ( q ) i ( q ) ,j + (cid:88) j ∈ H c i ( q ) ,p ( j ) · x (cid:48) ( q ) i ( q ) ,j + δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } (cid:88) j ∈D · c k,j · x (cid:48) ( q ) k,j + (cid:88) j ∈ H c k,p ( j ) · x (cid:48) ( q ) k,j + (cid:88) j ∈D · t (cid:48) j · α j · x ∗ i ( q ) ,j + (cid:88) j ∈ H · α j · x ∗ i ( q ) ,j . Proof of Lemma 31.
The lemma follows directly from the rounding process for clusters in C D (cid:48) .Consider the iteration for which q is formed and ready to be rounded. By the algorithm design,the total facility value that has been removed from F (cid:48) due to the rounding process for q is y (cid:48) ( q ) i ( q ) + δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } y (cid:48) ( q ) k = 12 = 12 · y ∗ i ( q ) , where in the first equality we apply the definition of δ i ( q ) . Attributing the cost of y ∗ i ( q ) to the facilityvalue that is removed from F (cid:48) due to cluster q proves the first part of this lemma.For the second part, by the definition of x ◦ and the way how the algorithm reroutes the assign-ments from facilities in B ( q ) \ { i ( q ) } to i ( q ), we have (cid:88) j ∈D c i ( q ) ,j · x ◦ i ( q ) ,j = (cid:88) j ∈D c i ( q ) ,j · t (cid:48) j · x ∗ i ( q ) ,j + (cid:88) (cid:96) ∈ H ( j ) x ∗ i ( q ) ,(cid:96) = (cid:88) j ∈D c i ( q ) ,j · t (cid:48) j · x (cid:48) ( q ) i ( q ) ,j + (cid:88) (cid:96) ∈ H c i ( q ) ,p ( (cid:96) ) · x (cid:48) ( q ) i ( q ) ,(cid:96) + δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } (cid:88) j ∈D c i ( q ) ,j · t (cid:48) j · x (cid:48) ( q ) k,j + (cid:88) (cid:96) ∈ H c i ( q ) ,p ( (cid:96) ) · x (cid:48) ( q ) k,(cid:96) . (34)By the algorithm setting, for any k ∈ B ( q ) \ { i ( q ) } and any (cid:96) ∈ H with x (cid:48) ( q ) k,(cid:96) >
0, we have c i ( q ) ,p ( (cid:96) ) ≤ c k,p ( (cid:96) ) + c k,j ( q ) + c i ( q ) ,j ( q ) ≤ c k,p ( (cid:96) ) + 2 α j ( q ) ≤ c k,p ( (cid:96) ) + 2 α (cid:96) , where in the second inequality we apply the fact that i ( q ) and k are in B ( q ), which implies that x (cid:48) ( q ) i ( q ) ,j ( q ) > x (cid:48) ( q ) k,j ( q ) >
0, and max (cid:0) c i ( q ) ,j ( q ) , c k,j ( q ) (cid:1) ≤ α j ( q ) by complementary slackness, and in thelast inequality we apply the assumption that x (cid:48) ( q ) k,(cid:96) >
0, which implies that (cid:96) ∈ H (cid:48) ( q ) and α j ( q ) ≤ α (cid:96) by the way j ( q ) is selected. By a similar argument, we have c i ( q ) ,j ≤ c k,j + 2 α j for any k ∈ B ( q ) \ { i ( q ) } and any j ∈ D with x (cid:48) ( q ) k,j >
0. 36y the above conclusion, Corollary 24, and the way the assignment x ∗ is formed during therounding process, we have δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } , j ∈D c i ( q ) ,j · t (cid:48) j · x (cid:48) ( q ) k,j ≤ δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } , j ∈D ( c k,j + 2 · α j ) · t (cid:48) j · x (cid:48) ( q ) k,j ≤ δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } , j ∈D · c k,j · x (cid:48) ( q ) k,j + (cid:88) j ∈D · t (cid:48) j · α j · x ∗ i ( q ) ,j . Similarly, we have δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } , (cid:96) ∈ H c i ( q ) ,p ( (cid:96) ) · x (cid:48) ( q ) k,(cid:96) ≤ δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } , (cid:96) ∈ H c k,p ( (cid:96) ) · x (cid:48) ( q ) k,(cid:96) + (cid:88) (cid:96) ∈ H · α (cid:96) · x ∗ i ( q ) ,(cid:96) . Combining the above two inequalities with (34) and further applying Corollary 24, we have (cid:88) j ∈D c i ( q ) ,j · x ◦ i ( q ) ,j ≤ (cid:88) j ∈ D · c i ( q ) ,j · x (cid:48) ( q ) i ( q ) ,j + (cid:88) (cid:96) ∈ H c i ( q ) ,p ( (cid:96) ) · x (cid:48) ( q ) i ( q ) ,(cid:96) + δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } (cid:88) j ∈D · c k,j · x (cid:48) ( q ) k,j + (cid:88) (cid:96) ∈ H c k,p ( (cid:96) ) · x (cid:48) ( q ) k,(cid:96) + (cid:88) j ∈D · t (cid:48) j · α j · x ∗ i ( q ) ,j + (cid:88) (cid:96) ∈ H · α (cid:96) · x ∗ i ( q ) ,(cid:96) . Take summation on the cost given in Lemma 31 over all clusters in C D (cid:48) (in the reverse orderthey are formed) and consider the contribution of each facility in F ∗ D (cid:48) ∪ G into the RHS of thesummation. We have the following lemma which bounds the cost incurred by x ◦ | F ∗ D (cid:48) , D and y ∗ | F ∗ D (cid:48) . Lemma 32. (i) (cid:88) i ∈ F ∗ D (cid:48) y ∗ i ≤ · (cid:88) i ∈ I \ G y (cid:48) i + 2 · (cid:88) i ∈ G (cid:16) y (cid:48) i − y (cid:48) (II) i (cid:17) . (35)(ii) (cid:88) i ∈ F ∗ D (cid:48) , j ∈D c i,j · x ◦ i,j ≤ (cid:88) i ∈ F ∗ D (cid:48) , j ∈D · t (cid:48) j · α j · x ∗ i,j + (cid:88) i ∈ F ∗ D (cid:48) , j ∈ H · α j · x ∗ i,j + (cid:88) i ∈ G, j ∈D · c i,j · x (cid:48) i,j − (cid:88) (cid:96) ∈ H ( j ) x (cid:48) i,(cid:96) − x (cid:48) (II) i,j + (cid:88) i ∈ G, j ∈ H c i,p ( j ) · (cid:16) x (cid:48) i,j − x (cid:48) (II) i,j (cid:17) + (cid:88) i ∈ I \ G, j ∈D · c i,j · x (cid:48) i,j − (cid:88) (cid:96) ∈ H ( j ) x (cid:48) i,(cid:96) + (cid:88) i ∈ I \ G, j ∈ H c i,p ( j ) · x (cid:48) i,j . (36) Proof of Lemma 32.
Consider the first part of Lemma 31. We have (cid:88) i ∈ F ∗ D (cid:48) y ∗ i = (cid:88) q ∈C D (cid:48) y ∗ i ( q ) ≤ (cid:88) q ∈C D (cid:48) y (cid:48) ( q ) i ( q ) + 2 δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } y (cid:48) ( q ) k . (37)37e charge the cost of each item in the RHS to the facilities in I . By the design of the scaled-downoperation for rounding each q ∈ C D (cid:48) , we know that, for each k ∈ ( I \ G ) ∩ D (cid:48) ( q ) , the facility value y (cid:48) ( q ) k decreases exactly by δ i ( q ) · y (cid:48) ( q ) k if k (cid:54) = i ( q ) and decreases by y (cid:48) ( q ) k followed by taken from considerationotherwise. For each k ∈ G ∩ D (cid:48) ( q ) , the facility value y (cid:48) ( q ) k decreases exactly by δ i ( q ) · y (cid:48) ( q ) k . Hence, bycharging the cost of each item in the RHS of (37) to the value decreased for each facility, we obtain (cid:88) i ∈ F ∗ D (cid:48) y ∗ i ≤ · (cid:88) i ∈ I \ G y (cid:48) i + 2 · (cid:88) i ∈ G (cid:16) y (cid:48) i − y (cid:48) (II) i (cid:17) . Note that, for each i ∈ G , its facility value y (II) i is preserved for the second phase when it is includedinto a cluster in C H (cid:48) , and thereby receives no charge from the RHS of (37) by the charging scheme.The second part of this lemma follows from an analogous charging argument. Consider thesecond part of Lemma 31. We have (cid:88) i ∈ F ∗ D (cid:48) , j ∈D c i,j · x ◦ i,j = (cid:88) q ∈C D (cid:48) (cid:88) j ∈D c i ( q ) ,j · x ◦ i ( q ) ,j ≤ (cid:88) q ∈C D (cid:48) (cid:88) j ∈ D · c i ( q ) ,j · x (cid:48) ( q ) i ( q ) ,j + (cid:88) j ∈ H c i ( q ) ,p ( j ) · x (cid:48) ( q ) i ( q ) ,j + (cid:88) q ∈C D (cid:48) δ i ( q ) · (cid:88) k ∈ B ( q ) \{ i ( q ) } (cid:88) j ∈D · c k,j · x (cid:48) ( q ) k,j + (cid:88) j ∈ H c k,p ( j ) · x (cid:48) ( q ) k,j + (cid:88) q ∈C D (cid:48) (cid:88) j ∈D · t (cid:48) j · α j · x ∗ i ( q ) ,j + (cid:88) j ∈ H · α j · x ∗ i ( q ) ,j . (38)For the last item in (38), we have (cid:88) q ∈C D (cid:48) (cid:88) j ∈D · t (cid:48) j · α j · x ∗ i ( q ) ,j + (cid:88) j ∈ H · α j · x ∗ i ( q ) ,j = (cid:88) i ∈ F ∗ D (cid:48) , j ∈D · t (cid:48) j · α j · x ∗ i,j + (cid:88) i ∈ F ∗ D (cid:48) , j ∈ H · α j · x ∗ i,j by definition. For the remaining items in the RHS of (38), we charge the cost to the assignmentvalues decreased due to the scaled-down operation when rounding each q ∈ C D (cid:48) . For each i ∈ I, j ∈ D , the assignment value lost due to the creation of outlier clients, if present, can be partlycharged except for those that become the demand of outlier clients, which is exactly (cid:80) (cid:96) ∈ H ( j ) x (cid:48) i,(cid:96) .For i ∈ G, j ∈ C ∪ H , the assignment value that is included into a cluster in C H (cid:48) and preservedfor the second phase, does not receive charges from the RHS of (38). Therefore, the second part ofthis lemma follows. 38 he Overall Guarantee. In the following we establish the overall guarantee. Combining In-equality (33), Inequality (35), and Inequality (36), we obtain ψ ( x ◦ , y ∗ ) = ψ (cid:16) x ◦ | U, D , y ∗ | U (cid:17) + ψ (cid:16) x ◦ | F ∗ D (cid:48) , D , y ∗ | F ∗ D (cid:48) (cid:17) + ψ (cid:16) x ◦ | G, D , y ∗ | G (cid:17) ≤ · (cid:88) i ∈ U y (cid:48) i + 2 · (cid:88) i ∈ I y (cid:48) i + (cid:88) i ∈ U, j ∈D c i,j · x (cid:48) i,j + (cid:88) i ∈ I, j ∈D · c i,j · x (cid:48) i,j − (cid:88) (cid:96) ∈ H ( j ) x (cid:48) i,(cid:96) + (cid:88) i ∈ I, j ∈ H c i,p ( j ) · x (cid:48) i,j + (cid:88) j ∈D · t (cid:48) j · α j + (cid:88) j ∈ H · α j · (cid:88) i ∈ F ∗ D (cid:48) x ∗ i,j + (cid:88) i ∈ G x (cid:48) (II) i,j . (39)Consider the item (cid:80) i ∈ I, j ∈ H c i,p ( j ) · x (cid:48) i,j in (39). By the definition of H ( j ) for each j ∈ D , we have (cid:88) i ∈ I, j ∈ H c i,p ( j ) · x (cid:48) i,j = (cid:88) i ∈ I, j ∈D , (cid:96) ∈ H ( j ) c i,j · x (cid:48) i,(cid:96) . (40)Consider the last item in (39). Applying the definition of t (cid:48) j for each j ∈ D with (cid:80) i ∈ F ∗ D (cid:48) x ∗ i,j + (cid:80) i ∈ G x (cid:48) (II) i,j >
0, we have (cid:88) j ∈D · t (cid:48) j · α j · (cid:88) i ∈ F ∗ D (cid:48) x ∗ i,j + (cid:88) i ∈ G x (II) i,j = (cid:88) j ∈D · α j · (cid:32) − (cid:88) i ∈ U x (cid:48) i,j − r (cid:48) j (cid:33) . (41)By the algorithm design, we have α j = α p ( j ) + c w ( j ) ,p ( j ) for any j ∈ H . Further applying thefact that the demand d j of any outlier client j ∈ H is fully-assigned when created and remainsfully-assigned during the rounding process, it follows that (cid:88) j ∈ H · α j · (cid:88) i ∈ F ∗ D (cid:48) x ∗ i,j + (cid:88) i ∈ G x (II) i,j = (cid:88) j ∈ H · (cid:0) α p ( j ) + c w ( j ) ,p ( j ) (cid:1) · d j ≤ (cid:88) j ∈D · α j · r (cid:48) j + (cid:88) j ∈ H · c w ( j ) ,p ( j ) · x (cid:48) w ( j ) ,p ( j ) ≤ (cid:88) j ∈D · α j · r (cid:48) j + 2 · (cid:88) i ∈ U, j ∈D c i,j · x (cid:48) i,j , (42)where in the second last inequality follows from the fact that (cid:88) k ∈ H ( j ) α p ( j ) · d k = α j · (cid:88) k ∈ H ( j ) d k = α j · (cid:88) k ∈ H ( j ) r (cid:48) j · x (cid:48) w ( j ) ,p ( j ) (cid:80) i ∈ U x (cid:48) i,p ( j ) = α j · r (cid:48) j for all j ∈ D and the fact that d j ≤ x (cid:48) w ( j ) ,p ( j ) for any j ∈ H by the definition of d j .39ombining (41) and (42), we have (cid:88) j ∈D · t (cid:48) j · α j + (cid:88) j ∈ H · α j · (cid:88) i ∈ F ∗ D (cid:48) x ∗ i,j + (cid:88) i ∈ G x (II) i,j ≤ (cid:88) j ∈D · (cid:32) − (cid:88) i ∈ U x (cid:48) i,j (cid:33) · α j + 2 · (cid:88) i ∈ U, j ∈D c i,j · x (cid:48) i,j . (43)Further combining (40) and (43) with (39), we obtain ψ ( x ◦ , y ∗ ) ≤ · (cid:88) i ∈ U y (cid:48) i + 3 · (cid:88) i ∈ U, j ∈D c i,j · x (cid:48) i,j + 2 · (cid:88) i ∈ I y (cid:48) i + 2 · (cid:88) i ∈ I, j ∈D c i,j · x (cid:48) i,j + (cid:88) j ∈D · (cid:32) − (cid:88) i ∈ U x (cid:48) i,j (cid:33) · α j . (44)The following lemma follows from complementary slackness between ( x (cid:48) , y (cid:48) ) and ( α , β , Γ , η ), andthe fact that 0 < y (cid:48) i < i ∈ I . Lemma 33. (cid:88) j ∈D (cid:32) − (cid:88) i ∈ U x (cid:48) i,j (cid:33) · α j ≤ (cid:88) i ∈ I y (cid:48) i + (cid:88) i ∈ I, j ∈D c i,j · x (cid:48) i,j . Proof of Lemma 33.
Consider any i ∈ I and the cost incurred. From the fact that ( x (cid:48) , y (cid:48) ) and( α , β , Γ , η ) are optimal primal and dual solutions for LP-(N) and LP-(DN), by complementaryslackness conditions, we have y (cid:48) i + (cid:88) j ∈D c i,j · x (cid:48) i,j = y (cid:48) i · u i · β i + (cid:88) j ∈D Γ i,j + (cid:88) j ∈D c i,j · x (cid:48) i,j since y (cid:48) i > y (cid:48) i < η i mustbe zero. By the fact that β i > i,j > β i · (cid:88) j ∈D x (cid:48) i,j + (cid:88) j ∈D Γ i,j · x (cid:48) i,j + (cid:88) j ∈D c i,j · x (cid:48) i,j . Finally, applying the fact that x (cid:48) i,j > y (cid:48) i + (cid:88) j ∈D c i,j · x (cid:48) i,j = (cid:88) j ∈D α j · x (cid:48) i,j . Taking summation over all facilities in I , we obtain (cid:88) i ∈ I y (cid:48) i + (cid:88) i ∈ I, j ∈D c i,j · x (cid:48) i,j = (cid:88) j ∈D , i ∈ I α j · x (cid:48) i,j ≥ (cid:88) j ∈D (cid:32) − (cid:88) i ∈ U x (cid:48) i,j (cid:33) · α j . where in the last equality we apply constraint (N-1) for each j ∈ D .Applying Lemma 33 on Inequality (44), we have ψ ( x ◦ , y ∗ ) ≤ · (cid:88) i ∈F y (cid:48) i + 4 · (cid:88) i ∈F , j ∈D c i,j · x (cid:48) i,j , and Theorem 5 is proved. 40 cknowledgement The author thanks Kai-Min Chung for inspiring discussion in early stages of this work and theanonymous reviewers for helpful comments on the presentation of this work.
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