Approximating Two-Stage Stochastic Supplier Problems
Brian Brubach, Nathaniel Grammel, David G. Harris, Aravind Srinivasan, Leonidas Tsepenekas, Anil Vullikanti
aa r X i v : . [ c s . D S ] A ug Approximation Algorithms for Radius-Based, Two-Stage StochasticClustering Problems with Budget Constraints
Brian Brubach ∗ Nathaniel Grammel † David G. Harris ‡ Aravind Srinivasan § Leonidas Tsepenekas ¶ Anil Vullikanti ‖ Abstract
The main focus of this paper is radius-based clustering problems in the two-stage stochasticsetting with recourse, where the inherent stochasticity of the model comes in the form of a budgetconstraint. We also explore a number of variants where additional constraints are imposed onthe first-stage decisions, specifically matroid and multi-knapsack constraints. Further, we showthat our problems have natural applications to allocating healthcare testing centers.The eventual goal is to provide results for supplier-like problems in the most general dis-tributional setting, where there is only black-box access to the underlying distribution. Ourframework unfolds in two steps. First, we develop algorithms for a restricted version of eachproblem, in which all possible scenarios are explicitly provided; second, we exploit structuralproperties of these algorithms and generalize them to the black-box setting. These key proper-ties are: (1) the algorithms produce “simple” exponential families of black-box strategies, and (2) there exist efficient ways to extend their output to the black-box case, which also preservethe approximation ratio exactly. We note that prior generalization approaches, i.e., variants ofthe
Sample Average Approximation method, can be used for the problems we consider, howeverthey would yield worse approximation guarantees. ∗ University of Maryland, College Park. Supported in part by NSF awards CCF-1422569 and CCF-1749864, andby research awards from Adobe. Email: [email protected] † University of Maryland, College Park. Supported in part by NSF awards CCF-1749864 and CCF-1918749, andby research awards from Amazon and Google. Email: [email protected] ‡ University of Maryland, College Park. Email: [email protected] § University of Maryland, College Park. Supported in part by NSF awards CCF-1422569, CCF-1749864 andCCF-1918749, and by research awards from Adobe, Amazon, and Google. Email: [email protected] ¶ University of Maryland, College Park. Supported in part by NSF awards CCF-1749864 and CCF-1918749, andby research awards from Amazon and Google. Email: [email protected] ‖ University of Virginia. Email: [email protected] Introduction
Stochastic optimization, first introduced in the work of Beale [3] and Dantzig [6], provides a wayfor modeling uncertainty in the realization of the input data. In this paper, we give approximationalgorithms for a family of problems in stochastic optimization, and more precisely in the 2- stagerecourse model [27]. This paradigm evolves in two stages. In the first, we are only given accessto a distribution D that describes possible realizations of future data. Given that knowledge, wetake some stage-I actions and commit to an anticipatory part of the solution x , incurring somecost c ( x ). In the second stage, an input instance (“scenario”) A is sampled from the distribution D , and we can take some stage-II recourse actions y A with cost f A ( x, y A ). If X is the set ofstage-I actions and Y the set of recourse actions, the goal is to find a solution x ⋆ ∈ X to minimize f ( x ) = c ( x ) + E A ∼D [ q A ( x )], where q A ( x ) = min y ∈ Y { f A ( x, y ) | ( x, y ) is a valid solution for A } .There are three main models proposed in the literature for how to represent knowledge of thedistribution D : the black-box model [25, 11, 22, 18, 26], the polynomial-scenarios model [24, 15, 21,9], and the independent-activations one [15, 7, 19]. We deal with all three of them in this paper.We later present the necessary details and definitions in the context of the problems we study. We are given a set of clients C and a set of facilities F , in a metric space with distance function d satisfying the triangle inequality. We let n = |C| and m = |F | . Our paradigm unfolds in two stages.In the first, each facility i ∈ F has a cost c Ii , but at that time we do not know which clients from C will need service, and we only have access to a distribution D which describes the potential arrivalsof clients later on. In the second stage, a set A ⊆ C (the “scenario”) is realized with probability p A according to D , and each facility i has a cost c Ai . The goal is to open a set of facilities F I in the firststage, and when A arrives in the second stage, to open some additional facilities F A , so that theexpected opening cost of the algorithm is at most some budget B : P i ∈ F I c Ii + E A ∼D [ P i ∈ F A c Ai ] ≤ B .Finally, note that by rescaling (divide all costs by the budget), we may always assume that B = 1. Modeling the Stage-I Distributional Knowledge:
We consider the following models.1. The most general way of representing knowledge of the distribution D is the black-box model,where we only have access to an oracle that can sample scenarios A according to D . Everytime a scenario A is revealed, either through the oracle or through an actual data realization,we also learn the facility-cost vector c A associated with it.2. In the polynomial-scenarios model, all scenarios A , together with their occurrence probabili-ties p A and their corresponding facility-cost vectors c A , are explicitly provided.3. In the independent-activation model, each j ∈ C arrives independently with some knownprobability p j . We also adopt an additional standard assumption for this model (see, e.g.,[15]), namely that the stage-II cost of a facility i is not scenario-dependent, and is proportionalto its stage-I cost. In other words, c IIi = λ · c Ii for some known parameter λ ≥ n, m . For the polynomial-scenario s case, the runtime should also be polynomial in the number of explicitly provided scenarios.Given this framework, we examine a family of problems, which may impose additional con-straints on the set F I and may also include different objectives. Letting d ( j, S ) = min i ∈ S d ( i, j ) forany j ∈ C and for any S ⊆ F , we summarize our problems of interest as follows.1 wo-Stage Stochastic (Matroid/Multi-Knapsack) Supplier: The objective here is to pick F I and F A , such that d ( j, F I ∪ F A ) ≤ R for every A that materializes and all j ∈ A , for theminimum R possible. We refer to this problem as . If the input also consists of a matroid M = ( F , I ), where I ⊆ F the family of independent sets of M , we require F I ∈ I and referto the problem as . On the other hand, if the input involves L additional knapsackconstraints on F I , we call the problem . Specifically, in this case we are given budgets W ℓ ≥ f ℓi ≥ i ∈ F and every integer ℓ ∈ [1 , L ], such that the stage-Ifacilities should satisfy P i ∈ F I f ℓi ≤ W ℓ for every ℓ . We call an instance of discrete , ifall weights f ℓi are integers, and also define parameter Λ = Q Lℓ =1 W ℓ . Two-Stage Stochastic Non-Uniform (Matroid) Supplier:
Here we let each client j ∈ C have a radius demand R j >
0, and we rather try to find a solution with d ( j, F I ∪ F A ) ≤ R j , forevery scenario A that materializes and every client j ∈ A . We refer to this problem as .When a matroid constraint is also imposed on F I , we call the resulting problem .We define a ρ -approximation for these problems as a solution which satisfies the budget constraintexactly, and also ensures that d ( j, F I ∪ F A ) ≤ ρR j for every A and j ∈ A .We use the suffixes BB, Poly , and IA to specify the distributional knowledge for our problems.For example, is the previously defined in the black-box model. and are natural extensions of well-known deterministic supplier problems [14]to the two-stage stochastic setting, and they have not been studied before. An important practicalapplication that these models capture, can be found in healthcare facility location for mitigating adisease outbreak, through the preventive placement of testing sites.Suppose that F corresponds to potential testing center locations, C to populations that canbe affected by a possible disease outbreak, and each scenario A ∈ D to which populations sufferthe outbreak. Since immediate testing is of utmost importance in preventing further spread of thedisease, a decision maker fixes a budget B , and sets up testing sites, such that under every A eachinfected point has close access to a testing center. Assembling these sites in advance, i.e., in stage-I,has multiple benefits; for example, the necessary equipment and materials might be much cheaperand easier to obtain, since during the outbreak a tremendous scarcity of them is to be expected.In many cases, there may be further constraints imposed on F I without regard to stage-II,which cannot be directly reduced to the budget B . Such constraints stem from the fact that stage-Ifacilities are paid for ahead of time, but needed only after the scenario realization. For instance,in our previous example we might have an additional constraint on the total number of personnelwe want to occupy, assuming that facility i requires f i people to keep it operational. Similarly,another constraint might be monthly expenses the central planner is required to pay for the set F I .To the extent of our knowledge, this is the first time additional stage-I constraints are studied inthe two-stage stochastic regime. Our ultimate goal is to devise algorithms in the black-box setting, and to that end we adopt atwo-step approach. First, we develop algorithms for the polynomial-scenarios model, which is lesscomplicated and can be viewed as essentially non-stochastic. Second, we formulate a samplingscheme that generalizes the latter to the black-box case.2o explain the issues here and compare our approach with previous results, consider the mini-mization of f ( x ) = c ( x ) + E A ∼D [ q A ( x )] in the black-box model, with X the set of stage-I decisions, Y the set of recourse actions, and q A ( x ) = min y ∈ Y { f A ( x, y ) | ( x, y ) is a valid solution for A } .There is a close connection between the black-box and the polynomial-scenarios model, viaan important result known as Sample Average Approximation (SAA) [5, 2]. In this method, apolynomial-sized set of scenarios S is sampled from D , so that f (¯ x ) ≤ α (1 + ǫ ) f ( x ⋆ ) holds withhigh probability, where x ⋆ is the minimizer of f ( x ), and ¯ x is an α -approximate minimizer of theempirical estimate of f ( x ), i.e., ˆ f ( x ) = c ( x ) + (cid:0) P A ∈ S q A ( x ) (cid:1) / | S | . Generalizing based on this resultfirst requires finding an α -approximate minimizer ¯ x of ˆ f ( x ) (where ˆ f ( x ) represents a polynomial-scenarios instance), and treating ¯ x as the stage-I actions. Then, given any A , the generalizationrequires re-solving the problem using any ρ -approximation algorithm, while assuming that ¯ x is afixed part of the solution. This yields a final approximation ratio of αρ + ǫ , where ρ ≤ α .Although this approach can capture any computational problem, it has a serious disadvantage:the problem needs to be re-solved in stage-II, and this leads to an additional multiplicative overhead ρ in the ratio. More advanced methods [22, 23, 26] can sometimes side-step the dependence on α ,by exploiting structural properties of the exponential LP describing f ( x ). These techniques canachieve an approximation ratio 2 ρ + ǫ for some problems, e.g., Set Cover and Facility Location.Since the inherent stochasticity appears only in the budget constraint for our problems, a naturalway to fit them within the existing frameworks is to use the opening cost as the objective function f .Consider our simplest problem , and assume the optimal radius R ∗ is known. The openingcost would then be C ( F I ) = P i ∈ F I c Ii + E A ∼D [ q A ( F I )] , where q A ( F I ) = min F A { P i ∈ F A c Ai | d ( j, F I ∪ F A ) ≤ R ∗ , ∀ j ∈ A } . Note that if we do not allow any violation in the distance-covering constraint,the best opening cost we can achieve is ρB where ρ is the approximation ratio for the underlyingnon-stochastic problem. In our example, this non-stochastic problem is simply to find facility set ofminimum cost covering all clients within distance R ∗ . This is as hard as approximating Set Cover(Appendix A), and hence ρ = Θ(log n ).Alternatively, to avoid this log n factor, we may allow a violation of the covering requirement.For this, a slight modification to the framework of [5, 2], yields a solution with opening cost atmost (1 + ǫ ) B , but also a multiplicative overhead in the radius ratio (e.g., in we would get α = ρ = 3 and eventually a 9 R ∗ coverage guarantee). The reason for this is that in this approachthe problem always requires the problem need to be re-solved in stage-II. Our Generalization Approach:
To get around these obstacles, we develop a novel generaliza-tion framework. The high-level idea is to use polynomial-scenarios approximation algorithms whichadditionally have specific structural properties. We begin by sampling N = poly( n, m ) scenarios S , S , . . . , S N , and then run the approximation algorithm on this collection. This gives a stage-Iset F I , and a stage-II solution F S v for each S v in the sample. To handle a new arriving scenario A ,we use the following key properties of the polynomial-scenarios algorithm: • Given F I and a stage-II arriving scenario A that is not among those initially sampled, thereis an efficient process that extends the polynomial-scenarios algorithm’s output to a stage-IIsolution F A ⊆ F , while also preserving the approximation ratio for the radius. • Given S , S , . . . , S N , the total number of black-box outcomes we can possibly produce, isonly exponential in n and m . By default, this number might be doubly-exponential in n, m .Our generalization scheme yields black-box results that (i) do not incur any increase in theopening cost, and (ii) preserve the radius approximation ratio α of the polynomial-scenarios variant(without even a constant-factor blowup). Unlike the method of [5, 2], we do not need to re-solve the3roblem in stage-II. Also, unlike the approach of [22, 23, 26], we do not pay an additional factorof 2 (either in the radius or the opening cost) due to LP rounding. For example, for our mostfundamental problem , we get a ratio of α = 3, exactly matching the lower bound for thenon-stochastic problem, while also achieving a near-optimal opening cost of (1 + ǫ ) B .Our approach also has a secondary benefit compared to the approach of [22, 23, 26]: the compu-tational core of our approximation algorithms involves solving an explicitly-presented, polynomial-size problem. By contrast, [22, 23, 26] require solving an implicitly-presented exponential-sized LP,which can only be done by computationally expensive algorithms based on separation oracles. In Section 2, we present the generalization framework capturing all the radius-based, budget con-strained problems we consider. We begin by formally defining the necessary properties required forthe underlying polynomial-scenarios algorithm, and we call such algorithms efficiently generalizable .Then, a simplified version of our main generalization result is the following:
Theorem 1.1 (Informal) . Suppose we have an efficiently generalizable, η -approximation for thepolynomial-scenarios variant of any of the problems we consider, and let also T be the maximumpossible stage-II cost. Then, for any ǫ ∈ (0 , and using O ( T ǫ · poly( n, m )) samples, we get an η -approximation for the black-box version of the problem, that has opening cost at most (1 + ǫ ) B . The dependence on T is natural in this line of work, and a similar issue is encountered in allrelevant literature. However, we also provide a modified variant of our scheme that removes thisdependence, at the expense of sometimes failing to return an η -approximate solution.The next four sections present efficiently generalizable algorithms for a variety of problems in thepolynomial-scenarios setting. Our algorithmic results are summarized in the following theorems. Theorem 1.2.
There exists an efficiently generalizable / / / -approximation for / / / . Theorem 1.3.
When the instance is discrete , there exists an efficiently generalizable -approximationalgorithm for , with runtime poly( n, m, Λ) . The 3-approximation algorithm for is presented in Section 3, and it relies ona novel type of correlated LP-rounding, not used in clustering problems before. Notably, forthis problem we get an approximation ratio which exactly matches the lower bound for the non-stochastic counterpart [14], something not common in the two-stage stochastic model.The 5-approximation for is given in Section 4, and is based on solving anauxiliary LP, whose optimal solution we prove to be integral. In Section 5, we extend this methodto and and provide an 11-approximation that capturesboth. This result is based on iterative rounding methods as in [16, 12]. Also, this is the firsttime iterative rounding has been applied to two-stage problems, offering the major advantage ofsimultaneously rounding stage-I and stage-II LP-variables. The algorithm for ispresented in Section 6, and is based on a reduction to a deterministic outliers-supplier problem.Finally, in Section 7 we develop an algorithm for , by extending the thresholdingtechnique of [15] to supplier problems. We get the following result:
Theorem 1.4.
For a provided feasible radius R , there is an efficient algorithm for ,that has opening cost at most B , and covers all arriving clients within distance R . Note that the algorithm of Section 3 and the framework of Section 2 could also be used for , but that would only give a pseudo-approximation, i.e. an opening cost of at most(1 + ǫ ) B . Also, this solution would be significantly more expensive computationally.4 lgorithm 1: Filter( Q , r , g ) H ← ∅ ; for each j ∈ Q in non-increasing order of g ( j ) do H ← H ∪ { j } ; for each j ′ ∈ Q with G j,r ( j ) ∩ G j ′ ,r ( j ′ ) = ∅ do π ( j ′ ) ← j, Q ← Q \ { j ′ } ; end Return H , π ; The non-stochastic counterpart of is the well-known
Knapsack-Supplier , which has a 3-approximation [14]. Another closely-related deterministic problem is
Matroid-Supplier , which alsohas a 3-approximation [4, 12]. This is the best ratio possible for these problems, unless P=NP.An excellent and in-depth survey of algorithmic results in the two-stage model is [27]. As forclustering problems in this setting, most prior work has focused on
Facility Location [24, 25, 21,22, 10, 18, 26]. On similar lines, [1] studies a stochastic k -center variant, where points arriveindependently, and the objective is to minimize the maximum distance of a point to its assignedcenter, while also ensuring that each point gets covered with some given probability.A common criticism of the two-stage model is that it provides guarantees only in expectation,hence there is always the risk of a significantly large stage-II cost. Research towards alleviatingthis issue focuses on risk-averse models that minimize the stochastic objective, while ensuring thatunder each arriving scenario, the stage-II cost does not exceed a certain threshold B II [25, 9]. We use [ n ] to denote the set { , , . . . , n } . Also, for a vector α = ( α , α , . . . , α n ) and a subset X ⊆ [ n ] we use α ( X ) to denote P i ∈ X α i . For a client j and R ≥
0, we define G j,R = { i ∈ F : d ( i, j ) ≤ R } , i Ij,R = arg min i ∈ G j,R c Ii and i Aj,R = arg min i ∈ G j,R c Ai for any A .We repeatedly use a key subroutine named Filter(), shown in Algorithm 1. Its input is a set ofclients Q , a function r giving a target radius for each j ∈ Q , and an ordering function g : Q 7→ R .Its output is a set H ⊆ Q along with a mapping π : Q 7→ H . For a scalar R , we also writeFilter( Q , R, g ) as shorthand for Filter( Q , r, g ) with the constant vector r ( j ) = R . Observation 1.5. If H, π is the output of Filter( Q , r, g ), then the following hold. • ∀ j, j ′ ∈ H with j = j ′ , we have G j,r ( j ) ∩ G j ′ ,r ( j ′ ) = ∅ • ∀ j ∈ Q and j ′ = π ( j ), we have G j,r ( j ) ∩ G j ′ ,r ( j ′ ) = ∅ and g ( j ′ ) ≥ g ( j ) Let P be any of the stochastic problems we consider, with polynomial-scenarios variant P - Poly and black-box variant P - BB . Moreover, suppose that we have an η -approximation algorithm Alg P for P - Poly , which we intend to use to solve P - BB .As a starting point, we will assume the radius demands are given to us; we later discuss how tooptimize them. Thus, we denote a P - BB problem instance by the tuple I = ( C , F , M I , c I , B, ~R ),where C is the set of clients, F the set of facilities i , each with stage-I cost c Ii , M I ⊆ F is the setof acceptable stage-I openings (representing the stage-I specific constraints of P ), B the allowed5udget, and ~R a vector of covering demands for each client. In addition, there is an underlyingdistribution D , and let p A the occurrence probability of scenario A according to it. When a scenario A ∈ D is revealed, we also learn the facility costs c Ai . Definition 2.1.
A strategy s is a set of actions that dictates a set of facilities F sI ⊆ F that openin stage-I, and for every A ∈ D a set F sA ⊆ F that opens when A arrives in stage-II.We say that instance I is feasible for P - BB if there exists some strategy s ∗ satisfying: F s ∗ I ∈ M I , c I ( F s ∗ I ) + X A ∈D p A c A ( F s ∗ A ) ≤ B, ∀ A ∈ D , j ∈ A d ( j, F s ∗ I ∪ F s ∗ A ) ≤ R j For P - Poly , consider an instance J = ( C , F , M I , Q, ~q, ~c, B, ~R ), where C , F , M I , B, ~R are as inthe P - BB setting, Q is the set of provided scenarios, ~c the vector of stage-I and stage-II explicitlygiven costs, and ~q the vector of occurrence probabilities q A of each A ∈ Q . We say the instance J is feasible for P - Poly , if there exist sets F I ⊆ F and F A ⊆ F for every A ∈ Q , such that: F I ∈ M I , c I ( F I ) + X A ∈ Q q A c A ( F A ) ≤ B, ∀ A ∈ Q, j ∈ A d ( j, F I ∪ F A ) ≤ R j We write F for the overall collection of sets F I and F A : A ∈ Q . Definition 2.2.
An algorithm
Alg P is a valid η -approximation algorithm for P - Poly , if given anyproblem instance J = ( C , F , M I , Q, ~q, ~c, B, ~R ) one of the following two cases holds: A1 If J is feasible for P - Poly , then
Alg P returns a collection of sets F with F I ∈ M I , c I ( F I ) + P A ∈ Q q A c A ( F A ) ≤ B and ∀ A ∈ Q, j ∈ A d ( j, F I ∪ F A ) ≤ ηR j . A2 If J is not feasible for P - Poly , then the algorithm either returns “INFEASIBLE”, or returnsa collection of sets F satisfying the properties presented in A1 . Definition 2.3.
A valid η -approximation algorithm Alg P for P - Poly is efficiently generalizable , iffor every instance J = ( C , F , M I , Q, ~q, ~c, B, ~R ) where it returns a solution F , there is a polynomialtime procedure that extends this to a strategy ˆ s such that: F ˆ sI = F I , F ˆ sA = F A for every A ∈ Q , and for every A ∈ D and j ∈ A we have d ( j, F ˆ sI ∪ F ˆ sA ) ≤ ηR j .Furthermore, let S the set of all possible strategies that are potentially achievable using thegiven instance J . The extension procedure must also ensure that |S| ≤ t P ( n, m ) for some function t P ( n, m ), which does not depend on Q , and satisfies log( t P ( n, m )) = poly( n, m ). (Note that bydefault |S| ≤ m |D| , where |D| the size of the support of D )Our generalization is based on sampling independently a set Q = { S , . . . , S N } of scenarios from D , and then applying an efficiently generalizable Alg P on Q . This will give us a set F I of facilities toopen in stage-I, and a set F S v of facilities to open if S v materializes in stage-II, but no instructions onhow to handle a scenario A outside Q . Nonetheless, if Alg P is efficiently generalizable, we will useits corresponding extension procedure, and thus get a set F A for each arriving A ∈ D . Algorithm 2demonstrates the sampling process that turns an efficiently generalizable approximation algorithmfor P - Poly into an approximation for P - BB .We need a technical lemma concerning concentration bounds for sums of random variables. Lemma 2.4. [8] Let X , . . . , X K be non-negative independent random variables, with expectations µ , . . . , µ K respectively, where µ k ≤ for every k . Let X = P Kk =1 X i , and let µ = P Kk =1 µ i = E [ X ] .Then for every δ > we have Pr[
X < µ + δ ] ≥ min { δ δ , } . lgorithm 2: Sampling Process for P - BB . Input:
Parameters ǫ, γ ∈ (0 , N ≥ P - BB instance I = ( C , F , M I , c I , B, ~R ). for h = 1 , . . . , l log (1 /γ ) m do Draw N independent samples from the black-box, obtaining set Q = { S , . . . , S N } ;Let ~c the vector containing c I and the stage-II facility-cost vectors of all S v ∈ Q ;For every S v ∈ Q set q S v ← /N ; if Alg P ( C , F , M I , Q, ~q, ~c, (1 + ǫ ) B, ~R ) returns F I and F S v for all S v ∈ Q then Return F I and F S v for all S v ∈ Q ; endend Return “INFEASIBLE”
Lemma 2.5.
If instance I is feasible for P - BB and N ≥ /ǫ , then with probability at least − γ Algorithm 2 does not terminate with “INFEASIBLE”.Proof.
We first assume by rescaling that B = 1. The cost of any strategy s over D is given by C ( s ) = c I ( F sI ) + P A ∈D p A c A ( F sA ). For any specific execution of the while loop in Algorithm 2, let Y sv be a random variable representing the second-stage cost of s on sample S v . For fixed s , therandom variables Y sv are independent. The empirical cost of s on Q is ˆ C ( s ) = c I ( F sI ) + N P Nv =1 Y sv .Let s ⋆ be a feasible strategy for P - BB . By definition we have F s ⋆ I ∈ M I , and d ( j, F s ⋆ I ∪ F s ⋆ A ) ≤ R j for every A ∈ Q and j ∈ A . We will also show that with probability at least 1 /
13, we have ˆ C ( s ⋆ ) ≤ (1 + ǫ ) B . In this case, the restriction of s ⋆ to Q shows that instance ( C , F , M I , Q, ~q, ~c, (1 + ǫ ) B, ~R )is feasible for P - Poly . Therefore, because
Alg P is a valid η -approximation for P - Poly , it will notreturn “INFEASIBLE”.Since C ( s ⋆ ) ≤ B , we have E [ Y s ⋆ v ] = P A ∈D p A · c A ( F s ⋆ A ) ≤ B = 1 for all v . Thus, usingLemma 2.4 with δ = ǫBN gives Pr h P Nv =1 Y s ⋆ v < E [ P Nv =1 Y s ⋆ v ] + ǫBN i ≥ min n ǫBN ǫBN , o . When N ≥ Bǫ = ǫ we see that ǫBN/ (1 + ǫBN ) ≥ /
13. Hence, with probability at least 1 /
13 we have P Nv =1 Y s ⋆ v < E [ P Nv =1 Y s ⋆ v ] + ǫBN , in which case we get ˆ C ( s ∗ ) ≤ (1 + ǫ ) B as shown below.ˆ C ( s ⋆ ) = c I ( F s ⋆ I ) + 1 N N X v =1 Y s ⋆ v ≤ c I ( F s ⋆ I ) + 1 N N X v =1 E [ Y s ⋆ v ] + ǫB ≤ c I ( F s ⋆ I ) + X A ∈D p A · c A ( F s ⋆ A ) + ǫB So each iteration terminates successfully with probability at least . To bring the error prob-ability down to at most γ as desired, we simply repeat for l log (1 /γ ) m iterations.If Algorithm 2 returns a solution F and Alg P is efficiently generalizable, then we can applythe extension procedure to generate a strategy ˆ s with F ˆ sI ∈ M I and d ( j, F ˆ sI ∪ F ˆ sA ) ≤ ηR j for every A ∈ D and j ∈ A . The only thing left to analyze is the opening cost C (ˆ s ) of ˆ s over D . Theorem 2.6.
Suppose that c A ( F sA ) ≤ T · B for all scenarios A ∈ D and strategies s . Then, for N = O (cid:16) T ǫ · log( t P ( n,m ) γ ) (cid:17) , our approach satisfies the following:1. If I is feasible, then it has a probability of at most γ of outputting “INFEASIBLE”.2. Conditioned on Algorithm 2 not outputting “INFEASIBLE”, the constructed strategy ˆ s has C (ˆ s ) ≤ (1 + 2 ǫ ) B with probability at least − γ . roof. The first statement follows directly from Lemma 2.5, and the fact that the given N is atleast 1 /ǫ . Focus now on any iteration h of the loop in Algorithm 2, and consider any arbitrary setof strategies W . For any strategy s , let Y sh,v be the random variable representing the stage-II costof s on the v th sample. The actual cost of s on D and its empirical cost for the iteration at handare then C ( s ) = c I ( F sI ) + P A ∈D p A c A ( F sA ), ˆ C h ( s ) = c I ( F sI ) + N P Nv =1 Y sh,v . Moreover, we have E [ ˆ C h ( s )] = C ( s ), where the expectation is over the random choice of the samples. Via Hoeffding’sinequality we get Pr[ | ˆ C h ( s ) − C ( s ) | > ǫB ] ≤ e − Nǫ /T , and by a union bound over all s ∈ W :Pr[ ∃ s ∈ W : | ˆ C h ( s ) − C ( s ) | > ǫB ] ≤ |W| e − Nǫ /T (1)If Algorithm 2 did not return “INFEASIBLE” and terminated at some iteration h , then ˆ C h (ˆ s ) ≤ (1 + ǫ ) B (properties 1 , S h be the set of potentiallyachievable strategies, if termination occurs at iteration h . If we further have | ˆ C h ( s ) − C ( s ) | ≤ ǫB for all s ∈ S h , then we get C (ˆ s ) ≤ ˆ C h (ˆ s ) + ǫB ≤ (1 + 2 ǫ ) B .Let E h the event of not having | ˆ C h ( s ) − C ( s ) | ≤ ǫB for all s ∈ S h , at any iteration h . Using (1)and taking N = T ǫ · log (cid:16) γ t P ( n, m )(log ( γ ) + 1) (cid:17) samples, results in Pr[ E h ] ≤ γ/ (log ( γ ) + 1)because |S h | ≤ t P ( n, m ) and t P is independent of N . Finally, let T the event that Algorithm2 terminates without “INFEASIBLE”, and T h the event that Algorithm 2 terminates without“INFEASIBLE” at iteration h . We thus see that the failure probability is:Pr[fail | T ] = X h Pr[ T h ] Pr[fail | T h ] ≤ X h Pr[ T h ] Pr[ E h | T h ] ≤ X h Pr[ E h ] ≤ γ By Definition 2.3 and Theorem 2.6, the number N would be polynomial in T, /ǫ, log(1 /γ ) aswell as input parameters n, m . One downside of this is that the value of T may be arbitrarilylarge. In previous literature on similar questions [22, 23, 5, 2], this issue was handled by assuminga bounded inflation property in the stage-II costs. In our problems, this means that the stage-IIcost of any facility i will be at most λc Ii , for a given parameter λ ≥
1. Since w.l.o.g. all stage-Icosts are at most B , we get T = λm and consequently N = O (cid:16) λ m ǫ · log( t P ( n,m ) γ ) (cid:17) , which is similarto the generalization bounds appearing in the previous literature. Optimizing over the Radius:
In what we described, each j has some fixed demand R j . How-ever, in some of the problems we consider, all clients have the same radius demand R , where R issome scalar to be minimized. Because the optimal R ∗ is always the distance between some facilityand some client, there are at most n · m alternatives for it. Therefore, we can run Algorithm 2for all possible n · m target radius values, each represented through the vector ~R , and use errorparameter γ ′ = γnm . In the end, we keep the smallest radius that resulted in a solution. By a unionbound over possible radius choices, the probability of keeping R > R ∗ is at most γ . Moreover, forwhichever R we kept, the guarantee of Theorem 2.6 still ensures an opening cost over D of at most(1 + 2 ǫ ) B , with probability at least 1 − O ( γ ). Because of this generic search step for theradius R , we assume for all our P -poly problems that a radius R is given explicitly.Removing the Dependency on T : For this, we make a minor change to our algorithm: giventhe constructed strategy ˆ s and any α >
1, if the scenario A that materializes has c A ( F ˆ sA ) > αB ,we perform no stage-II opening. Hence we get a new strategy ˆ s ′ , with F ˆ s ′ I = F ˆ sI , and F ˆ s ′ A = ∅ when αB < c A ( F ˆ sA ), F ˆ s ′ A = F ˆ sA otherwise. Theorem 2.7.
For any ǫ, γ ∈ (0 , and N = O (cid:16) α ǫ · log( t P ( n,m ) γ ) (cid:17) , our modified approach satisfies: . If I is feasible, then it has a probability of at most γ of outputting “INFEASIBLE”.2. Conditioned on Algorithm 2 not returning “INFEASIBLE”, there is a probability of at least − γ that the constructed strategy ˆ s ′ has the following two properties: (i) C (ˆ s ′ ) ≤ (1 + 2 ǫ ) B and (ii) Pr A ∼D [ d ( j, F ˆ s ′ I ∪ F ˆ s ′ A ) ≤ ηR j , ∀ j ∈ A ] ≥ − O (1 /α ) Proof.
The first statement follows again directly from Lemma 2.5, and the fact that N ≥ /ǫ .Now, for a scenario A and strategy s , define ˜ c A ( F sA ) = min( αB, c A ( F sA )). Observe that the secondstatement of Theorem 2.6 works for the modified cost vectors ˜ c A as it does for c A . Also, since thestage-II cost of any strategy for ˜ c A is at most αB (with T = α ), the number of required samples isnow O ( α ǫ · log( t P ( n,m ) γ )). Therefore, we either get “INFEASIBLE”, or an ˆ s , for which the modifiedcost ˜ C (ˆ s ) on D satisfies Pr[ ˜ C (ˆ s ) ≤ (1 + 2 ǫ ) B ] ≥ − γ .Let us now assume that it holds that ˜ C (ˆ s ) ≤ (1 + 2 ǫ ) B . For the output strategy ˆ s ′ , the coston a scenario A is c A ( F ˆ sA ) if c A ( F ˆ sA ) ≤ αB and 0 otherwise; in particular, it is at most ˜ c A ( F ˆ sA ),and hence C (ˆ s ′ ) ≤ ˜ C (ˆ s ). Thus, we get Pr[ C (ˆ s ′ ) ≤ (1 + 2 ǫ ) B ] ≥ − γ . Also, ˆ s ′ discards scenarios A only if c A ( F ˆ sA ) > αB , which also occurs if ˜ c A ( F ˆ sA ) ≥ αB . By Markov’s inequality appliedto the random process of drawing A ∼ D , the probability that the latter happens is at most˜ C (ˆ s ) / ( αB ) ≤ (1 + 2 ǫ ) B/ ( αB ) ≤ − O (1 /α ). In this section we tackle , by first designing a 3-approximation algorithm for , and then proving that the latter is efficiently generalizable . We are given a list of scenarios Q together with their probabilities p A and cost vectors c A , a targetradius R , and let G j = G j,R , i Ij = i Ij,R , i Aj = i Aj,R for every j ∈ C and A ∈ Q . Consider LP (2)-(4). X i ∈F y Ii · c Ii + X A ∈ Q p A X i ∈F y Ai · c Ai ≤ B (2) X i ∈ G j ( y Ii + y Ai ) ≥ , ∀ A ∈ Q, ∀ j ∈ A (3)0 ≤ y Ii , y Ai ≤ R , then LP (2)-(4) has a fractionalsolution. Moreover, constraint (2) captures the total expected cost, and constraint (3) the fact thatfor all A ∈ Q , every j ∈ A must have an open facility within distance R from it.Algorithm 3 begins with two filtering steps, one for each stage. The stage-I filtering step ensures y I ( G π I ( j ) ) ≥ y I ( G j ) for all j ∈ C , and the stage-II filtering guarantees that y I ( G π I ( π A ( j )) ) ≤ y I ( G π I ( j ) ) for all A and j ∈ A . Given the results of the filtering steps, we order the clients of H I as j , . . . , j h , such that y I ( G j ) ≤ y I ( G j ) ≤ · · · ≤ y I ( G j h ). For each integer ℓ = 0 , . . . , h , we considera putative solution F ℓI , F ℓA as follows. The set F ℓI will contain the minimum cost facility i Ij k of G j k for every k > ℓ . In the second-stage, for any scenario A , the set F ℓA contains the minimum-costfacility i Aj inside G j for any client j ∈ H A that had no stage-I opening in G π I ( j ) . Our algorithmreturns the computed sets F ℓI , F ℓA with the smallest opening cost S ℓ = c I ( F ℓI ) + P A ∈ Q p A · c A ( F ℓA ). Theorem 3.1.
For every scenario A ∈ Q and every j ∈ A we have d ( j, F ℓ ∗ I ∪ F ℓ ∗ A ) ≤ R . lgorithm 3: Correlated Rounding Algorithm for
Solve LP (2)-(4) to get a feasible solution y I , y A for all A ∈ Q ; if no feasible LP solution then return “INFEASIBLE”;( H I , π I ) ← Filter( C , R, g I ) where g I ( j ) = y I ( G j ) ; // Stage-I Filtering for each scenario A ∈ Q do ( H A , π A ) ← Filter(
A, R, g A ) where g A ( j ) = − y I ( G π I ( j ) ) ; // Stage-II Filtering end Order the clients of H I as j , . . . , j h such that y I ( G j ) ≤ y I ( G j ) ≤ · · · ≤ y I ( G j h ); for all integers ℓ = 0 , , . . . , h do F ℓI ← { i Ij k | j k ∈ H I and k > ℓ } ; for each scenario A ∈ Q do F ℓA ← { i Aj | j ∈ H A and F ℓI ∩ G π I ( j ) = ∅} ; S ℓ ← c I ( F ℓI ) + P A ∈ Q p A · c A ( F ℓA ); end Return F ℓ ∗ I , F ℓ ∗ A such that ℓ ∗ = arg min ℓ S ℓ ; Proof.
Focus on any A ∈ Q . For j ∈ H A this is clear, because there is either an open facility in G π I ( j ) from stage-I, or an open facility in G j in stage-II. So consider some j ∈ A \ H A . If we opena facility i ∈ G π A ( j ) in stage-II, then i will be within distance 3 R from j . If we do not open such i , then G π I ( π A ( j )) must have an open facility from stage-I. Recall now that the stage-II filteringensures that y I ( G π I ( j ) ) ≥ y I ( G π I ( π A ( j )) ). Therefore, from the way we formed F ℓ ∗ I we know that G π I ( j ) also has an open facility from stage-I, which has distance at most 3 R to j . Theorem 3.2.
The opening cost S ℓ ∗ of Algorithm 3 is at most B .Proof. Consider the following process to generate a random solution: we draw a random variable β uniformly from [0 , F βI = { i Ij | j ∈ H I and y I ( G j ) ≥ β } , F βA = { i Aj | j ∈ H A and F I ∩ G π I ( j ) = ∅} for all A ∈ Q . Note that for each possible draw for β , the resulting sets F βI , F βA correspond to sets F ℓI , F ℓA for some value ℓ . Hence, in order to show that there exists some ℓ with S ℓ ≤ B , it suffices to show that E β ∼ [0 , [ c I ( F βI ) + P A ∈ Q p A · c A ( F βA )] ≤ B .We start by calculating the probability of opening a given facility i Ij with j ∈ H I in stage-I.This will occur if β ≤ y I ( G j ), and so Pr[ i Ij is opened at stage-I] ≤ min( y I ( G j ) , E β ∼ [0 , [ c I ( F βI )] ≤ X j ∈ H I c Ii Ij · y I ( G j ) ≤ X i ∈F y Ii · c Ii (5)Next, Pr[ i Aj is opened at stage-II | A ] = 1 − min( y I ( G π I ( j ) ) , ≤ − min( y I ( G j ) , ≤ y A ( G j ) forany j ∈ H A and any scenario A ∈ Q . The first inequality is due to the filtering order of stage-Ithat gives y I ( G π I ( j ) ) ≥ y I ( G j ), and the second follows from (3). Thus: E β ∼ [0 , [ c A ( F βA )] ≤ X j ∈ H A c Ai Aj · y A ( G j ) ≤ X i ∈F y Ai · c Ai (6)Finally, combining (5), (6) and (2) gives E β ∼ [0 , [ c I ( F βI )] + P A ∈ Q p A · E β ∼ [0 , [ c A ( F βA )] ≤ B . To show that Algorithm 3 fits the framework of Section 2, we need to prove that it is efficientlygeneralizable as in Definition 2.3. Hence, we need an efficient process that extends its output to any10 lgorithm 4:
Generalization Procedure for
Suppose scenario A ∈ D arrived in the second stage;For every j ∈ A set g ( j ) ← − y I ( G π I ( j ) ), where y I is the LP solution vector and π I is thestage-I mapping, both obtained through Algorithm 3;( H A , π A ) ← Filter(
A, R, g );Open the set F A = { i Aj | j ∈ H A and F I ∩ G π I ( j ) = ∅} ;arriving scenario A . This process, which actually defines a strategy ˆ s , is demonstrated in Algorithm4, and it mimics the stage-II actions of Algorithm 3. Here we crucially exploit the fact that thestage-II decisions of Algorithm 3 only depend on information from the LP about stage-I variables.The arguments in Theorem 3.1 guarantee d ( j, F I ∪ F A ) ≤ R for all j ∈ A and any A ∈ D (notjust those in Q ). To conclude, we need to bound the number of potentially achievable strategies. Lemma 3.3.
Let S K the set of strategies achievable via Algorithm 4. Then |S K | ≤ ( n + 1)! .Proof. The constructed strategy is determined by 1) the sorted order of y I ( G j ) for all j ∈ C , and2) the chosen threshold ℓ . Given those, we know exactly which clients will constitute H I and H A for every A ∈ D , as well as which openings will be performed in both stages. Observe now thatthere are n ! possible total possible orderings for the y I ( G j ) values, and the threshold parameter ℓ can take at most n + 1 values. Therefore, |S K | ≤ ( n + 1)!. The outline of this section is similar to that of Section 3. We begin by providing a 5-approximationalgorithm for , and then move on to show that it satisfies all necessary propertiesof our generalization framework. Hence, we eventually get results for the black-box model.
Given an explicit list of scenarios Q , together with their occurrence probabilities p A and facility-costvectors c A , and a target radius R , we once more define G j = G j,R for every j ∈ C . If r M is therank function of the input matroid M , then consider LP (7)-(10). X i ∈F y Ii · c Ii + X A ∈ Q p A X i ∈F y Ai · c Ai ≤ B (7) X i ∈ G j ( y Ii + y Ai ) ≥ , ∀ A ∈ Q, ∀ j ∈ A (8) X i ∈ U y Ii ≤ r M ( U ) , ∀ U ⊆ F (9)0 ≤ y Ii , y Ai ≤ lgorithm 5: Rounding Algorithm for
Solve LP (7)-(10) to get a feasible solution y I , y A for all A ∈ Q ; if no feasible LP solution then return “INFEASIBLE”; end ( H I , π I ) ← Filter( C , R, g I ) where g I ( j ) = y I ( G j ) ; // Stage-I Filtering Choose an arbitrary permutation g II of C ; for each scenario A ∈ Q do ( H A , π A ) ← Filter(
A, R, g II ) ; // Stage-II Filtering end Solve LP (11)-(14) and get an optimal solution z ∗ ,I and z ∗ ,A for every A ∈ Q ; F I ← { i ∈ F | z ∗ ,Ii = 1 } , F A ← { i ∈ F | z ∗ ,Ai = 1 } for every A ∈ Q .Initially our algorithm, presented in Algorithm 5, solves the above LP to obtain solution vectors y I , y A for every A . There are then two filtering steps, one for each stage, that produce sets H I , H A with corresponding mappings π I and π A . Given those sets and mappings, we use the auxiliary LPshown in (11)-(14); critically, the optimal solution of the latter LP is always integral.minimize X i ∈F z Ii · c Ii + X A ∈ Q p A X i ∈F z Ai · c Ai (11)subject to z I ( G π I ( j ) ) + z A ( G j ) ≥ , ∀ A ∈ Q, ∀ j ∈ H A (12) z I ( U ) ≤ r M ( U ) , ∀ U ⊆ F (13)0 ≤ z Ii , z Ai (14) Lemma 4.1.
If the original LP (7)-(10) is feasible, then there exists a feasible solution for theauxiliary LP (11)-(14) of objective function value at most B .Proof. Let y I , y A be a solution for the original LP (7)-(10); we will show this provides the desiredsolution to (12)-(14) with the required bound on (11). The only non-obvious constraint is (12); forthis, note that for any A ∈ Q , and any j ∈ H A we have y I ( G π I ( j ) ) + y A ( G j ) ≥ y I ( G j ) + y A ( G j ) ≥ z ∗ ,I , z ∗ ,A be the optimal solution of (11)-(14). W.l.o.g. we assume that this is also avertex solution. To show integrality of z ∗ ,I , z ∗ ,A , we use the following characterization of the tightconstraints of a matroid polytope. Lemma 4.2. [13]. Consider a matroid M = (Ω , I ) with rank function r M , and the set of con-straints P M = { x ( U ) ≤ r M ( U ) ∀ U ⊆ Ω , x i ≥ ∀ i ∈ Ω } . The set D M of tight constraints of P M can be expressed as D M = { x ( O ) = q O ∀ O ∈ O , x i = 0 ∀ i ∈ J } , where J ⊆ Ω , and O is a familyof k pairwise disjoint sets O , O , . . . , O k ⊆ Ω with q O , q O , . . . , q O k ∈ Z > . Lemma 4.3.
The optimal vertex solution of (11)-(14) is integral, i.e., z ∗ ,Ii , z ∗ ,Ai ∈ { , } for allfacilities i ∈ F and scenarios A ∈ Q .Proof. At first, we clearly have z ∗ ,Ii , z ∗ ,Ai ≤ i, A , since we are dealing with a minimizationproblem. Define m = |{ i ∈ F : z ∗ ,Ii > }| and m = |{ ( i, A ) ∈ F × Q : z ∗ ,Ai > }| . Because z ∗ ,I , z ∗ ,A is a vertex solution, the number of linearly independent tight constraints (12),(13) should12e at least m + m . By Lemma 4.2, the tight constraints (13) can be expressed as D = { z ∗ ,I ( O ) = q O , ∀ O ∈ O} , where O is a family of k disjoint sets (all subsets of F ), and every q O is a positiveinteger. Also, let L be the set of tight constraints (12) for which all stage-II variables are 0, and L the set of tight constraints (12) that include at least one non-zero stage-II variable. Lettingrank() denote the number of linearly independent constraints, we have: m + m ≤ rank( D ∪ L ∪ L ) = rank( D ∪ L ) + rank( L ) = rank( D ∪ L ) + | L | (15)The second and third equalities are due to the definitions of D, L , L , and the fact that the sets G j , for j ∈ H A and any A ∈ Q , are disjoint.Initially assume there exists some i with z ∗ ,Ii ∈ (0 , z ∗ is a vertex solution and the sets G j (for j ∈ H A and any A ∈ Q ) are disjoint, this i must participate in some constraint in either D or L . Focus now on any such tight constraint where i is present, and notice that this shouldalways involve another facility i ′ = i , with z ∗ ,Ii ′ ∈ (0 , q O are integers, and the constraints in L sum up to exactly 1. In general, if a tight constraint in L ∪ D includes a fractional facility, it should certainly include another one as well. In addition, thesets in O are pairwise disjoint, and the sets G j with j ∈ H I are pairwise disjoint too. The previoustwo arguments imply that rank( D ∪ L ) ≤ m −
1. Finally, because each constraint in L involvesat least one non-zero stage-II variable and the sets G j for j ∈ H A are disjoint, we have | L | ≤ m .Using (15) gives m + m ≤ m + m −
1, which is a contradiction.For the second case, assume that there exists some i with z ∗ ,Ai ∈ (0 , L , and see that because we cannot have a fractional stage-I variable, the stage-I part of theconstraint must be 0 and the stage-II part 1 (the constraints in L are tight). At first, we triviallyhave rank( D ∪ L ) ≤ m . Further, because for every H A the sets G j with j ∈ H A are disjoint, andeither L does not involve all non-negative stage-II variables, or some constraint in L includes atleast 2 fractional variables, we have | L | ≤ m −
1. The contradiction results from using (15).Given z ∗ ,I , z ∗ ,A , our algorithm chooses F I = { i ∈ F | z ∗ ,Ii = 1 } and F A = { i ∈ F | z ∗ ,Ai = 1 } .Because of Lemma 4.1 we know that the budget constraint is not violated, and since the vector z ∗ ,I satisfies (13), we also have F I ∈ I . Theorem 4.4.
For any A ∈ Q and any j ∈ A , we have d ( j, F I ∪ F A ) ≤ R .Proof. Constraint (12) ensures that for all A ∈ Q and for every j ∈ H A , there is an open stage-Ifacility in G π I ( j ) , which is at distance at most 3 R from j , or there is an open stage-II facility in G j , which is at most R away from it. Therefore, all j ∈ A \ H A will be satisfied with a coveringdistance of at most 5 R , because of the triangle inequality and the fact that d ( j, π A ( j )) ≤ R . We need to show that Algorithm 5 is efficiently generalizable as in Definition 2.3. Algorithm 6demonstrates how to handle any scenario A , by mimicking the actions of Algorithm 5 (note thatAlgorithm 5 would also open F A = { i Aj,R | j ∈ H A and F I ∩ G π I ( j ) = ∅} , since it minimizes (11)).Also, a reasoning similar to that in Theorem 4.4 guarantees d ( j, F I ∪ F A ) ≤ R for all j ∈ A and A ∈ D . To conclude, we need to bound the total number of achievable strategies. Lemma 4.5.
Let S M the set of strategies achievable via Algorithm 6. Then |S M | = 2 m · n ! .Proof. The constructed strategy is a result of 1) the set F I returned by the polynomial-scenariosalgorithm, and 2) the arbitrary order g II we chose for clients of C , which eventually governs thestage-II filtering. The total number of possible outcomes for F I is 2 m , and the total number oforderings for the clients of C is n !. Hence, |S M | = 2 m · n !.13 lgorithm 6: Generalization Procedure for
Suppose A ∈ D arrived in the second stage;Let π I the stage-I mapping and g II the permutation of C , both obtained in Algorithm 5;Set ( H A , π A ) ← Filter(
A, R, g II );Open the set F A = { i Aj,R | j ∈ H A and F I ∩ G π I ( j ) = ∅} ; We first give an 11-approximation for ; note that this automatically covers as well. We then show that our algorithm is also efficiently generalizable.
Our algorithm for this problem is inspired by the iterative rounding algorithms of [16, 12]. It beginsby solving LP (7)-(10) for a set of explicitly provided scenarios Q , with G j = G j,R j for each j ∈ C ,getting a fractional solution y I and y A . As explained in Section 4, this LP constitutes a validrelaxation of the original problem. Next, there is a stage-II filtering step that creates sets H A andmappings π A , for every provided A ∈ Q . See Algorithm 7 for the full details.We then use an iterative rounding strategy to round y I , y A to an integral solution. Let us firstdefine some important client sets. The set C s will contain clients for which we have not yet decidedif we will satisfy them with a stage-I or a stage-II opening. The set C It will contain clients j forwhich a stage-I opening will be performed in G j . Finally, the set C IIt will contain clients j whichare to be covered in stage-II, i.e., for every scenario A ∈ Q and j ∈ C IIt ∩ H A , we open some stage-IIfacility in G j . During the iterative process, we preserve the following invariants on the sets: S.1
For all j, j ′ ∈ C It , with j = j ′ , we have G j ∩ G j ′ = ∅ . S.2 C s , C It , C IIt are pairwise disjoint.The iterative rounding is based on the auxiliarly LP defined in (16)-(22).minimize X i ∈F z Ii · c Ii + X A ∈ Q p A X i ∈F z Ai · c Ai (16)subject to z I ( G j ) + z A ( G j ) ≥ , ∀ A ∈ Q, ∀ j ∈ H A ∩ ( C s ∪ C It ∪ C IIt ) (17) z I ( G j ) ≥ , ∀ j ∈ C It (18) z I ( G j ) = 0 , ∀ j ∈ C IIt (19) z I ( G j ) ≤ , ∀ j ∈ C s (20) z I ( U ) ≤ r M ( U ) , ∀ U ⊆ F (21)0 ≤ z Ii , z Ai ∀ i ∈ F , ∀ A ∈ Q (22)Constraint (17) represents the covering requirement, and is only defined for clients j ∈ H A for some A ∈ Q . (For other clients j , we will use the facility serving π A ( j ) to cover them). Constraints (18),(19) and (20) capture the definitions of the sets C It , C IIt and C s respectively, and constraint (21)enforces the matroid requirement. Finally, the objective function (16) measures the opening cost. Lemma 5.1.
Let z ∗ ,I , z ∗ ,A be an optimal vertex solution of (16)-(22) when C s , C It , C IIt satisfy
S.1 , S.2 . Then if C s = ∅ , there exists at least one j ∈ C s with z ∗ ,I ( G j ) ∈ { , } . Moreover, if C s = ∅ then the solution is integral, i.e., for all i ∈ F and for all A ∈ Q we have z ∗ ,Ii , z ∗ ,Ai ∈ { , } . lgorithm 7: Iterative Rounding for
Solve LP (7)-(10) to get y I and y A for every A ∈ Q ; if no feasible LP solution then return “INFEASIBLE”; end For every j ∈ C set g ( j ) ← − R j and r ( j ) ← R j ; // Stage-II Filtering for each scenario A ∈ Q do ( H A , π A ) ← Filter(
A, r, g );( C It , π It ) ← Filter( { j ∈ C | y I ( G j ) > } , r, g ); // Sets Initialization C IIt ← ∅ ; C s ← { j ∈ C | y I ( G j ) ≤ ∀ j ′ ∈ C It : ( G j ∩ G j ′ = ∅ ∨ R j < R j ′ / } ; while C s = ∅ do // Iterative Rounding Solve LP (16)-(22), using the current C s , C It , C IIt , and get a solution z I , z A ;Find a j ∈ C s with z I ( G j ) ∈ { , } and set C s ← C s \ { j } ; if z I ( G j ) = 0 then C IIt ← C IIt ∪ { j } ; else C It ← C It ∪ { j } ; for any client j ′ ∈ C It ∪ C s with G j ∩ G j ′ = ∅ and R j ′ ≥ R j / do C s ← C s \ { j ′ } , C It ← C It \ { j ′ } ; endendend Solve LP (16)-(22) once more, using the current C s , C It , C IIt , and get z I, final , z A, final ; F I ← { i ∈ F | z I, final i = 1 } , F A ← { i Aj,R j | j ∈ C IIt ∩ H A } for every A ∈ Q ; Proof.
This proof is very similar to that of Lemma 4.3, and therefore is moved to Appendix A.Algorithm 7 shows the main iterative rounding process. We use z I, ( h ) , z A, ( h ) to denote thesolution obtained in iteration h , and C ( h ) s , C I, ( h ) t , C II, ( h ) t for the client sets at the end of the h th iteration. We also use z I, ( T +1) and z A, ( T +1) for z I, final and z A, final , where T is the total number ofiterations of the main while loop. Moreover, let z I, (0) = y I and z A, (0) = y A for every A ∈ Q , and C (0) s , C I, (0) t , C II, (0) t be the client sets before the start of the loop. Finally, we can assume w.l.o.g.that for every h ∈ [ T + 1], z I, ( h ) , z A, ( h ) is a vertex solution. Lemma 5.2.
For every h = 0 , , . . . T , the vectors z I, ( h ) , z A, ( h ) are a feasible solution for (17)-(22)with objective function value (16) at most B , when the LP is defined using C ( h ) s , C I, ( h ) t , C II, ( h ) t .Moreover, the latter three client sets satisfy invariants S.1 and
S.2 .Proof.
We prove this by induction on h . For the base case h = 0, constraints involving C II, (0) t are trivially satisfied because C IIt is initially empty. Constraint (17) is satisfied because of (8).Constraint (18) holds, since the clients j we chose to put in C I, (0) t have y I ( G j ) >
1. Similarly,constraint (20) is satisfied, because all clients j that we placed in C (0) s have y I ( G j ) ≤
1. Further,constraint (21) holds due to y I already satisfying (9). The filtering step for C I, (0) t ensures property S.1 . Since C II, (0) t = ∅ , C (0) s contains only clients j with y I ( G j ) ≤ C I, (0) t only clients j with y I ( G j ) > S.2 also holds. Finally, the objective value bound holds due to (7).15or iteration h >
0, note that by the inductive hypothesis the solution z I, ( h − , z A, ( h − isfeasible for the auxiliary LP defined using sets C ( h − s , C I, ( h − t , C II, ( h − t . Since the latter threesets satisfy S.1 and
S.2 , we can find an optimal vertex solution z I, ( h ) and z A, ( h ) satisfying the C s = ∅ case of Lemma 5.1; this new solution can only decrease the objective value. We now needto make sure that z I, ( h ) , z A, ( h ) remain feasible after updating C s , C It , C IIt , and these new sets stillsatisfy the proper invariants.Let j h the chosen client of iteration h , with z I, ( h ) ( G j h ) ∈ { , } . If z I, ( h ) ( G j h ) = 0, then C ( h ) s = C ( h − s \ { j h } , C II, ( h ) t = C II, ( h − t ∪ { j h } . Thus, we only need to check constraint (19) for j h ; thisholds trivially because z I, ( h ) ( G j h ) = 0. Next suppose z I, ( h ) ( G j h ) = 1, so that C ( h ) s = C ( h − s \ { j h } , C I, ( h ) t = C I, ( h − t ∪{ j h } . Here we only need to verify (18) for j h , but this is true since z I, ( h ) ( G j h ) = 1.Further, S.2 remains true, because we just moved j h from C ( h − s to either C I, ( h − t or C II, ( h − t . S.1 can only be violated if j h ∈ C I, ( h ) t , and there was a j ∈ C I, ( h − t with G j ∩ G j h = ∅ and R j < R jh .However, this is impossible, because when j first entered C It it should have removed j h from C s . Theorem 5.3.
Algorithm 7 terminates in at most n iterations, the set F I satisfies the matroidconstraint, and the solution F I , F A has opening cost at most B .Proof. Lemma 5.2 guarantees that the auxiliary LP is feasible and satisfies the conditions of Lemma5.1 at each iteration h . Hence, we can always find a j ∈ C s with z I, ( h ) ( G j ) ∈ { , } , and removeit from C ( h +1) s . Thus the loop must terminate after at most n iterations. At this point, when C s = ∅ , Lemmas 5.1 and 5.2 ensure that the final solution z final is integral. Moreover, Lemma’s 5.2statement about the cost ensures that P i ∈F z I, ( T +1) i · c Ii + P A p A P i ∈F z A, ( T +1) i · c Ai ≤ B .Since z I, final is integral, the cost of F I is precisely the cost P i ∈F z I, final i c Ii . Likewise, since z I, final satisfies the matroid polytope constraints (21), the set F I is an independent set of the matroid.Now consider the stage-II opening costs. For each j ∈ C IIt ∩ H A , the solution F A opens i Aj,R j ,while constraint (17) ensures that z A, final ( G j ) = 1. Since i Aj,R j has the lowest stage-II cost in G j ,the solution F A only reduces the opening cost compared to the LP solution. Since the opening costof z final is at most B , the cost of F I , F A is also at most B .Next we bound the approximation ratio. Some of the following arguments are very similar to[12], but for the sake of completeness we present them in full detail. Lemma 5.4. If j ∈ C I, ( h ) t for any h , then d ( j, F I ) ≤ R j .Proof. We show this by induction on h . When h = T this clearly holds, since then we will open afacility in G j and have d ( j, F I ) ≤ R j . Otherwise, suppose that j was removed from C It in iteration h . This occurs because j h , the client chosen in iteration h , entered C It . Therefore, G j ∩ G j h = ∅ .Moreover, because j h ∈ C (0) s and also j h was not removed from C s when j first entered C It , we have R j h ≤ R j /
2. Finally, since j h is present in C I, ( h ) t , the inductive hypothesis holds for it and yields d ( j h , F I ) ≤ R j h . Overall we get d ( j, F I ) ≤ R j + R j h + d ( j h , F I ) ≤ R j . Lemma 5.5.
For all A ∈ Q and all j ∈ H A we have d ( j, F I ∪ F A ) ≤ R j .Proof. Consider any A and j ∈ H A , and first suppose that y I ( G j ) >
1. In this case, the filtering stepto form C I, (0) t ensures that G j ∩ G j ′ = ∅ for some j ′ ∈ C I, (0) t with R j ′ ≤ R j . Furthermore, Lemma 5.4guarantees d ( j ′ , F I ) ≤ R j ′ . Overall, d ( j, F I ) ≤ d ( j, j ′ ) + d ( j ′ , F I ) ≤ R j + R j ′ + 3 R j ′ ≤ R j .Next, suppose that y I ( G j ) ≤
1. There are a number of cases to consider here. To begin ourreasoning, first assume that j was placed into C (0) s .16 lgorithm 8: Generalization Procedure for
Suppose A ∈ D arrived in the second stage;For every j ∈ A set g ( j ) ← − R j and r ( j ) ← R j ;( H A , π A ) ← Filter(
A, r, g );Open the set F A = { i Aj,R j | j ∈ C IIt ∩ H A } , where C IIt = C II, ( T ) t from Algorithm 7;Suppose that j was removed from C s due to being selected as having z I ( G j ) = 1 at time h . Inthis case, j is placed into C I, ( h ) t . By Lemma 5.4, we therefore have d ( j, F I ) ≤ R j .Suppose that j was removed from C s and was placed into C IIt . Thus, it will remain in C IIt atthe algorithm’s termination. We then have i Aj,R j ∈ F A , which ensures that d ( j, F A ) ≤ R j .Suppose that j was removed from C s , because G j ∩ G j h = ∅ and R j ≥ R j h /
2, where client j h was placed into C It at time h . By Lemma 5.4, we have d ( j h , F I ) ≤ R j h . So d ( j, F I ) ≤ d ( j, j h ) + d ( j h , F I ) ≤ R j + R j h + 3 R j h = R j + 4 R j h . Since R j h ≤ R j , this is at most 9 R j . Finally,notice that the arguments of this paragraph also cover the case of j not making it into C (0) s . Theorem 5.6.
For all A ∈ Q and all j ∈ A we have d ( j, F I ∪ F A ) ≤ R j .Proof. Lemma 5.5 shows that d ( j, F I ∪ F A ) ≤ R j if j ∈ H A . Therefore, all j ∈ A \ H A will besatisfied with a covering distance of at most 11 R j , because of the triangle inequality and the factthat the greedy stage-II filtering ensures R π A ( j ) ≤ R j , and thus d ( j, π A ( j )) ≤ R j . We show that Algorithm 7 is indeed efficiently generalizable as in Definition 2.3. Algorithm 8demonstrates how to handle any stage-II scenario A and yield the desired strategy ˆ s , by mimickingthe stage-II actions of Algorithm 7.For the approximation ratio, notice that based on Algorithm 7, if j / ∈ C IIt then d ( j, F I ) ≤ R j .Hence, reasoning similar to Theorem 5.6 ensures that d ( j, F I ∪ F A ) ≤ R j for every j ∈ A and A ∈ D . To conclude we again only need to bound the total number of strategies that are potentiallyachievable though this extension. Lemma 5.7.
Let S NU the set of strategies achievable via Algorithm 8. Then |S NU | = 2 m + n .Proof. The returned strategy depends on the sets F I and C IIt given by the polynomial-scenariosalgorithm. There are 2 m choices for F I and 2 n choices for C IIt . To tackle this, we construct an efficiently generalizable algorithm for , via anintriguing reduction to a non-stochastic clustering problem with outliers. Specifically, if we viewstage-I as consisting of a deterministic robust problem, stage-II can be interpreted as covering alloutliers left over by the first. Formally, we will use the following robust supplier problem:
Robust Weighted Multi-Knapsack-Supplier:
We are given a set of clients C and a set offacilities F , in a metric space with distance d . The input also includes parameters V, R ∈ R ≥ ,and for every client j ∈ C an associated weight v j ∈ R ≥ . In addition, there are the same types ofmulti-knapsack constraints as in : there are budgets W ℓ , and every facility i ∈ F has17 lgorithm 9: Approximation Algorithm for
Choose an arbitrary permutation g II of C ; // Filtering Step for each scenario A do ( H A , π A ) ← Filter(
A, R, g II );Construct instance I ′ of Robust Weighted Multi-Knapsack-Supplier as discussed; if RW ( I ′ ) = “INFEASIBLE” then return “INFEASIBLE”; F I ← RW ( I ′ ); // Stage-I facilities for all scenarios A ∈ Q do F A ← { i Aj | j ∈ H A with d ( j, F I ) > ρR } ; // Stage-II facilities end costs f ℓi for ℓ ∈ [ L ]. The goal is to choose a set of facilities S ⊆ F such that P j ∈C : d ( j,S ) >R v j ≤ V ,and f ℓ ( S ) ≤ W ℓ for every ℓ ∈ [ L ]. Clients j with d ( j, S ) > R are called outliers. We say that theinstance is discrete if the values f ℓi are all integers.We first show how any ρ -approximation for Robust Weighted Multi-Knapsack-Supplier can be used to get an efficiently generalizable ( ρ + 2)-approximation for . Wenext show that existing work [4, 20] gives a 3-approximation for discrete instances of RobustWeighted Multi-Knapsack-Supplier , leading to a 5-approximation for . to Robust Weighted Multi-Knapsack-Supplier
We first suppose that the costs c Ii as well as the budget B are polynomially bounded integers. Thisrestriction can be removed when we generalize to the black-box setting. Once more, let Q be aset of provided scenarios, R a target radius, and G j = G j,R , i Aj = i Aj,R for all j ∈ C and A ∈ Q .Furthermore, suppose that we have a ρ -approximation algorithm RW for Robust WeightedMulti-Knapsack-Supplier . For a feasible instance I ′ of the latter problem, RW should return asolution S satisfying all knapsack constraints and also P j ∈C : d ( j,S ) >ρR v j ≤ V .If the provided instance I of is feasible, the first step in tackling the problemis figuring out the portion of the budget B I used in the first stage of a feasible solution. Since thecosts c Ii are polynomially bounded integers, we can guess B I in polynomial time through solvingthe problem for all different alternatives for it. So from this point on, assume w.l.o.g. that we havethe correct B I , and also let B II = B − B I .Algorithm 9 shows how to use RW to approximate . It begins with filteringsteps for each A , and given H A , π A it constructs an instance I ′ of Robust Weighted Multi-Knapsack-Supplier . C , F , d , and R are the same for both problems. For all j ∈ C we set v j = X A ∈ Q : j ∈ H A p A · c Ai Aj and also V = B II . Finally, the instance I ′ has L ′ = L + 1 knapsack constraints, where the first L are the stage-I constraints of ( f ℓ ( S ) ≤ W ℓ ), and the last is c I ( S ) ≤ B I . Lemma 6.1.
If the original instance I is feasible, then the Robust WeightedMulti-Knapsack-Supplier instance I ′ is also feasible.Proof. Consider some feasible solution F ⋆I , F ∗ A for . We claim that F ⋆I is a validsolution for I ′ . It clearly satisfies the required L + 1 knapsack constraints, including the additional18 lgorithm 10: Generalization Procedure for
Suppose A arrived in the second stage;( H A , π A ) ← Filter(
A, R, g II ), where g II is the permutation of C chosen in Algorithm 9;Open the set F A = { i Aj | j ∈ H A and d ( j, F I ) > ρR } ;constraint c I ( F ⋆I ) ≤ B I . Also, for any A ∈ Q , any client j ∈ H A with d ( j, F ⋆I ) > R must be coveredby some facility x Aj ∈ G j ∩ F ⋆A . Since G j ′ ∩ G j ′′ = ∅ for all distinct j ′ , j ′′ ∈ H A we have: B II ≥ X A p A X i ∈ F ⋆A c Ai ≥ X A p A X j ∈ H A : d ( j,F ⋆I ) >R c Ax Aj ≥ X A p A X j ∈ H A : d ( j,F ⋆I ) >R c Ai Aj = X j ∈C : d ( j,F ⋆I ) >R v j This implies that S = F ⋆I satisfies the constraint P j : d ( j,S ) >R v j ≤ B II of instance I ′ . Theorem 6.2.
Algorithm 9 is a valid ( ρ + 2) -approximation for .Proof. First of all, Lemma 6.1 guarantees that if the given instance of is feasible,we will get a solution F I , F A . By specification of RW (), this satisfies the L knapsack constraintsand the constraint on the stage-I cost. The stage-II cost C II is given by: C II = X A p A X j ∈ H A : d ( j,F I ) >ρR c Ai Aj = X j ∈C : d ( j,F I ) >ρR v j ≤ B II , where the last inequality follows because F I is the output of RW ( I ′ ).Consider now a j ∈ A . The distance to its closest facility will be at most d ( π A ( j ) , F I ∪ F A ) + d ( j, π A ( j )). Since π A ( j ) ∈ H A , there will either be a stage-I open facility within distance ρR fromit, or we perform a stage-II opening in G π ( j ) , which results in a covering distance of at most R .Also, by the filtering step, we have d ( j, π A ( j )) ≤ R . So d ( j, F I ∪ F A ) ≤ ( ρ + 2) R .By combining this algorithm with existing 3-approximation algorithms for Robust WeightedMulti-Knapsack-Supplier , we get the following result:
Theorem 6.3.
There is a -approximation algorithm for discrete instances of ,with runtime poly( n, m, Λ) .Proof. The results of [4] give a 3-approximation for discrete instances of
Robust Weighted Multi-Knapsack-Supplier , when v j = 1 for all j . The work of [20] extends this to allow arbitrary v j values. Note that by our assumption that the values c I are polynomially bounded integers, thisalgorithm of [20] can be combined with Algorithm 9 and hence give a 5-approximation for . Finally, given the results of [4, 20], the runtime will be poly( n, m, Λ).
We need to show that the algorithm of Section 6.1 is efficiently generalizable as in Definition 2.3.Algorithm 10 demonstrates how to handle any stage-II scenario A , by mimicking the actions ofAlgorithm 9, and thus eventually yielding the desired strategy ˆ s .Theorem 6.2 ensures d ( j, F I ∪ F A ) ≤ ( ρ + 2) R for every j ∈ A and A ∈ D . To conclude, we againneed to bound the total number of strategies that are potentially achievable though this extension. Lemma 6.4.
Let S MK the set of strategies achievable via Algorithm 10. Then |S MK | = 2 m · n ! . roof. Proof is identical to that of Lemma 4.5.Recall now that our algorithm for requires the values c Ii to be polynomiallybounded integers. However, this assumption can be removed by a standard rescaling trick, at theexpense of a small loss of ǫB in the opening cost. Since our generalization procedure already causesa loss of O ( ǫB ) in the budget, this additional loss from Theorem 6.5 is then negligible. Theorem 6.5.
Suppose that the first-stage costs are arbitrary real numbers. By appropriate cost-quantization and any ǫ ∈ (0 , , Algorithm 9 gives a ( ρ + 2) -approximation for ,whose opening cost is at most (1 + ǫ ) B .Proof. For convenience, let us assume that B = 1, and suppose that all facilities have c Ii ≤ B = 1(as otherwise they can never be opened). Given some ǫ >
0, let us define q = ǫ/m , and form newcosts by ˜ c Ii = ⌈ c Ii /q ⌉ , ˜ c Ai = c Ai /q, B ′ = B (1 + ǫ ) /q . The costs ˜ c Ii are at most ⌈ /q ⌉ , and hence arepolynomially-bounded integers. Therefore, the reduction of Section 6.1 can be applied.Suppose now that F I , F A is a solution to the original instance with opening cost at most B .We then have ˜ c I ( F I ) + P A p A ˜ c A ( F A ) ≤ ( c I ( F I ) + P A p A c A ( F A )) /q + P i ∈F ≤ B/q + m ≤ B ′ .Thus, F I , F A is also a solution to the modified instance with opening cost at most B ′ . Further,consider any solution ˜ F I , ˜ F A to the modified instance, that we would get after running Algorithm 9with the new costs; its opening cost in the original instance is c I ( ˜ F I ) + P A p A c A ( ˜ F A ) ≤ q ˜ c I ( F I ) + P A p A q ˜ c A ( F A ) ≤ qB ′ = B (1 + ǫ ). Suppose we define our non-stochastic robust problem as having one knapsack and one matroidconstraint, instead of L knapsack constraints. Then the reduction of Section 6.1 would yield a( ρ + 2)-approximation for in the exact same manner, where ρ the ratio of thealgorithm used to solve the corresponding deterministic outliers problem.A result of [4, Theorem 16] gives a 3-approximation algorithm for this outliers problem, whichin turn would give a 5-approximation algorithm for . However, the algorithmobtained in this way would be randomized (in the sense that its returned solution may not be avalid one), and would be significantly more complex than the algorithm of Section 4. Our algorithm for is based on the thresholding technique of [15]. According to thismethod, an element should be selected in the first stage iff the probability that we need it is greaterthan 1 /λ . For our problem, in order to identify necessary facilities, we will iteratively sparsify thegiven instance by making locally approximate-optimal decisions.Consider a problem instance I = ( C , F , B, ~c, ~p, R ) where C the set of clients, F the set offacilities, B the given budget, ~c the vector of stage-I costs (recall that the stage-II cost of i is λc Ii ),and ~p the vector of arrival probabilities, i.e., every j ∈ C arrives independently with probability p j ∈ [0 , I is feasible if there is a solution that satisfies the expected budget constraintand where every arriving client has an open facility within distance R ; note that the radius R hereis included in the instance.Algorithm 11 describes our first-stage actions. For each facility i ∈ F it considers the set ofclients B i within distance R from it, and the probability P i = 1 − Q j ∈ B i (1 − p j ) that a client in B i will arrive in stage-II. Let also N i denote the facilities i ′ with B i ∩ B i ′ = ∅ , and S i = S i ′ ∈ N i B i ′ . Ifthere does not exist an i with P i ≥ /λ , then we are done with the stage-I actions. Otherwise, find20 lgorithm 11: First-Stage Decisions for true do ∀ i ∈ F : set B i ← { j ∈ C | d ( i, j ) ≤ R } and P i = 1 − Q j ∈ B i (1 − p j );Find an i ∈ F with P i ≥ /λ ; if no such i exists then break ; N i ← { i ′ ∈ F | B i ∩ B i ′ = ∅} and S i ← S i ′ ∈ N i B i ′ ;Open i m = arg min i ′ ∈ N i c Ii ′ in stage-I; C ← C \ S i and F ← F \ N i ; end such an i , and open the cheapest facility, say i m , in N i . Given this action, we assume that all clientsin S i get assigned to i m , and thus we remove S i from C . Also, i m now serves as a representative ofall facilities in N i , and since we made a local decision for N i and the clients dependent on it, wealso remove N i from F . In the next iteration, we proceed with the updated sets C and F .Let C k , F k denote the client and facility sets at the beginning of iteration k . Also, let i k , i km theselected facility with P i k ≥ λ and the facility chosen to open at iteration k respectively, given thatthe loop was not broken. The following lemma gives a key bound on the final facility opening cost. Lemma 7.1.
If the initial instance ( C , F , B, ~c, ~p, R ) of the problem is feasible, then for every iter-ation k ≥ , the instance ( C k , F k , B − P ℓ 21o get the second inequality, we use the fact that an actual arrival event is independent of whatactions are taken in stage-I. For the last one, we simply use two union bounds. To conclude, if wecombine (23) and (24) we see that the expected opening cost of S k restricted to F k +1 , C k +1 is atmost B − P ℓ ≤ k c Ii ℓm , which is exactly what we wanted. Proposition 7.2. Suppose that ( F , C , B, ~c, ~p, R ) is feasible, and P i < /λ, ∀ i ∈ F , where P i isthe probability that some client within distance R from i will arrive in stage-II. Then w.l.o.g. anyfeasible solution of radius at most R , never performs a stage-I opening.Proof. Assume that facility i is opened in stage-I, which means that we end up paying c Ii for it. Ifwe wait until stage-II and open it only if there is some arrival in B i , we will pay λc Ii · P i < c Ii .We conclude by describing the stage-II actions for a given scenario A ⊆ C . Notice that at thispoint we have a residual problem instance I ′ = ( F ′ , C ′ , B ′ , ~c, ~p, R ), where by Proposition 7.2 wemay assume that no stage-I openings occur. As we mentioned earlier, each client in A ∩ ( C \ C ′ )will be assigned to the stage-I open facility that was chosen by Algorithm 11, in the iteration thatthe client was discarded from C . Hence, we only have to deal with A ′ = A ∩ C ′ , and we interpretthat set as a stage-II realization for I ′ .Hence, let B ∗ A ′ the second-stage cost of a feasible solution of I ′ , when the arriving scenario is A ′ . To tackle A ′ , we use the standard non-stochastic 3-approximation for Knapsack-Supplier [14], with target radius R . The properties of this algorithm ensure that all clients of A ′ are coveredwithin distance 3 R , with an opening cost of at most B ∗ A ′ . Therefore, our overall opening cost forhandling I ′ is P A ′ p A ′ B ∗ A ′ ≤ B ′ , where p A ′ = Q j ∈ A ′ p j Q j ∈C ′ \ A ′ (1 − p j ). Theorem 7.3. If I is feasible, then our described solution has expected opening cost at most B ,and maximum distance to an assigned facility R .Proof. Let K the total number of iterations of Algorithm 11. Then the corresponding stage-Iopening cost is P K − k =1 c Ii km . Given that B ′ = B − P K − k =1 c Ii km , and as argued earlier the second-stagecost of our solution is at most B ′ , we have that the total opening cost is at most B .The clients that get satisfied through a stage-I opening will be within distance at most 5 R fromtheir closest facility. To see this focus on an arbitrary j ∈ B i , where i ∈ N i k for some iteration k .This client eventually gets assigned to i km and d ( j, i km ) ≤ R , because i km ∈ N i k , B i k ∩ B i = ∅ and B i km ∩ B i k = ∅ . On the other hand, due to the non-stochastic approximation algorithm used, theclients that get covered in stage-II are within distance 3 R from their closest facility. A Appendix: Omitted Details Hardness Result of Section 1.3: Take any Set Cover instance with E the universe of elements,and S , S , . . . , S m the collection of sets, where S i ⊆ E for every i ∈ [ m ]. For every set S i constructa facility i with the opening cost 1. All facilities will be within distance 2 of each other. In addition,for every element e of the universe have a client j e . All clients will also be within distance 2 ofeach other. Finally, for every S i and element e , have d ( i, j e ) = 1 if e ∈ S i , d ( i, j e ) = 3 otherwise.The constructed instance for the clustering problem forms a valid metric. The solutions to theclustering problem for R ∗ = 1 correspond to optimal Set Cover solutions, and vice versa. Proof of Lemma 5.1: We clearly have z ∗ ,Ii , z ∗ ,Ai ≤ i, A . Let us now define m = |{ i ∈F : z ∗ ,Ii > }| and m = |{ ( i, A ) ∈ F × Q : z ∗ ,Ai > }| . Since z ∗ ,I , z ∗ ,A is a vertex solution, thenumber of linearly independent tight constraints (17),(18),(20),(21) is at least m + m . Note that22onstraints (19) do not affect the above equality. Once more, using the characterization of Lemma4.2, the tight constraints (21) are D = { z ∗ ,I ( O ) = q O , ∀ O ∈ O} , where O is a family of k disjointsets (all subsets of F ), and every q O is a positive integer. Also, let L be the set of tight constraints(17) for which all stage-II variables are 0, L the set of tight constraints (17) that include at leastone non-zero stage-II variable, and L the set of tight constraints (18).First, suppose that C s = ∅ . Then, as in the proof of Lemma 4.3, we have: m + m ≤ rank( L ∪ L ∪ L ∪ D ) = rank( L ∪ L ∪ D ) + | L | (25)If there exists a fractional stage-I facility, then again as in the proof of Lemma 4.3 we have | L | ≤ m .With similar arguments as those presented in that Lemma, we can also get rank( L ∪ L ∪ D ) ≤ m − 1. This yields the desired contradiction. If on the other hand there exists a fractional stage-IIfacility, we trivially have rank( L ∪ L ∪ D ) ≤ m , and | L | ≤ m − C s = ∅ , and for the sake of contradiction we have z ∗ ,I ( G j ) ∈ (0 , 1) for every j ∈ C s . This immediately implies that there are no tight constraints (20), and hencewe can still use (25). Also, all tight constraints (17) for j ∈ C s are in L . At first, we triviallyhave | L | ≤ m . Finally, we can still show that rank( L ∪ L ∪ D ) ≤ m − L ∪ L ∪ D does not involve all stage-Inon-zero variables, getting the m − j ∈ C s . Using that, all previous reasoning presented in Lemma 4.3 goesthrough in the exact same manner. References [1] Shipra Agrawal, Amin Saberi, and Yinyu Ye. Stochastic Combinatorial Optimization underProbabilistic Constraints . 2008. arXiv: .[2] Alok Baveja, Amit Chavan, Andrei Nikiforov, Aravind Srinivasan, and Pan Xu. “ImprovedBounds in Stochastic Matching and Optimization”. In: APPROX/RANDOM 2015 . 2015,pp. 124–134.[3] E. M. L. Beale. “On Minimizing A Convex Function Subject to Linear Inequalities”. In: Journal of the Royal Statistical Society. Series B (Methodological) (1955), pp. 173–184.[4] Deeparnab Chakrabarty and Maryam Negahbani. “Generalized Center Problems with Out-liers”. In: ACM Trans. Algorithms (2019).[5] Moses Charikar, Chandra Chekuri, and Martin Pal. “Sampling Bounds for Stochastic Opti-mization”. In: Approximation, Randomization and Combinatorial Optimization. Algorithmsand Techniques . 2005, pp. 257–269.[6] George B. Dantzig. “Linear Programming under Uncertainty”. In: Management Science (1955),pp. 197–206.[7] Brian C. Dean, Michel X. Goemans, and Jan Vondrak. “Approximating the Stochastic Knap-sack Problem: The Benefit of Adaptivity”. In: Mathematics of Operations Research (2008),pp. 945–964.[8] Uriel Feige. “On Sums of Independent Random Variables with Unbounded Variance andEstimating the Average Degree in a Graph”. In: SIAM Journal on Computing . 2004, pp. 218–227.[10] Anupam Gupta, Martin Pal, R. Ravi, and Amitabh Sinha. “Sampling and Cost-Sharing:Approximation Algorithms for Stochastic Optimization Problems”. In: SIAM J. Comput. (2011), pp. 1361–1401.[11] Anupam Gupta, Martin Pal, Ramamoorthi Ravi, and Amitabh Sinha. “Boosted sampling:Approximation algorithms for stochastic optimization”. In: 2004, pp. 417–426.[12] David G. Harris, Shi Li, Thomas Pensyl, Aravind Srinivasan, and Khoa Trinh. “Approxima-tion Algorithms for Stochastic Clustering”. In: Proceedings of the 32nd International Confer-ence on Neural Information Processing Systems . 2018, pp. 6041–6050.[13] David G. Harris, Thomas Pensyl, Aravind Srinivasan, and Khoa Trinh. “A Lottery Model forCenter-Type Problems with Outliers”. In: Approximation, Randomization, and CombinatorialOptimization. Algorithms and Techniques (APPROX/RANDOM 2017) .[14] Dorit S. Hochbaum and David B. Shmoys. “A Unified Approach to Approximation Algorithmsfor Bottleneck Problems”. In: J. ACM (1986).[15] Nicole Immorlica, David Karger, Maria Minkoff, and Vahab S. Mirrokni. “On the Costs andBenefits of Procrastination: Approximation Algorithms for Stochastic Combinatorial Op-timization Problems”. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium onDiscrete Algorithms . 2004, pp. 691–700.[16] Ravishankar Krishnaswamy, Shi Li, and Sai Sandeep. “Constant Approximation for K-Medianand k-Means with Outliers via Iterative Rounding”. In: Proceedings of the 50th Annual ACMSIGACT Symposium on Theory of Computing . 2018, pp. 646–659.[17] Lap-Chi Lau, R. Ravi, and Mohit Singh. Iterative Methods in Combinatorial Optimization .1st. USA: Cambridge University Press, 2011. isbn : 0521189438.[18] Andre Linhares and Chaitanya Swamy. “Approximation Algorithms for Distributionally-Robust Stochastic Optimization with Black-Box Distributions”. In: Proceedings of the 51stAnnual ACM SIGACT Symposium on Theory of Computing . 2019, pp. 768–779.[19] Rolf H. Mohring, Andreas S. Schulz, and Marc Uetz. “Approximation in Stochastic Schedul-ing: The Power of LP-Based Priority Policies”. In: J. ACM (1999), pp. 924–942.[20] Andrea Pietracaprina, Geppino Pucci, and Federico Solda. Coreset-based Strategies for RobustCenter-type Problems . 2020. arXiv: .[21] R. Ravi and Amitabh Sinha. “Hedging Uncertainty: Approximation Algorithms for StochasticOptimization Problems”. In: Integer Programming and Combinatorial Optimization . 2004.[22] David Shmoys and Chaitanya Swamy. “An approximation scheme for stochastic linear pro-gramming and its application to stochastic integer programs”. In: J. ACM 53 (Nov. 2006),pp. 978–1012.[23] Anthony Man–Cho So, Jiawei Zhang, and Yinyu Ye. “Stochastic Combinatorial Optimizationwith Controllable Risk Aversion Level”. In: Approximation, Randomization, and Combinato-rial Optimization. Algorithms and Techniques . 2006, pp. 224–235.[24] Aravind Srinivasan. “Approximation Algorithms for Stochastic and Risk-Averse Optimiza-tion”. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algo-rithms . 2007, pp. 1305–1313. 2425] Chaitanya Swamy. “Risk-Averse Stochastic Optimization: Probabilistically-Constrained Mod-els and Algorithms for Black-Box Distributions: (Extended Abstract)”. In: Proceedings of theAnnual ACM-SIAM Symposium on Discrete Algorithms (2011), pp. 1627–1646.[26] Chaitanya Swamy and David Shmoys. “Sampling-Based Approximation Algorithms for Mul-tistage Stochastic Optimization”. In: 2005, pp. 357–366.[27] Chaitanya Swamy and David B. Shmoys. “Approximation Algorithms for 2-Stage StochasticOptimization Problems”. In: