Robust Fault Tolerant uncapacitated facility location
aa r X i v : . [ c s . D S ] F e b ROBUST FAULT TOLERANT UNCAPACITATED FACILITY LOCATION
SHIRI CHECHIK AND DAVID PELEG Department of Computer Science and Applied MathematicsThe Weizmann Institute of ScienceRehovot 76100, Israel
E-mail address : {shiri.chechik,david.peleg}@weizmann.ac.il Abstract.
In the uncapacitated facility location problem, given a graph, a set of demandsand opening costs, it is required to find a set of facilities R , so as to minimize the sumof the cost of opening the facilities in R and the cost of assigning all node demands toopen facilities. This paper concerns the robust fault-tolerant version of the uncapacitatedfacility location problem (RFTFL). In this problem, one or more facilities might fail, andeach demand should be supplied by the closest open facility that did not fail. It is requiredto find a set of facilities R , so as to minimize the sum of the cost of opening the facilitiesin R and the cost of assigning all node demands to open facilities that did not fail, afterthe failure of up to α facilities. We present a polynomial time algorithm that yields a 6.5-approximation for this problem with at most one failure and a 1 . . α -approximationfor the problem with at most α > RF T F L problem isNP-hard even on trees, and even in the case of a single failure.
Introduction
The robust fault-tolerant facility location problem
For a given optimization problem, the robust fault-tolerant version of the problem callsfor finding a solution that is still valid even when some components of the system fail.We consider the robust fault-tolerant version of the uncapacitated facility location (UFL) problem. In this problem, given a graph G , a demand ω ( v ) for every node v and a cost f ( v )for opening a facility at v , it is required to find a set of facilities R , so as to minimize thesum of the costs of opening the facilities in R and of shipping the demands of each nodefrom the nearest open facility (at a cost proportional to the distance). In the robust fault-tolerant version of this problem (RFTFL), one or more facilities might fail. Subsequently,each demand should be supplied by the closest open facility that did not fail. It is requiredto select a set of facilities R , so as to minimize the sum of the costs of opening the facilitiesin R and the costs of assigning all node demands to open facilities that did not fail, afterthe failure of up to α facilities. We present a polynomial time algorithm that yields a 6.5-approximation for this problem with at most one failure and a 1 . . α -approximation Key words and phrases: facility location, approximation algorithms, fault-tolerance. c (cid:13) S. Chechik and D. Peleg CC (cid:13) Creative Commons Attribution-NoDerivs License
92 S. CHECHIK AND D. PELEG for the problem with at most α >
RF T F L problem isNP-hard even on trees, and even in the case of a single failure.
Related Work
Many papers deal with approximating the
U F L problem, cf. [3, 4, 7, 9, 12, 13]. Thebest approximation ratio known for this problem is 3/2, shown by Byrka in [2].A fault-tolerant version of the facility location problem was first introduced by Jainand Vazirani [10], who gave it an approximation algorithm with ratio dependent on theproblem parameters. The approximation ratio was later improved by Guha et al. to 2.41[8] and then by Swamy and Shmoys to 2.076 [14]. However, the variant of the problemstudied in these papers is different from the one studied here. In that version, every node j is assigned in advance to a number of open facilities, and pays in advance for all of them.More explicitly, every node j is assigned to r j open facilities, and its shipping cost is someweighted linear combination of the costs of shipping its demand from all the facilities towhich it is assigned. It is required to find a set of facilities R that minimizes the sum of thecosts of opening the facilities in R and the sum of costs of shipping the demand of each node j from its r j facilities in R . This approach is used to capture the expected cost of supplyingall clients demand when some of the facilities fail. In contrast, in our definition a node j does not pay in advance for shipping its demand from a number of open facilities. Rather,it pays only for the cost of shipping its demand from the surviving facility that actuallysupplied its demand. Hence our definition for the fault-tolerant facility location problemrequires searching for a set of facilities R that minimizes the sum of the costs of openingthe facilities in R and the costs of assigning the demands of each node to one open facilitythat did not fail, for any failure of up to α facilities. Our approach is used to capture theworst case cost of supplying all clients demand when some of the facilities fail. We arguethat our definition may be more natural in some cases, where after the failure of somefacilities, each demand should still be supplied by a single supplier, preferably the closestsurviving open facility, and each client should pay only for the cost of shipping its demandfrom that surviving facility, and not for all the other (possibly failed) facilities to which itwas assigned originally. On the technical level, the approach taken in [8, 10, 14] is based onapplying randomized rounding techniques and primal-dual methods to the correspondinginteger linear program. This approach does not readily apply to our version of the problem,and we use a direct combinatorial algorithmic approach instead.Two other closely related types of problems are the 2-stage stochastic and robust opti-mization problems (cf. [5, 6]). Both of these models involve two decision stages. In the firststage, some facilities may be purchased. This stage is followed by some scenario dependingon the specifics of the problem at hand (in a facility location problem for example, thescenario may specify the clients and their corresponding demands). Subsequently, a secondstage is entered, in which it is allowed to purchase additional facilities (whose cost mightbe much higher than in the first stage). In stochastic optimization there is a distributionover all possible scenarios and the goal is to minimize the expected total cost. In robustoptimization the goal is to minimize the cost of the first stage plus the cost of the worst casescenario in the second stage. In contrast with these two models, in our variant the facilitiesmust be selected and opened in advance, and these advance decisions must be adequateunder all possible future scenarios. OBUST FAULT TOLERANT UNCAPACITATED FACILITY LOCATION 193
Billionnet and Costa [1] showed a polynomial time algorithm for solving the ordinary(non-fault-tolerant) UFL problem on trees. In contrast, we show that the fault-tolerantvariant RFTFL is NP-hard on trees.
1. Preliminaries
Let us start with common notation to be used later on. Consider an optimizationproblem Π over a universe V , which given an instance I , requires finding a solution consistingof a set of elements R ⊆ V . Denote by C Π ( I, R ) the cost of the solution R on the instance I of Π. Let R ∗ Π ( I ) denote the optimal solution to the problem Π on instance I , and let C ∗ Π ( I ) = C Π ( I, R ∗ Π ( I )) be the cost of the optimal solution. We denote our algorithm foreach problem Π studied later by A Π ( I ). The solution returned by the algorithm is referredto as R alg Π ( I ) and its cost is C alg Π ( I ) = C Π ( I, R alg Π ( I )).Let us now define the uncapacitated facility location (UFL) problem. Let I = h G, l, f, ω i be an instance of the problem, where G = ( V, E ) is a graph with vertex set V = { , ..., n } and edge set E . Each node v ∈ V hosts a client in need of service, and may host a facility,providing service to clients in nearby nodes. Each edge e ∈ E has a positive length l ( e ).The distance d ( u, v ) between two points u and v on G is defined to be the length of theshortest path between them, where the length of a path is the sum of the lengths of itsedges. For each node v , let f ( v ) denote the opening cost associated with placing a facilityat v , and let ω ( v ) denote the demand of the node v . The shipping cost of assigning thedemand ω ( u ) of a client u to an open facility v is the product SC u,v = ω ( u ) d ( u, v ). Theshipping cost SC u,R from a set of open facilities R to a node u is the minimum cost ofassigning u to a server in R , namely, SC u,R = min { SC u,v | v ∈ R } . Defining the distance d ( v, R ) between a set of points R and a point v on G to be the minimum distance between v and any node in R , i.e., d ( v, R ) = min r ∈ R d ( v, r ), we also have SC v,R = ω ( v ) d ( v, R ).It is required to find a subset R ⊆ V that minimizes the sum of costs of opening thefacilities in R and the shipping costs from R to all other nodes. This problem can beformulated as searching for a subset R ⊆ { , ..., n } that minimizes the cost function C UF L ( I, R ) = C facil ( I, R ) + C ship ( I, R ) , (1.1)where C facil ( I, R ) = X r ∈ R f ( r ) and C ship ( I, R ) = n X u =1 SC u,R = n X u =1 ω ( u ) · d ( u, R ) . Given a set R of open facilities and a facility r ∈ R , let ϕ ( I, r, R ) denote the set of clientsthat are served by r under R , i.e., ϕ ( I, r, R ) = { u | d ( v, r ) ≤ d ( v, r ′ ) f or every r ′ ∈ R } , orin other words, the nodes u that satisfy d ( u, R ) = d ( u, r ), where ties are broken arbitrarily,i.e., if there is more than one open facility r such that d ( u, R ) = d ( u, r ), then just chooseone open facility r that satisfies d ( u, R ) = d ( u, r ) and add u to ϕ ( I, r, R ). (When the set R is clear from the context we omit it and write simply ϕ ( I, r ), or even ϕ ( r ) when the instance I is clear as well.)The robust fault-tolerant facility location (RFTFL) problem is defined as follows. Eachclient is supplied by the nearest open facility, and in case this facility fails - it is suppliedby the next nearest open facility. We would like to find a solution that is tolerant against afailure of one node. This problem can be formulated as searching for a subset R ⊆ { , ..., n } that minimizes the cost function
94 S. CHECHIK AND D. PELEG C RF T F L ( I, R ) = C facil ( I, R ) + C ship ( I, R ) + C backup ( I, R ) , (1.2)where C facil ( I, R ) and C ship ( I, R ) are defined as above and C backup ( I, R ) = max r ∈ R X v ∈ ϕ ( I,r,R ) ω ( v ) · ( d ( v, R \{ r } ) − d ( v, r )) . (1.3)Note that C RF T F L ( I, R ) = C facil ( I, R ) + max r ∈ R { C ship ( I, R \ { r } ) } = C facil ( I, R ) + max r ∈ R ( n X v =1 SC v,R \{ r } ) = C facil ( I, R ) + max r ∈ R ( n X v =1 ω ( v ) · d ( v, R \{ r } ) ) . (1.4)Again, when the instance I is clear from the context we omit it and write simply C RF T F L ( R ), C facil ( R ), C ship ( R ), C backup ( R ), etc.We also consider the robust α -fault-tolerant facility location ( α RFTFL) problem, forinteger α ≥
1, where the solution should be resilient against a failure of up to α nodes. Wedefine the α RFTFL as follows. Each client is supplied by the nearest open facility whichdid not fail. We are looking for a subset R ⊆ { , ..., n } that minimizes the cost function C α RF T F L ( I, R ) = C facil ( I, R ) + max | R ′ |≤ α ( n X v =1 ω ( v ) · d ( v, R \ R ′ ) ) . (1.5)
2. A constant approximation algorithm for RFTFL
Towards developing a constant ratio approximation algorithm for RFTFL, we firstconsider a different problem, named concentrated backup (conc bu) , defined as follows. Aninstance of the problem consists of a pair h I, R i where I = h G, l, f, ω i is defined as beforeand R = { r , ..., r k } is a set of nodes. In this version, the nodes of R act as both clientsand servers (with open facilities), and all other nodes v / ∈ R have zero demands. Informally,it is assumed that we have already paid for opening the facilities in R , and each r ∈ R serves itself, at zero shipping cost. The problem requires to assign each client r ∈ R to abackup server v = r , which may be either some server in R or a new node from V \ R . Fora set of nodes R , define the backup cost C bu ( I, R , R ) = max r ∈ R (cid:8) SC r,R ∪ R \{ r } (cid:9) = max r ∈ R { ω ( r ) d ( r, R ∪ R \{ r } ) } . We are looking for a set R minimizing C conc bu ( I, R , R ) = C facil ( R ) + C bu ( R , R ) . (2.1)We denote this minimum cost by C ∗ conc bu ( I, R ). We show a 2-approximation algorithm forthe concentrated backup problem. OBUST FAULT TOLERANT UNCAPACITATED FACILITY LOCATION 195
The problems studied in this section and in section 3.1 are closely related to thoseconsidered in [11], and to solve them we use methods similar to the ones presented in [11].Let us consider a simpler variant of the backup problem, named the bounded backup (bb) problem, which is defined on h I, R , M i and requires looking for a solution R minimizing C bb ( I, R , M, R ) = C facil ( R )subject to the constraint C bu ( R , R ) ≤ M , for integer M . We now present a relaxationalgorithm that finds a set R satisfying C facil ( R ) ≤ C ∗ bb ( R , M ) but obeying only therelaxed constraint C bu ( R , R ) ≤ M instead C bu ( R , R ) ≤ M .Algorithm A bb ( I, R , M ) (1) R algbb ← ∅ (2) For i = 1 to k do: • S i ← { v | ω ( r i ) d ( v, r i ) ≤ M }\{ r i } /* “relaxed” backup servers for r i */ • If S i ∩ ( R ∪ R algbb ) = ∅ then add to R algbb the node v in S i with theminimum facility cost f ( v ).(3) Return R algbb .Let us now prove the properties of algorithm A bb . For every r i ∈ R let the set offeasible backup servers be T i = { v | ω ( r i ) d ( v, r i ) ≤ M }\{ r i } . Let the set of relaxed backupservers selected by the algorithm (namely, the final set R algbb it returns) be R algbb ( R , M ) = { q alg , ..., q algJ } . Let ℓ j be the phase in which the algorithm adds the new facility q algj to R algbb ,for 1 ≤ j ≤ J . Lemma 2.1. T ℓ i ∩ T ℓ j = ∅ for ≤ i, j ≤ J . Proof:
Assume otherwise, and let v ∈ T ℓ i ∩ T ℓ j for some 1 ≤ i, j ≤ J, i = j . Assumewithout loss of generality that ω ( r ℓ i ) ≤ ω ( r ℓ j ). Since ω ( r ℓ j ) d ( v, r ℓ j ) ≤ M , necessarily ω ( r ℓ i ) d ( v, r ℓ j ) ≤ M as well, and by the definition of T ℓ i , also ω ( r ℓ i ) d ( v, r ℓ i ) ≤ M , hence ω ( r ℓ i ) d ( r ℓ i , r ℓ j ) ≤ ω ( r ℓ i )( d ( v, r ℓ i ) + d ( v, r ℓ j )) ≤ M, implying that r ℓ j ∈ S ℓ i ∩ R , so the algorithm should not have opened a new facility inphase ℓ i , contradiction. Lemma 2.2. C facil ( R algbb ( R , M )) ≤ C ∗ bb ( R , M ) . Proof:
Notice that there must be at least one node from every T i in the optimal solution R ∗ bb ( R , M ). By Lemma 2.1 the sets T ℓ , ..., T ℓ J are disjoint, so there are at least J distinctnodes q ∗ j ∈ R ∗ bb ( R , M ), one from each T ℓ j , for 1 ≤ j ≤ J . In each phase i , the algorithmselects the cheapest node in S i ⊇ T i . Therefore, f ( q algj ) ≤ f ( q ∗ j ) for every 1 ≤ j ≤ J . Hence C facil ( R algbb ( R , M )) = J P j =1 f ( q algj ) ≤ J P j =1 f ( q ∗ j ) ≤ C ∗ bb ( R , M ) . Lemma 2.3. C bu ( R , R algbb ( R , M )) ≤ M .
96 S. CHECHIK AND D. PELEG
Proof:
For each server r i in R , the algorithm ensures that there is at least one open facilityfrom the set S i , so ω ( r i ) d ( r i , R ∪ R algbb ( R , M ) \ { r i } ) ≤ M .Now we present an approximation algorithm A conc bu for the concentrated backup prob-lem using the relaxation algorithm A bb for the bounded backup problem. First note thatthere can be at most nk possible values for the shipping costs SC u,v = ω ( u ) d ( u, v ).Algorithm A conc bu ( I, R ) (1) For every M ∈ { SC u,v | u, v ∈ V } do: • let R algbb ( R , M ) ← A bb ( I, R , M ).(2) Return the set R algbb ( R , M ) with the minimum cost C conc bu ( R , R algbb ( R , M )). Lemma 2.4. C algconc bu ( I, R ) ≤ C ∗ conc bu ( I, R ) . Proof:
Recall that, letting R ∗ = R ∗ conc bu ( R ), C ∗ conc bu ( I, R ) = C conc bu ( I, R , R ∗ ) = C facil ( R ∗ ) + C bu ( I, R , R ∗ ) . Let u ∈ R be the node that attains the maximum shipping cost SC u,R ∪ R \{ u } , i.e., satisfies ω ( u ) d ( u, R ∪ R ∗ \{ u } ) = C bu ( I, R , R ∗ ), and let v ∈ R ∪ R ∗ \{ u } be its backup, i.e., theclosest node to u . Then C ∗ conc bu ( I, R ) = C conc bu ( I, R , R ∗ ) = C facil ( R ∗ )+ SC u,v . Since thealgorithm examines all possible values of M , it tests also M = SC u,v . For this value, thereturned set R algbb ( R , M ) has opening cost at most C ∗ bb ( R , M ) = C facil ( R ∗ ) and backupcost at most C bu ( I, R , R algbb ( R , M )) ≤ M by Lemmas 2.2 and 2.3. Since the algorithm takes theminimum cost C conc bu ( R , R algbb ( R , M )) over all possible values of M , the resulting costsatisfies C algconc bu ( I, R ) ≤ C facil ( R ∗ ) + 2 SC u,v ≤ C ∗ conc bu ( I, R ), namely, an approximationratio of 2. We now present a polynomial time algorithm A RF T F L that yields 6.5-approximationfor the robust fault-tolerant uncapacitated facility location problem RFTFL. Consider aninstance I = h G, l, f, ω i of the problem. The algorithm consists of three stages. Stage 1:
Apply the 1.5-approximation algorithm of [2] to the original UFL problem inorder to find an initial subset R of servers. Notice that the cost of this solution satisfies C UF L ( R ) ≤ . C ∗ UF L ≤ . C ∗ RF T F L . (2.2)Each node is now assigned to a server in R . Next, we need to assign to each node a backupserver which will serve it in case its original server fails. Stage 2:
Transform the given instance I = h V, l, ω, f i of the problem into an instance I ′ = h V, l, ω ′ , f ′ i as follows. First, change the facility cost f by setting f ′ ( r ) = 0 for r ∈ R .Next, for each server r ∈ R , relocate all the demands of the nodes that are served by r ,and place them at the server r itself, that is, set ω ′ ( r ) = ( P v ∈ ϕ ( I,r,R ) ω ( v ) , f or r ∈ R , , f or r / ∈ R . (2.3) OBUST FAULT TOLERANT UNCAPACITATED FACILITY LOCATION 197
Stage 3:
Invoke the 2-approximation algorithm A conc bu for the concentrated backup prob-lem on the new instance I ′ and the set R . The approximation algorithm returns a new set R . We then return the set R ∪ R as the final set of open facilities. Lemma 2.5.
For every instance I and set R ⊆ V , C ∗ conc bu ( I ′ , R ) ≤ C ∗ RF T F L ( I ) + C UF L ( I, R ) . Proof:
Consider some vertex r ∈ R and let ϕ ( I, r, R ) = { v r , ..., v rk r } be the nodes itserves. Consider the optimal solution R ∗ RF T F L ( I ) to the RFTFL problem. Let d ri be thedistance from r to v ri for 1 ≤ i ≤ k r , and also let x ri be the distance from v ri to its optimalbackup server, which is also its distance to R ∗ r ≡ R ∪ R ∗ RF T F L ( I ) \{ r } , i.e., x ri = d ( v ri , R ∗ r ).By the triangle inequality, d ( r, R ∗ r ) ≤ d ri + x ri , for every 1 ≤ i ≤ k r , so ω ′ ( r ) · d ( r, R ∗ r ) = k r X l =1 ω ( v rl ) · d ( r, R ∗ r ) ≤ k r X l =1 ω ( v rl )( d rl + x rl )= k r X l =1 ω ( v rl ) d ( v rl , R ) + k r X l =1 ω ( v rl ) x rl ≤ n X v =1 ω ( v ) · d ( v, R ) + n X v =1 ω ( v ) · d ( v, R ∗ r ) . Therefore, C bu ( I ′ , R , R ∗ RF T F L ( I )) = max r ∈ R (cid:8) ω ′ ( r ) · d ( r, R ∗ r ) (cid:9) ≤ C ship ( I, R ) + max r ∈ R ( n X v =1 ω ( v ) · d ( v, R ∗ r ) ) . Using (1.4) and (2.1) we now bound the cost of the optimal solution for problem conc bu by C ∗ conc bu ( I ′ , R ) ≤ C conc bu ( I ′ , R , R ∗ RF T F L ( I ))= C facil ( I ′ , R ∗ RF T F L ( I )) + C bu ( I ′ , R , R ∗ RF T F L ( I )) ≤ C facil ( I ′ , R ∗ RF T F L ( I )) + max r ∈ R ( n X v =1 ω ( v ) d ( v, R ∗ r ) ) + C ship ( I, R ) ≤ C ∗ RF T F L ( I ) + C ship ( I, R ) ≤ C ∗ RF T F L ( I, R ) + C UF L ( I, R ) . Lemma 2.6.
For every instance I and sets R , R ⊆ V , C RF T F L ( I, R ∪ R ) ≤ C UF L ( I, R ) + C conc bu ( I ′ , R , R ) . Proof:
The cost of opening the facilities in R ∪ R is clearly at most the cost of opening thefacilities in R plus the cost of opening the facilities in R . For every facility r ∈ R ∪ R ,in order to bound C ship ( I, R ∪ R \ { r } ), note that one can first move each client v to itsclosest open facility in R , and then move all the clients assigned to r (if r ∈ R ) to thebackup facility of r in R . The inequality follows. More formally we have the following.Recall that by (1.4), C RF T F L ( I, R ∪ R ) = C facil ( I, R ∪ R ) + max r ∈ R ∪ R { C ship ( I, R ∪ R \ { r } ) } .
98 S. CHECHIK AND D. PELEG
Consider first the case that max r ∈ R ∪ R { C ship ( I, R ∪ R \ { r } ) } is attained for some r ′ ∈ R .In this case, we get by (1.1) that C RF T F L ( I, R ∪ R ) = C facil ( I, R ∪ R ) + C ship ( I, R ∪ R \ { r ′ } ) ≤ C facil ( I, R ∪ R ) + C ship ( I, R )= C UF L ( I, R ) + C facil ( I, R ) ≤ C UF L ( I, R ) + C conc bu ( I ′ , R , R ) . So now assume that max r ∈ R ∪ R { C ship ( I, R ∪ R \ { r } ) } is attained for some r ′ ∈ R . There-fore, C RF T F L ( I, R ∪ R ) = C facil ( I, R ∪ R ) + C ship ( I, R ∪ R \ { r ′ } )= C facil ( I, R ) + C facil ( I, R ) + n X v =1 SC v,R ∪ R + X v ∈ ϕ ( I,r ′ ,R ∪ R ) ω ( v ) · ( d ( v, R ∪ R \{ r ′ } ) − d ( v, r ′ )) ≤ C UF L ( I, R ) + C facil ( I, R )+ max r ∈ R X v ∈ ϕ ( I,r,R ) ω ( v ) · ( d ( r, R ∪ R \{ r } )) = C UF L ( I, R ) + C facil ( I, R ) + max r ∈ R (cid:8) w ′ ( r ) · ( d ( r, R ∪ R \{ r } )) (cid:9) = C UF L ( I, R ) + C conc bu ( I ′ , R , R ) . Lemma 2.7.
Algorithm A RF T F L yields a 6.5-approximation for the RFTFL problem.
Proof:
Consider the set of opened facilities R ∪ R . By Lemma 2.4, R is a 2-approximationof the concentrated backup problem on the instance I ′ , so C conc bu ( I ′ , R , R ) ≤ C ∗ conc bu ( I ′ , R ) . By Lemma 2.5, C ∗ conc bu ( I ′ , R ) ≤ C ∗ RF T F L ( I ) + C UF L ( I, R ), hence C conc bu ( I ′ , R , R ) ≤ C ∗ RF T F L ( I ) + 2 C UF L ( I, R ) . Using Lemma 2.6 we get C RF T F L ( I, R ∪ R ) ≤ C UF L ( I, R ) + 2 C ∗ RF T F L ( I ) , and by (2.2), C RF T F L ( I, R ∪ R ) ≤ . C ∗ RF T F L ( I ).
3. An approximation algorithm for α RFTFL α backup problem As in the case of a single failure, we first consider a different problem, named concen-trated α backup ( conc α bu ) , defined as follows. An instance of the problem consists of apair h I, R i where I = h G, l, f, ω i is defined as before and R is a set of nodes. The nodesof R act as both clients and servers (with open facilities), and all other nodes v / ∈ R havezero demands. We are looking for a set R minimizing C conc α bu ( I, R , R ) = C facil ( R ) + C α bu ( I, R , R ) , (3.1) OBUST FAULT TOLERANT UNCAPACITATED FACILITY LOCATION 199 where C α bu is the maximum α backup cost for a set of nodes R , defined as C α bu ( I, R , R ) = max | F |≤ α X r ∈ ( F ∩ R ) ω ( r ) · d ( r, R ∪ R \ F ) . We will shortly present a 3 α -approximation algorithm for the concentrated α -backupproblem.Towards this, let us first consider a simpler variant of the backup problem, named the α -bounded backup ( α bb ) problem, which is defined on h I, R , M i and requires looking fora solution R minimizing C α bb ( R , M, R ) = C facil ( R )subject to the constraint C light α bu ( R , R ) ≤ M for some integer M , where C light α bu ( R , R ) = max r ∈ R , | F |≤ α { ω ( r ) d ( r, R ∪ R \ F ) } . We now present a relaxation algorithm that finds a set R satisfying C facil ( R ) ≤ C ∗ α bb ( R , M ) but allowing the relaxed constraint C light α bu ( R , R ) ≤ M instead of C light α bu ( R , R ) ≤ M .Algorithm A α bb ( I, R , M ) (1) R algα bb ← ∅ (2) Let r , ..., r k be the servers in R sorted by nonincreasing order of demands.(3) Z ← ∅ /* The set of servers r i where the algorithm opens facilities in phase i */(4) For i = 1 to k do:(5) • S i ← { v | ω ( r i ) d ( v, r i ) ≤ M }\{ r i } . • T i ← { v | ω ( r i ) d ( v, r i ) ≤ M }\{ r i }• If S i ∩ Z = ∅ then: – Add to R algα bb , the α − | T i ∩ ( R ∪ R algα bb ) | nodes in T i \ ( R ∪ R algα bb )with the lowest facility costs. – Z ← Z ∪ { r i } (6) Return R algα bb .Let us now prove the properties of Alg. A α bb . Let { ℓ j | ≤ j ≤ J } be the phases inwhich the algorithm adds new facilities to R algα bb . By a proof similar to that of Lemma 2.1,we have the following. Lemma 3.1. T ℓ i ∩ T ℓ j = ∅ for ≤ j < i ≤ J . Lemma 3.2. C facil ( R algα bb ( R , M )) ≤ C ∗ α bb ( R , M ) . Proof:
There must be at least α nodes in every T ℓ j in the optimal solution R ∗ α bb ( R , M ).By Lemma 3.1 the sets T ℓ j for 1 ≤ j ≤ J are disjoint, so the only nodes that the algorithmadds to R algα bb from the set T ℓ j are added at phase ℓ j . The algorithm selects the cheapestnodes in T ℓ j in order to complete to α nodes. Therefore, C facil ( R algα bb ( R , M ) ∩ T ℓ j ) ≤
00 S. CHECHIK AND D. PELEG C facil ( R ∗ α bb ( R , M ) ∩ T ℓ j ) for every 1 ≤ j ≤ J . Hence C facil ( R algα bb ( R , M )) = J X j =1 C facil ( R algα bb ( R , M ) ∩ T ℓ j ) ≤ J X j =1 C facil ( R ∗ α bb ( R , M ) ∩ T ℓ j ) ≤ C ∗ α bb ( R , M ) . Lemma 3.3. C α bu ( R , R algα bb ( R , M )) ≤ M . Proof:
For each server v i ∈ R , the algorithm ensures that either there are at least α openfacilities from the set T i or v i is at distance at most 2 M from another v j ∈ R that has α open facilities from the set T j . In the first case the distance is at most M and in the second- at most 3 M .Now we present an approximation algorithm A conc α bu for the concentrated α backupproblem, using the relaxation algorithm A α bb for the α bounded backup problem.Algorithm A conc α bu ( I, R ) (1) For every subset T ⊆ { SC v,u | v, u ∈ V } such that | T | ≤ α do: • M ( T ) ← P m ∈ T m • let R algα bb ( R , M ( T )) ← A α bb ( I, R , M ( T )).(2) Return the set R algα bb ( R , M ( T )) with the minimum cost C conc α bu ( R , R algα bb ( R , M ( T ))). Lemma 3.4. C algconc α bu ( I, R ) ≤ αC ∗ conc α bu ( I, R ) . Proof:
Denote the optimal solution for conc α bu on h I, R i by R ∗ = R ∗ conc α bu ( R ). Then C ∗ conc α bu ( I, R ) = C conc α bu ( I, R , R ∗ ) = C facil ( R ∗ ) + C α bu ( I, R , R ∗ ) . Let { u , ..., u j } ⊆ R and { v , ..., v j } ⊆ R ∪ R ∗ for some j ≤ α be the sets ofnodes that attain the maximum shipping cost, i.e., satisfy C α bu ( I, R , R ∗ ) = M for M = j P i =1 SC u i ,v i = j P i =1 ω ( u i ) d ( u i , v i ). Then C ∗ conc α bu ( I, R ) = C facil ( R ∗ ) + M . Noticethat there must be at least α nodes in the set R ∗ ∪ R at distance at most M fromevery server r in R . Clearly C facil ( R ∗ α bb ( R , M )) ≤ C facil ( R ∗ ). Since the algorithmexamines all possible values of M ( T ), it tests also M . For this value, the returned set R algα bb ( R , M ) has opening cost at most C ∗ α bb ( R , M ) ≤ C facil ( R ∗ ) and backup cost atmost C α bu ( I, R , R algα bb ( R , M )) ≤ M by Lemmas 3.2 and 3.3. Since the algorithm takesthe minimum cost C conc α bu ( R , R algα bb ( I, R , M ( T ))) over all possible subsets T , the result-ing cost is at most C algconc α bu ( I, R ) ≤ C conc α bu ( I, R , R algα bb ( R , M )) ≤ C facil ( R ∗ ) + max | F |≤ α X r ∈ ( F ∩ R ) ω ( r ) d ( r, R ∪ R algα bb ( R , M ) \ F ) ≤ C facil ( R ∗ ) + 3 αM ≤ αC ∗ conc α bu ( I, R ) . OBUST FAULT TOLERANT UNCAPACITATED FACILITY LOCATION 201 (1 . . α ) -approximation algorithm to the α RFTFL
We now present a polynomial time algorithm named A α RF T F L , yielding a (1 . . α )-approximation for the robust fault-tolerant uncapacitated facility location prob-lem α RFTFL against a failure of α nodes, for constant α >
1. Consider an instance I = h G, l, f, ω i of the problem. The algorithm is similar to Algorithm RFTFL, except forthe third stage. Instead of invoking the 2-approximation algorithm A conc bu for the concen-trated backup problem on the new instance I ′ and the set R , invoke the 3 α -approximationalgorithm A conc α bu for the concentrated α backup problem on the new instance I ′ and theset R . Algorithm A conc α bu returns a new set R alg . Algorithm A α RF T F L now returns theset R ∪ R alg . Proof of the following lemma is deferred to the full paper. Lemma 3.5.
Algorithm A α RF T F L yields a (1 . . α ) -approximation for the α RFTFLproblem.
4. Robust Fault-tolerant uncapacitated facility location on trees
In this section we show that the RFTFL problem is NP-hard even on trees. The claimholds even in the case where only the edge lengths or only the node demands are variableand the other parameters are uniform. An instance of the RFTFL problem is h T, l, f, ω, P i ,where T is a tree, l, f and ω are defined as before and P is an integer. It is required todecide if the cost of the optimal solution to the RFTFL problem on the instance h T, l, f, ω i is P or less.The proofs, via reductions from subset sum and from a variant of the partition problem,are deferred to the full paper. The following results are established. Theorem 4.1.
RF T F L on trees is NP-complete even with (1) unit edge lengths and opening costs (but variable node demands), (2) unit node demands and opening costs (but variable edge lengths).
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Combina-torica , (1982), 385–393. This work is licensed under the Creative Commons Attribution-NoDerivs License. To view acopy of this license, visit http://creativecommons.org/licenses/by-nd/3.0/http://creativecommons.org/licenses/by-nd/3.0/