Generalized Center Problems with Outliers

We study the $\mathcal{F}$-center problem with outliers: given a metric space $(X,d)$, a general down-closed family $\mathcal{F}$ of subsets of $X$, and a parameter $m$, we need to locate a subset $S\in \mathcal{F}$ of centers such that the maximum distance among the closest $m$ points in $X$ to $S$ is minimized. Our main result is a dichotomy theorem. Colloquially, we prove that there is an efficient $3$-approximation for the $\mathcal{F}$-center problem with outliers if and only if we can efficiently optimize a poly-bounded linear function over $\mathcal{F}$ subject to a partition constraint. One concrete upshot of our result is a polynomial time $3$-approximation for the knapsack center problem with outliers for which no (true) approximation algorithm was known.


Introduction
The k-center problem is a classic discrete optimization problem with numerous applications. Given a metric space (X, d) and a positive integer k, the objective is to choose a subset S ⊆ X of at most k points such that max v∈X d(v, S) is minimized, where d(v, S) = min u∈S d(v, u). Informally, the problem is to open k centers to serve all points, minimizing the maximum distance to service. This problem has been studied for at least 50 years [Hak64,Hak65], is NP-hard to approximate to a factor better than 2 [HN79], and has a simple 2-approximation algorithm [Gon85,HS85].
In many applications one is interested in a nuanced version of the problem where instead of serving all points in X, the objective is to serve at least a certain number of points. This is the so-called k-center with outliers version, or the robust k-center problem. This problem was first studied by Charikar et al. in [CKMN01] which gives a 3-approximation for the problem. A best possible 2-approximation algorithm was recently given by Chakrabarty et al. in [CGK16] (see also the paper [HPST17] by Harris et al. ).
Another generalization of the k-center problem arises when the location of centers has more restrictions. For instance, if each point in X has a different weight and the constraint is that the total weight of centers opened is at most k. This problem, now called the knapsack center problem, was studied by Hochbaum and Shmoys in [HS86] which gives a 3-approximation for the problem. To take another instance, X could be vectors in high dimension and the centers picked need to be linearly independent vectors. This motivates the matroid center problem where the set of centers must be an independent set in a matroid. Chen et al. give a 3-approximation for this problem in [CLLW13].
Naturally, the two aforementioned generalizations can be taken together. Indeed, for the robust matroid center problem, that is, the problem of picking centers which are an independent set and can choose a representation that works best for the F -PCM solver.
A series of corollaries follow from the above theorem. These are summarized in Table 1.
• When F = {A : w(A) ≤ k}, the F -PCM problem can be solved in polynomial time via dynamic programming. This crucially uses that the val is poly-bounded. Therefore we get a 3approximation for the robust knapsack center problem. (Theorem 12) • When F is the independent set of a matroid, then the F -PCM problem is a matroid intersection problem. Therefore we get a 3-approximation for the robust matroid center problem recovering the result from [HPST17]. (Theorem 14) • When F = {A : w 1 (A) ≤ k 1 , w 2 (A) ≤ k 2 , . . . , w d (A) ≤ k d } is defined by d weight functions and each weight function w i is poly-bounded, then F -PCM can be solved efficiently using dynamic programming. Therefore we get a 3-approximation algorithm for the robust multi-knapsack center problem, extending the result in [HS86] to the case with outliers. (Theorem 13) • When F is given by the intersection of a single knapsack and a single matroid constraint, then we don't know the complexity. However, when the weight function w(·) is poly-bounded and the matroid is representable, then we can give a randomized algorithm for the F -PCM problem via a reduction to the exact matroid intersection problem. Therefore, we get a randomized 3approximation for this special case of robust knapsack-and-matroid center problem (Theorem 16).
Remark 1: The Zero Outlier Case. At this juncture, the reader may wonder about the complexity of the F -center problem which doesn't allow any outliers. This is related to the following decision problem. Given (X, F ) and an arbitrary sub-partition P of X, the problem asks whether there is a set S ∈ F such that S contains exactly one element from each part of P. We call this the F -feasibility under partition constraints or simply the F -PCF problem. Analogous to the informal theorem from earlier, the F -center problem (without outliers) has an efficient 3-approximation algorithm if the F -PCF problem can be solved efficiently; otherwise the F -center problem has no non-trivial approximation algorithm. Indeed, this theorem is much simpler to prove and arguably the roots of this lie in [HS86].
This raises the main open question from our paper: what is the relation between the F -PCF and the F -PCM problem? Clearly, the F -PCF problem is as easy as the F -PCM problem (set all values equal to one in the latter). But is there an F such that F -PCM is "hard" while F -PCF is "easy"? One concrete example is the corollary discussed in the last bullet point above. When F is a single knapsack constraint and a single matroid constraint, then the F -PCF problem is solvable in polynomial time by minimizing a linear function over a matroid polytope and another partition matroid base polytope. As noted above, we don't know the complexity of the F -PCM problem in this case. Remark 2: Handling Approximations. If the F -PCM problem is NP-hard, then the robust Fcenter has no non-trivial approximation algorithm. However approximation algorithms for F -PCM translate to bi-criteria approximation algorithms for the robust F -center problem. More precisely, if we have a ρ-approximation for the F -PCM problem (ρ ≤ 1), then we get a (3, ρ)-bi-criteria approximation algorithm for the robust F -center problem. That is, we return a solution S ∈ F such that the maximum distance among the closest ρ · m points is at most 3 times the optimum value.
There could be a different notion of approximation possible for the F -PCM problem. Given an instance, there may be an algorithm which returns a set S whose value is at least the optimum value but S ∈ F R for some F R ⊇ F which is a 'relaxation' of F . For instance, if F is the intersection of multiple (constant) knapsack constraints which are not poly-bounded, then for any constant ε > 0 the F -PCM problem can be solved [CVZ11,GRSZ14] returning a set with value at least the optimum but violating each constraint by multiplicative (1 + ε). We can use the same to get a polynomial time 3-approximation for the robust multiple knapsack-center problem if we are allowed to violate the knapsack constraints by (1 + ε). Our Technique Although our theorem statement is quite general, the proof is quite easy. Let us begin with the F -center problem without outliers. For this, we follow the algorithmic 'partitioning' idea outlined in paper [HS86] by Hochbaum and Shmoys. As is standard, we guess the optimum distance which we assume to be 1 by scaling. Initially, all points are marked uncovered. Subsequently, we pick any uncovered point x and consider a subset B x of points within distance 1 from it. Note that the optimum solution must pick at least one point from each B x to serve x. Next, we call x "responsible" for all uncovered points within a distance 2 from it, and mark all these points covered. Observe that all the newly covered points are within distance 3 from any point in B x . We continue the above procedure till all points are marked covered. Also observe that the B x 's form a sub-partition P of the universe where each part has a responsible point. By the above two observations, we see that the F -PCF problem must have a feasible solution with respect to P, and any solution to the F -PCF problem gives a 3-approximation to the F -center problem.
Handling outliers is a bit trickier. The above argument doesn't work since the 'responsible' point may be an outlier in the optimal solution and we can no longer assert that the optimal solution must contain a point from each part. Indeed, the nub of the problem seems to be figuring out which points should be outliers. The 3-approximation algorithm in [CKMN01] by Charikar et al. (see also paper [APF + 10]) cleverly chooses the partitioning via a greedy procedure, but their argument seems hard to generalize to other constraints.
A different attack used in the algorithm in [CGK16] by Chakrabarty et al. and that in [HPST17] by Harris et al. is by writing an LP relaxation and using the solution of the LP to recognize the outliers. At a high level, the LP assigns each point x a variable (in this paper we call it cov(x)) that indicates the extent to which x is served. Subsequently, the partitioning procedure described in the first paragraph is run, except the responsible points are considered in decreasing order of cov(x). The hope is that points assigned higher cov(x) in the LP solution are less likely to be outliers, and therefore the partition returned by the procedure can be used to recover a 3-approximate solution. This idea does work for the natural LP relaxation of the robust matroid center problem but fails for the natural LP relaxation of the robust knapsack center problem. Indeed, the latter has unbounded integrality gap.
Our solution is to use the round-or-cut framework that has recently been a powerful tool in designing many approximation algorithms (see [CKK17,Li16,Li15,ASS14,CFLP00]). We consider the following "coverage polytope" for the robust F -center problem: the variables are cov(x) denoting the extent to which x is covered by a convex combination of sets S ∈ F . Of course, we cannot hope to efficiently check whether a particular cov lies in this polytope. Nevertheless, we show that for any cov in the coverage polytope, the partitioning procedure when run in the decreasing order of cov, has the property that there exists a solution S ∈ F intersecting each part at most once which covers at least m points. We can then use the algorithm for F -PCM to find this set. Furthermore, and more crucially, if the partitioning procedure does not have this property, then we can efficiently find a hyperplane separating cov from the the coverage polytope. Therefore, we can run the ellipsoid algorithm on the coverage polytope each time either obtaining a separating hyperplane, or obtaining a cov that leads to a desired partition, and therefore a 3-approximation.

Preliminaries
In this section we give formal definitions and statements of our results. As mentioned in a footnote in the Introduction, we focus on the supplier version of the problem.
Definition 2 (Robust F -Supplier Problem). The input is an instance of the F -supplier problem along with an integer parameter m ∈ {0, 1, . . . , |C|}. The objective is to find S ∈ F and T ⊆ C for which |T | ≥ m, and max u∈T d(u, S) is minimized.
Thus an instance I of the robust F -supplier problem is defined by the tuple (F, C, d, m, F ). In the definitions above, F and C are often called the set of facilities and customers respectively. Given the set system F defined over F , we define the following optimization problem.
Definition 3 (F -PCM problem). The input is J = (F, F , P, val) where F is a finite set and F ⊆ 2 F is a down-closed family, P ⊆ 2 F is a sub-partition of F , and val : F −→ {0, 1, 2, · · · } is an integer-valued function with maximum range |val| satisfying: The objective is to find: The next theorem is the main result of the paper. The next theorem is the (easier) second part of the dichotomy theorem. We show that if F -PCM cannot be solved, then the corresponding Robust F -Supplier cannot be approximated. Proof. Given J we construct an instance I of the Robust F -Supplier problem. The set of facilities is F . We describe the set of customers C next. Extend P to a partition of F denoted by Q = P ∪ {{f } : f ∈ F, ∄A ∈ P : f ∈ A}. By definition of the F -PCM problem, for any A ∈ Q, there exists a number n A ∈ {0, 1, 2, · · · } such that val(f ) = n A , for all f ∈ A. For each A ∈ Q, we add n A customers to C and call this set φ(A).
We now describe the distance function. For each A ∈ Q, for each pair Observe that d satisfies the triangle inequality.
Finally, we let m be our guess of the value of opt(J ). This completes the description of I = (F, C, d, m, F ).
Suppose algorithm B finds S ∈ F and T ⊆ C such that |T | ≥ m and max v∈T d(v, S) ≤ αopt(I ) = α. Without loss of generality, we can assume |S ∩ A| ≤ 1 for all A ∈ P, which implies that S is a feasible solution for J . The reason is, if there exists f 1 , f 2 ∈ S for which f 1 , f 2 ∈ A ∈ P, then S\f 2 is still an α-approximate solution for I . To see why this is true, recall that F is down-closed so S\f 2 ∈ F and since d(f 1 , f 2 ) = 0 then S\f 2 covers all the customers that S covers. Next, we assert that val Finally, since val is poly-bounded which makes the value of opt(J ) to be bounded by poly(|J |), one can iterate over all the possible values for opt(J ) to guess m.
We end this section by setting a few notations used in the remainder of the paper. For any u ∈ F ∪ C we let B C (u, r) be the customers in a ball of radius r around u i.e. B C (u, r) = {v ∈ C : d(u, v) ≤ r}. Similarly, define B F (u, r) as the facilities in a ball of radius r around u i.e. for u ∈ F ∪ C, B F (u, r) = {f ∈ F : d(u, f ) ≤ r}.

Algorithm and Analysis : Proof of Theorem 1
We fix I = (F, C, d, F , m) the instance of the Robust F -Supplier problem. We use opt to denote our guess of the value of the optimal solution. Without loss of generality, we can always assume opt = 1 because if not, we could scale d to meet this criteria. Our objective henceforth is to either find a set S ∈ F such that |{v ∈ C : d(v, S) ≤ 1}| ≥ m, or prove that opt(I ) > 1.
There are two parts to our proof. The first part is a partitioning procedure which given an assignment cov(v) ∈ R ≥0 for every customer v ∈ C, constructs an instance J of F -PCM. We call cov valuable if J has optimum value ≥ m. Our procedure ensures that if cov is valuable, then we get a 3-approximate solution for I . This is described in Section 3.1. The second part contains the proof of Theorem 1. In particular we show how using the round-and-cut methodology using polynomially many calls to A (recall this is the algorithm for F -PCM) we can either prove opt(I ) > 1, or find a valuable cov. This is described in Section 3.2.

Reduction to F -PCM
Algorithm 1 inputs an assignment {cov(v) ∈ R ≥0 : v ∈ C}. It returns a sub-partition P of F and assigns val : F → {0, 1, · · · , |C|} such that all the facilities in the same part of P get the same val. That is, it returns an F -PCM instance J = (F, F , P, val) with |val| ≤ |C|.
The algorithm maintains a set of uncovered customers U ⊆ C initialized to C (Line 1). In each iteration, it picks the customer v ∈ U with maximum cov (Line 5) and adds it to set Reps cov (Line 6). We add the set of facilities B F (v, 1) at distance 1 from v to P (Line 7, 8). For each such v, we eke out the subset Chld(v) = B C (v, 2) ∩ U of currently uncovered clients "represented" by v (Line 9).
For every facility f ∈ B F (v, 1) we define its value to be: val(f ) = |Chld(v)| (Line 10). At the end of the iteration, Chld(v) is removed from U (Line 11) and the loop continues till U becomes ∅. This way, the algorithm partitions C into {Chld(v) : v ∈ Reps cov } (see fact (3)). Claim 5 shows that P is a sub-partition of F . Reps cov ← Reps cov ∪ v 7: ⊲ Facilities that can cover v with a ball of radius 1 8: Fact 4. For a v ∈ Reps cov and any u ∈ Chld(v) line 6 of the algorithm implies cov(v) ≥ cov(u).
Claim 5. P constructed by Algorithm 1 is a sub-partition of F .
Proof. By Line 11 of the algorithm, for each u, v ∈ Reps cov we have d(u, v) > 2 hence B F (u, 1) ∩ B F (v, 1) = ∅ implying P is a sub-partition of F .
Proof. For any u ∈ Chld(v), we have d(u, v) ≤ 2 and since d(f, v) ≤ 1, the fact that d is metric implies d(f, u) ≤ 3. Proof. For an f ∈ S, according to Line 10 of the algorithm, val(f ) > 0 only if f ∈ B F (v, 1) for some v ∈ Reps cov . Also, by definition of the F -PCM problem, |B F (v, 1)∩S| ≤ 1 for any v ∈ Reps cov . That is, there is exactly one f ∈ B F (v, 1) ∩ S for each v ∈ R(S) and again by line 10, val(f ) = |Chld(v)|. Summing this equality over all v ∈ R(S) and the corresponding f ∈ B F (v, 1) ∩ S proves the claim.
Claim 8. Let I = (F, C, d, m, F ) be a Robust F -Supplier instance and let cov : C → R ≥0 be a coverage function. Let J = (F, F , P, val) be the F -PCM instance returned by Algorithm 1 on input I and cov. Given any feasible solution S to J , we can cover at least val(S) customers of C by opening radius 3-balls around each facility in S.
Proof. By considering R(S) from Definition 4, Claim 7 gives: implying the size of the former is at least val(S), thus proving the lemma.
The above claim motivates the following definition of valuable cov assignments, and the subsequent lemma.

Definition 5. An assignment
where J is the F -PCM instance returned by Algorithm 1 from I and cov.
Lemma 9. Given an instance I of the Robust F -Supplier problem with opt(I ) = 1, and a valuable assignment cov with respect to it, we can obtain a 3-approximate solution in time poly(|I |) + T A (J ) where J is the instance constructed by Algorithm 1 from I and cov.
Proof. Since cov is valuable, opt(J ) ≥ m. We use solver A to return an optimal solution S ∈ F with val(S) ≥ m. Claim 8 implies that S is a 3-approximate solution to I .

The Round and Cut Approach
If the guess opt = 1 for I = (F, C, d, m, F ) is at least opt(I ), then the following polytope must be non-empty. To see this, if S * ∈ F is the optimal solution to I then set z S * := 1 and z S := 0 for S ∈ F \S * .
Even though P I cov has exponentially many auxiliary variables (z S for all S ∈ F ), its dimension is still |C|. The following gives a family of valid inequalities for P I cov via Farkas lemma.
Then any cov ∈ P I cov satisfies Proof. Given cov ∈ P I cov , there exists {z S : S ∈ F } such that together they satisfy (P I cov .1)- The next lemma shows that all cov's in P I cov are valuable.
Lemma 11. Suppose an assignment {cov(v) ∈ R ≥0 : v ∈ C} is not valuable with respect to I = (F, C, d, m, F ). Then there is a hyper-plane separating it from P I cov that can be constructed in polynomial time.
Proof. If v∈C cov(v) < m, this inequality itself is a separating hyper-plane and we are done. So we may assume v∈C cov(v) ≥ m.
Let J = (F, F , P, val) be the F -PCM instance constructed by Algorithm 1 from I and cov. Fix S ∈ F and recall from Definition 4 that R(S) = {v ∈ Reps cov : B F (v, 1) ∩ S = ∅}. Pick an arbitrary T ⊆ S for which |B F (v, 1) ∩ T | = 1, for all v ∈ R(S). Observe that by downclosedness of F , we have T ∈ F which implies T is a feasible solution for J , and since cov is not valuable val(T ) < m. Furthermore, Claim 7 applied to T gives val(T ) = v∈R(T ) |Chld(v)|. Since R(S) = R(T ) and |Chld(v)| is integer-valued, we get: for v ∈ C as: Now observe that for any S ∈ F : That is, λ(v)'s satisfy (V1). Now we prove (V2) is not satisfied thus it can be used to separate cov from P I cov . v∈C Proof of Theorem 1. Given the guess opt which is scaled to 1, we use the ellipsoid algorithm to check if P I cov is empty or not. Whenever ellipsoid asks if a given cov is in P I cov or not, run Algorithm 1 for this given cov to construct the corresponding F -PCM instance J and use algorithm A, promised in the statement of Theorem 1, to solve it. If opt(J ) ≥ m, then Lemma 9 implies that we have a 3-approximate solution. Otherwise, cov is not valuable, and we can use Lemma 11 to find a separating hyperplane. In polynomial time, either we get a cov ∈ P I cov which by Lemma 11 has to be valuable, or we prove P I cov is empty and we modify our opt guess. For the correct guess, the latter case won't occur and we get a 3-approximate solution.

Applications and Extensions
In this section we elaborate on the applications and extensions stated in the Introduction. We begin with looking at specific instances of F which have been studied in the literature, and some which have not.

Single and Multiple Knapsack Constraints. We look at
where there are d weight functions over F and k i 's are upper bounds on these weights. Of special interest is the case d = 1 in which we get the robust knapsack supplier problem also called the weighted k-supplier problem with outliers.
The F -PCM problem for the above F KN has the following complexity: When d = 1, the problem can be solved in polynomial time. Indeed, given a partition P, since val(u) = val(v) for all v in the same part, any solution which picks a facility from a part A ∈ P may as well pick the one with the smallest weight in that part. Thus, the problem boils down to the usual knapsack problem in which we have |P| items where the item corresponding to part A ∈ P has weight min v∈A w(v) and value val(v). Since the values are poly-bounded, this problem is solvable in polynomial time. Thus, we get the following corollary to Theorem 1 resolving the open question raised in [CLLW13] and [HPST17].
Theorem 12. There is a polynomial time 3-approximation to the robust knapsack center problem.
When d > 1, then the F -PCM problem is NP-hard even when val is poly-bounded. However, if the w i 's are also poly-bounded (actually one of them can be general), then the F -PCM problem can be solved in polynomial time using dynamic programming. This problem was in fact studied in [HS86] (the conference version) and is called the suitcase problem there. Thus, we get the following corollary to Theorem 1 extending the result in [HS86].
Theorem 13. There is a polynomial time 3-approximation to the robust multiple-knapsack center problem if the number of weights is a constant and all but possibly one weight function are poly-bounded.

Single and Multiple Matroid Constraints. We look at
When d = 1, we get the robust matroid center problem. The F -PCM paper reduces to finding a maximum value set in I M and a partition matroid induced by P. This is solvable in polynomial time even when val is general and not poly-bounded, and even when I M is given as an independent set oracle. Thus, we get the following corollary to Theorem 1 obtaining the result in [HPST17].

Theorem 14.
[Theorem 1.1 in [HPST17]] There is a polynomial time 3-approximation to the robust matroid center problem even when the matroid is described as an independent set oracle.
When there are d > 1 matroids, then the F -PCM problem is NP-hard. Therefore, Theorem 2 implies that for instance, we can have no unicriteria approximation for the robust matroidintersection center problem.

Single Knapsack and Single Matroid Constraint. We look at
which is the intersection of a single matroid and a single knapsack constraint. To the best of our knowledge, the resulting Robust F -Supplier problem has not been studied before. One natural instantiation is when F is a collection of high-dimensional vectors with weights and the constraint on the centers is to pick a linearly independent set with total weight at most k.
The corresponding F -PCM problem asks us, given a partition P and poly-bounded values val, to find a set S ∈ I M ∩ I P of maximum value such that w(S) ≤ k, where I P is the partition matroid induced by P. We don't know if this problem can be solved in polynomial time, even in the case when M is another partition matroid. However, the above problem is related to the exact matroid intersection problem. In this problem, we are given two matroids M and P, and a weight function w on each ground element and a budget W. The objective is to decide whether or not there is a set S ∈ I M ∩ I P such that w(S) = W. Understanding the complexity of this problem is a long standing challenge [CGM92,MVV87,PY82]. When the matroids are representable over the same field, then [CGM92] gives a randomized pseudopolynomial time algorithm for the problem. The following claim shows the relation between F -PCM and the exact matroid intersection problem; this claim is essentially present in [BBGS11].
Claim 15. Given an algorithm for the exact matroid intersection problem, one can solve the F -PCM problem in polynomial time when the weights w are poly-bounded.
Proof. We guess V * to be the optimum value of the F -PCM problem; since val is poly-bounded, there are only polynomially many guesses. We also guess k * ≤ k to be the total w of the optimum set. Again if w is poly-bounded, there are polynomially many guesses. We define a weight function w as follows. Let φ = w(F ) + 1 be a large enough upper-bound on the possible values of w(S), S ⊆ F . Define w(f ) = φval(f ) + w(f ) for all f ∈ F and W = φV * + k * .
We claim that there is a set S in I M ∩ I P with w(S) = W iff val(S) = V * and w(S) = k * . The if-direction is trivial.
On the other hand if w(S) = W we get k * = w(S) − φV * = φval(S) + w(S) − φV * . Now if val(S) = V * since val is integer-valued and since φ > w(S) for any S ⊆ F , the RHS is either negative or > w(F ). In any case it cannot be k * . Therefore, we must have val(S) = V * which implies w(S) = k * .
Armed with the non-trivial result about exact matroid intersection from [CGM92], we get the following.
Theorem 16. Given a linear matroid M and a poly-bounded weight function, there is a randomized polynomial time 3-approximation to the robust knapsack-and-matroid center problem.

The Case of No Outliers
The F -supplier problem, that is the case of m = |C|, may be of special interest. In this case the problem is easier and the complexity is defined by the complexity of the following decision problem.
Definition 6 (F -PCF problem). The input is J = (F, F , P) where F is a finite set, F ⊆ 2 F is a down-closed family and P ⊆ 2 F is an arbitrary sub-partition of F . The objective is to decide whether there exists a set S ∈ F such that |S ∩ A| = 1, ∀A ∈ P.
Theorem 17. If the F -PCF problem can be solved efficiently for any partition P, then the F -supplier problem has a polynomial time 3-approximation. Otherwise, there is no non-trivial approximation possible for the F -supplier problem.
Sketch. Run Algorithm 1 with an arbitrary assignment cov (and ignore the val's). Let J = (F, F , P) be the resulting F -PCF instance. If the guess opt = 1 is correct, then note that the optimum solution S * must satisfy S * ∩ A = ∅ for all A ∈ P; if not, then the corresponding v ∈ Reps cov can't be served. Conversely, any S satisfying S ∩ A = ∅ for all A ∈ P implies a 3-approximate solution. Therefore, an algorithm for F -PCF can either give a 3-approximate solution or prove the guess opt is too low.
Theorem 1 and Theorem 17 raise the question: is there any set of constraints for which the problem without outliers is significantly easier than the problem with outliers? We don't know the answer to this question, although we guess the answer is yes. For this, it suffices to design a set system for which F -PCF is easy but F -PCM is hard (perhaps NP-hard). To see the difference between these problems consider the F KN∩Mat family described in the previous subsection. We don't know if F -PCM is easy or hard, but F -PCF is easy: this amounts to minimizing w(S) over S ∈ I M ∩ B P where B P is the base polytope induced by P. This can be done in polynomial time, and therefore we get the following corollary.
Theorem 18. There is a polynomial time 3-approximation to the knapsack-and-matroid center problem.

Handling Approximation
The technique used to prove Theorem 1 is robust enough to translate approximation algorithms for the F -PCM problem to bi-criteria approximation algorithms for the Robust F -Supplier problem. There are two notions of approximation algorithms for the F -PCM problem and they lead to two notions of bi-criteria approximation.
The first is the standard notion: a ρ-approximation (for ρ ≤ 1) algorithm that takes instance J of F -PCM, returns a solution S ∈ F of value val(S) ≥ ρ·opt(J ). The corresponding bi-criteria approximation notion for the Robust F -Supplier problem is the following: an (α, β)-approximation algorithm for instance I of Robust F -Supplier returns a solution which opens centers at S ∈ F and the distance of at least βm customers to S is ≤ α · opt(I ). The proof of Theorem 1 in fact implies the following.
Theorem 19. Let A be a polynomial time ρ-approximate algorithm for the F -PCM problem. Then there is a polynomial time (3, ρ)-bi-criteria approximation algorithm for the Robust F -Supplier problem.
The second notion of approximation for the F -PCM problem is one which satisfies the constraints approximately. This notion is more problem dependent and makes sense only if there is a notion of an approximate relaxation F R for the set F . For example, an (1 + ε)-relaxation for F KN could be the subsets S with w i (S) ≤ (1+ε)·k i for all i. A ρ-violating algorithm for an instance J of F -PCM would then return a set S with val(S) ≥ opt(J ) but S ∈ F R which is an ρ-relaxation for F . This defines a different bi-criteria approximation notion for the Robust F -Supplier problem. An α-approximate β-violating algorithm for the Robust F -Supplier problem takes an instance I and returns a solution S ∈ F R which is a β-relaxation for F such that at least m customers in C are at distance at most α · opt(I ) to S. Theorem 20. Let A be a polynomial time ρ-violating algorithm for the F -PCM problem. Then there is a polynomial time 3-approximate-ρ-violating algorithm for the Robust F -Supplier problem.
When F is described by constant d knapsack constraints (with general weights) and a single matroid constraint, for any constant ε > 0 Chekuri et al. give an (1 + ε)-approximation algorithm for the F -PCM in [CVZ11]. Without the matroid constraint, Grandoni et al. give an (1+ε)-violating algorithm in [GRSZ14]. Together, we get the following corollary. The latter recovers a result from [CLLW13].