Robust k-Center with Two Types of Radii
RRobust k -Center with Two Types of Radii Deeparnab Chakrabarty ∗ Maryam Negahbani
Abstract
In the non-uniform k -center problem, the objective is to cover points in a metric space withspecified number of balls of different radii. Chakrabarty, Goyal, and Krishnaswamy [ICALP2016, Trans. on Algs. 2020] (CGK, henceforth) give a constant factor approximation when thereare two types of radii. In this paper, we give a constant factor approximation for the two radiicase in the presence of outliers. To achieve this, we need to bypass the technical barrier ofbad integrality gaps in the CGK approach. We do so using “the ellipsoid method inside theellipsoid method”: use an outer layer of the ellipsoid method to reduce to stylized instances anduse an inner layer of the ellipsoid method to solve these specialized instances. This idea is ofindependent interest and could be applicable to other problems. In the non-uniform k -center ( NU k C ) problem, one is given a metric space ( X, d ) and balls of differentradii r > · · · > r t , with k i balls of radius type r i . The objective is to find a placement C ⊆ X of centers of these (cid:80) i k i balls, such that they cover X with as little dilation as possible. Moreprecisely, for every point x ∈ X there must exist a center c ∈ C of some radius type r i such that d ( x, c ) ≤ α · r i and the objective is to find C with α as small as possible.Chakrabarty, Goyal, and Krishnaswamy [CGK20] introduced this problem as a generalizationto the vanilla k -center problem [Gon85, HS85, HS86] which one obtains with only one type ofradius. One motivation arises from source location and vehicle routing: imagine you have a fleetof t -types of vehicles of different speeds and your objective is to find depot locations so that anyclient point can be served as fast as possible. This can be modeled as an NU k C problem. Thesecond motivation arises in clustering data. The k -center objective forces one towards clusteringwith equal sized balls, while the NU k C objective gives a more nuanced way to model the problem.Indeed, NU k C generalizes the robust k -center problem [CKMN01] which allows the algorithm tothrow away z points as outliers. This is precisely the NU k C problem with two types of radii, r = 1, k = k , r = 0, and k = z .Chakrabarty et al. [CGK20] give a 2-approximation for the special case of robust k -centerwhich is the best possible [HS85, Gon85]. Furthermore, they give a (1 + √ NU k C problem with two types of radii (henceforth, the 2 -NU k C problem). [CGK20]also prove that when t , the number of types of radii, is part of the input, there is no constant factorapproximation algorithms unless P=NP. They explicitly leave open the case when the numberof different radii types is a constant, conjecturing that constant-factor approximations should be ∗ Partially supported by NSF grant a r X i v : . [ c s . D S ] F e b ossible. We take the first step towards this by looking at the robust -NU k C problem. That is,the NU k C problem with two kinds of radii when we can throw away z outliers. This is the case of3-radii with r = 0. Theorem 1.
There is a -approximation for the Robust -NU k C problem. Although the above theorem seems a modest step towards the CGK conjecture, it is in fact anon-trivial one which bypasses multiple technical barriers in the [CGK20] approach. To do so, ouralgorithm applies a two-layered round-or-cut framework, and it is foreseeable that this idea willform a key ingredient for the constantly many radii case as well. In the rest of this section, webriefly describe the [CGK20] approach, the technical bottlenecks one faces to move beyond 2 typesof radii, and our approach to bypass them. A more detailed description appears in Section 2.One key observation of [CGK20] connects NU k C with the firefighter problem on trees [FKMR07,CC10, ABZ18]. In the latter problem, one is given a tree where there is a fire at the root. Theobjective is to figure out if a specified number of firefighters can be placed in each layer of the tree,so that the leaves can be saved. To be precise, the objective is to select k i nodes from layer i ofthe tree so that every leaf-to-root path contains at least one of these selected nodes.Chakrabarty et al. [CGK20] use the integrality of a natural LP relaxation for the firefighterproblem on height-2 trees to obtain their constant factor approximation for 2 -NU k C . In particular,they show how to convert a fractional solution of the standard LP relaxation of the 2 -NU k C problemto a feasible fractional solution for the firefighter LP. Since the latter LP is integral for height-2trees, they obtain an integral firefighting solution from which they construct an O (1)-approximatesolution for the 2 -NU k C problem. Unfortunately, this idea breaks down in the presence of outliersas the firefighter LP on height-2 trees when certain leaves can be burnt (outlier leaves, so to speak)is not integral anymore. In fact, the standard LP-relaxation for Robust -NU k C has unboundedintegrality gap. This is the first bottleneck in the CGK approach.Although the LP relaxation for the firefighter problem on height-2 trees is not integral when someleaves can be burnt, the problem itself (in fact for any constant height) is solvable in polynomialtime using dynamic programming (DP). Using the DP, one can then obtain (see, for instance,[Kai11]) a polynomial sized integral LP formulation for the firefighting problem. This suggests thefollowing enhancement of the CGK approach using the ellipsoid method. Given a fractional solution x to Robust -NU k C , use the CGK approach to obtain a fractional solution y to the firefightingproblem. If y is feasible for the integral LP formulation, then we get an integral solution to thefirefighting problem which in turn gives an O (1)-approximation for the Robust -NU k C instancevia the CGK approach. Otherwise, we would get a separating hyperplane for y and the poly-sizedintegral formulation for firefighting. If we could only use this to separate the fractional solution x from the integer hull of the Robust -NU k C problem, then we could use the ellipsoid methodto approximate Robust -NU k C . This is the so-called “round-or-cut” technique in approximationalgorithms.Unfortunately, this method also fails and indicates a much more serious bottleneck in the CGKapproach. Specifically, there is an instance of Robust -NU k C and an x in the integer hull of itssolutions, such that the firefighting instance output by the CGK has no integral solution! Thus,one needs to enhance the CGK approach in order to obtain O (1)-approximations even for the Robust -NU k C problem. The main contribution of this paper is to provide such an approach. Weshow that if the firefighting instance does not have an integral solution, then we can tease outmany stylized Robust -NU k C instances on which the round-or-cut method provably succeeds, and2n O (1)-approximation to any one of them gives an O (1)-approximation to the original Robust -NU k C instance. Our Approach.
Any solution x in the integer hull of NU k C solutions gives an indication of wheredifferent radii centers are opened. As it turns out, the key factor towards obtaining algorithms forthe Robust -NU k C problem is observing where the large radii (that is, radius r ) balls are opened.Our first step is showing that if the fractional solution x tends to open the r -centers only on“well-separated” locations then in fact, the round-or-cut approach described above works. Moreprecisely, if the Robust -NU k C instance is for some reason forced to open its r centers on pointswhich are at least cr apart from each other for some constant c >
4, then the CGK approachplus round-or-cut leads to an O (1)-approximation for the Robust -NU k C problem. We stress thatthis is far from trivial and the natural LP relaxations have bad gaps even in this case. We use ourapproach from a previous paper [CN19] to handle these well-separated instances.But how and why would such well-separated instances arise? This is where we use ideas fromrecent papers on fair colorful clustering [BIPV19, JSS20, AAKZ20]. If x suggested that the r -radiicenters are not well-separated, then one does not need that many balls if one allows dilation. Inparticular, if p and q are two r -centers of a feasible integral solution, and d ( p, q ) ≤ cr , then justopening one ball at either p or q with radius ( c + 1) r would cover every point that they each coverwith radius r -balls. Thus, in this case, the approximation algorithm gets a “saving” in the budgetof how many balls it can open. We exploit this savings in the budget by utilizing yet anotherobservation from Adjiashvili, Baggio, and Zenklusen [ABZ18] on the natural LP relaxation for thefirefighter problem on trees. This asserts that although the natural LP relaxation for constantheight trees is not integral, one can get integral solutions by violating the constraints additively bya constant. The aforementioned savings allow us to get a solution without violating the budgetconstraints.In summary, given an instance of the Robust -NU k C problem, we run an outer round-or-cutframework and use it to check whether an instance is well-separated or not. If not, we straightawayget an approximate solution via the CGK approach and the ABZ observation. Otherwise, we useenumeration (similar to [AAKZ20]) to obtain O ( n ) many different well-separated instances and foreach, run an inner round-or-cut framework. If any of these well-separated instances are feasible,we get an approximate solution for the initial Robust -NU k C instance. Otherwise, we can assert aseparating hyperplane for the outer round-or-cut framework. Related Work. NU k C was introduced in [CGK20] as a generalization to the k -center prob-lem [Gon85, HS85, HS86] and the robust k -center problem [CKMN01]. In particular CGK reduce NU k C to the firefighter problem on trees which has constant approximations [FKMR07, CC10,ABZ18] and recently, a quasi-PTAS [RS20]. NU k C has also been studied in the perturbation re-silient [ABS12, AMM17, CG18] settings. An instance is ρ -perturbation resilient if the optimal clus-tering does not change even when the metric is perturbed up to factor ρ . Bandapadhyay [Ban20]gives an exact polynomial time algorithm for 2-perturbation resilient instances with constant num-ber of radii.As mentioned above, part of our approach is inspired by ideas from fair colorful k -center cluster-ing [BIPV19, JSS20, AAKZ20] problems studied recently. In this problem, the points are dividedinto t color classes and we are asked to cover m i , i ∈ { , . . . , t } many points from each color byopening k -centers. The idea of moving to well-separated instances are present in these papers. Weshould mention, however, that the problems are different, and their results do not imply ours.The round-or-cut framework is a powerful approximation algorithm technique first used in a3aper by Carr et al . [CFLP00] for the minimum knapsack problem, and since then has found use inother areas such as network design [CCKK15] and clustering [ASS17, Li15, Li16, CN19, AAKZ20].Our multi-layered round-or-cut approach may find uses in other optimization problems as well. In this section, we provide the necessary technical preliminaries required for proving Theorem 1 andgive a more detailed description of the CGK bottleneck and our approach. We start with notations.Let (
X, d ) be a metric space on a set of points X with distance function d : X × X −→ R ≥ satisfyingthe triangle inequality. For any u ∈ X we let B ( u, r ) denote the set of points in a ball of radius r around u , that is, B ( u, r ) = { v ∈ X : d ( u, v ) ≤ r } . For any set U ⊆ X and function f : U → R , weuse the shorthand notation f ( U ) := (cid:80) u ∈ U f ( u ). For a set U ⊆ X and any v ∈ X we use d ( v, U )to denote min u ∈ U d ( v, u ).The 2-radii NU k C problem and the robust version are formally defined as follows. Definition 1 (2 -NU k C and Robust -NU k C ) . The input to 2 -NU k C is a metric space ( X, d ) alongwith two radii r > r ≥ k , k ∈ N . The objective of 2 -NU k C is to findthe minimum ρ ≥ S , S ⊆ X such that (a) | S i | ≤ k i for i ∈ { , } ,and (b) (cid:83) i (cid:83) u ∈ S i B ( u, ρr i ) = X . The input to Robust -NU k C contains an extra parameter m ∈ N ,and the objective is the same, except that condition (b) is changed to | (cid:83) i (cid:83) u ∈ S i B ( u, ρr i ) | ≥ m .An instance I of Robust -NU k C is denoted as (( X, d ) , ( r , r ) , ( k , k ) , m ). As is standard, we willfocus on the approximate feasibility version of the problem. An algorithm for this problem takesinput an instance I of Robust -NU k C , and either asserts that I is infeasible , that is, there is nosolution with ρ = 1, or provides a solution with ρ ≤ α . Using binary search, such an algorithmimplies an α -approximation for Robust -NU k C . Linear Programming Relaxations.
The following is the natural LP relaxation for the feasibilityversion of
Robust -NU k C . For every point v ∈ X , cov i ( v ) denotes its coverage by balls of radius r i . Variable x i,u denotes the extent to which a ball of radius r i is open at point u . If instance I isfeasible, then the following polynomial sized system of inequalities has a feasible solution. { ( cov i ( v ) : v ∈ X, i ∈ { , } ) : (cid:88) v ∈ X cov ( v ) ≥ m ( Robust -NU k C LP) (cid:88) u ∈ X x i,u ≤ k i ∀ i ∈ { , } cov ( v ) = (cid:88) u ∈ B ( v,r ) x ,u , cov ( v ) = (cid:88) u ∈ B ( v,r ) x ,u ∀ v ∈ X cov ( v ) = cov ( v ) + cov ( v ) ≤ ∀ v ∈ Xx i,u ≥ ∀ i ∈ { , } , ∀ u ∈ X } For our algorithm, we will work with the following integer hull of all possible fractional coverages.Fix a
Robust -NU k C instance I = (( X, d ) , ( r , r ) , ( k , k ) , m ) and let F be the set of all tuplesof subsets ( S , S ) with | S i | ≤ k i . For v ∈ X and i ∈ { , } , we say F covers v with radius r i if d ( v, S i ) ≤ r i . Let F i ( v ) ⊆ F be the subset of solutions that cover v with radius r i . Moreover, wewould like F ( v ) and F ( v ) to be disjoint, so if S ∈ F ( v ), we do not include it in F ( v ). Thefollowing is the integer hull of the coverages. If I is feasible, there must exist a solution in P I cov .4 ( cov i ( v ) : v ∈ X, i ∈ { , } ) : (cid:88) v ∈ X ( cov ( v ) + cov ( v )) ≥ m ( P I cov ) ∀ v ∈ X, i ∈ { , } cov i ( v ) − (cid:88) S ∈ F i ( v ) z S = 0 ( P I cov .1) (cid:88) S ∈ F z S = 1 ( P I cov .2) ∀ S ∈ F z S ≥ } ( P I cov .3) Fact 1. P I cov lies inside Robust -NU k C LP.
Firefighting on Trees.
As described in Section 1, the CGK approach [CGK20] is via the firefighterproblem on trees. Since we only focus on
Robust -NU k C , the relevant problem is the weighted L andleaf nodes L . Each leaf v ∈ L has a parent p ( v ) ∈ L and an integer weight w ( v ) ∈ N . We use Leaf ( u ) to denote the leaves connected to a u ∈ L (that is, { v ∈ L : p ( v ) = u } ). Observe that { Leaf ( u ) : u ∈ L } partitions L . So we could represent the edges of the trees by this Leaf partition.Hence the structure is identified as ( L , L , Leaf , w ). Definition 2 (2-Level Fire Fighter ( ) Problem) . Given height-2 trees ( L , L , Leaf , w ) alongwith budgets k , k ∈ N , a feasible solution is a pair T = ( T , T ), T i ⊆ L i , such that | T i | ≤ k i for i ∈ { , } . Let C ( T ) := { v ∈ L : v ∈ T ∨ p ( v ) ∈ T } be the set of leaves covered by T . Theobjective is to maximize w ( C ( T )). Hence a instance is represented by (( L , L , Leaf , w ) , k , k ).The standard LP relaxation for this problem is quite similar to the Robust -NU k C LP. For eachvertex u ∈ L ∪ L there is a variable 0 ≤ y u ≤ u is included inthe solution. For a leaf v , Y ( v ) is the fractional amount by which v is covered through both itselfand its parent.max (cid:88) v ∈ L w ( v ) Y ( v ) : (cid:88) u ∈ L i y u ≤ k i , ∀ i ∈ { , } ; ( LP) Y ( v ) := y p ( v ) + y v ≤ , ∀ v ∈ L ; y u ≥ , ∀ u ∈ L ∪ L Remark 1.
The following figure shows an example where the above LP relaxation has an integralitygap. However, can be solved via dynamic programming in O ( n ) time and has similar sizedintegral LP relaxations.Figure 1: A instance with budgets k = k = 1. Multiplicity w is 1 for the circle leaves and 3for the triangles. The highlighted nodes have y = 1 / y = 0. Theobjective value for this y is 4 × / .1 CGK ’s Approach and its Shortcomings
Given fractional coverages ( cov ( v ) , cov ( v ) : v ∈ X ), the CGK algorithm [CGK20] runs theclassic clustering subroutine by Hochbaum and Shmoys [HS85] in a greedy fashion. In English,the Hochbaum-Shmoys (HS) routine partitions a metric space such that the representatives of eachpart are well-separated with respect to an input parameter. The CGK algorithm obtains a instance by applying the HS routine twice. Once on the whole metric space in decreasing order of cov ( v ) = cov ( v ) + cov ( v ), and the set of representatives forms the leaf layer L with weights beingthe size of the parts. The next time on L itself in decreasing order of cov and the representativesform the parent layer L . These subroutines and the subsequent facts form a part of our algorithmand analysis. Algorithm 1 HS Input:
Metric (
U, d ), parameter r ≥
0, and assignment { cov ( v ) ∈ R ≥ : v ∈ U } R ← ∅ (cid:46) The set of representatives while U (cid:54) = ∅ do u ← arg max v ∈ U cov ( v ) (cid:46) The first client in U in non-increasing cov order R ← R ∪ u Child ( u ) ← { v ∈ U : d ( u, v ) ≤ r } (cid:46) Points in U at distance r from u (including u itself) U ← U \ Child ( u ) end whileOutput: R , { Child ( u ) : u ∈ R } Algorithm 2
CGK
Input:
Robust -NU k C instance (( X, d ) , ( r , r ) , ( k , k ) , m ), dilation factors α , α >
0, and as-signments cov ( v ) , cov ( v ) ∈ R ≥ for all v ∈ X ( L , { Child ( v ) , v ∈ L } ) ← HS (( X, d ) , α r , cov = cov + cov ) ( L , { Child ( v ) , v ∈ L } ) ← HS (( L , d ) , α r , cov ) w ( v ) ← | Child ( v ) | for all v ∈ L Leaf ( u ) ← Child ( u ) for all u ∈ L Output: instance (( L , L , Leaf , w ) , ( k , k )) Definition 3 (Valuable instances) . We call an instance T returned by the CGK algorithmvaluable if it has an integral solution of total weight at least m . Using dynamic programming, thereis a polynomial time algorithm to check whether T is valuable. Fact 2.
The following are true regarding the output of HS : (a) ∀ u ∈ R, ∀ v ∈ Child ( u ) : d ( u, v ) ≤ r ,(b) ∀ u, v ∈ R : d ( u, v ) > r , (c) The set { Child ( u ) : u ∈ R } partitions U , and (d) ∀ u ∈ R, ∀ v ∈ Child ( u ) : cov ( u ) ≥ cov ( v ). Lemma 1 (rewording of Lemma 3.4. in [CGK20]) . Let I be a Robust -NU k C instance. If for anyfractional coverages ( cov ( v ) , cov ( v )) the instance created by Algorithm 2 is valuable, thenone obtains an ( α + α )-approximation for I .Lemma 1 suggests that if we can find fractional coverages so that the corresponding instance T is valuable, then we are done. Unfortunately, the example illustrated in Figure 2 shows that for any6 α , α ) there exists Robust -NU k C instances and fractional coverages ( cov ( v ) , cov ( v )) ∈ P I cov inthe integer hull, for which the CGK algorithm returns instances that are not valuable.Figure 2: At the top, there is a feasible Robust -NU k C instance with k = 2, k = 3, and m = 24.There are 6 triangles representing 3 collocated points each, along with 12 circles, each representingone point. The black edges are distance r > α r and the grey edges are distance α r . There aretwo integral solutions S and S (cid:48) each covering exactly 24 points. S = { u , u } , S = { u , v , u } , S (cid:48) = { u , u } , and S (cid:48) = { u , v , u } . Having z S = z S (cid:48) = 1 / P I cov , gives cov of 1 / cov of 1 / instance at thebottom. According to Proposition 1 the highlighted nodes have y = 1 / y = 0 with objective value 12 × / Although the instance obtained by Algorithm 2 from fractional coverages ( cov ( v ) , cov ( v ) : v ∈ X ) may not be valuable, [CGK20] proved that if these coverages come from ( Robust -NU k C LP),then there is always a fractional solution to (
LP) for this instance which has value at least m . Proposition 1 (rewording of Lemma 3.1. in [CGK20]) . Let ( cov ( v ) , cov ( v ) : v ∈ X ) be anyfeasible solution to Robust -NU k C LP. As long as α , α ≥
2, the following is a fractional solutionof
LP with value at least m for the instance output by Algorithm 2. y v = (cid:40) cov ( v ) v ∈ L min { cov ( v ) , − cov ( p ( v )) } v ∈ L . Therefore, the problematic instances are precisely instances that are integrality gap examplesfor (
LP). Our first observation stems from what Adjiashvili, Baggio, and Zenklusen [ABZ18]call “the narrow integrality gap of the firefighter LP”.
Lemma 2 (From Lemma 6 of [ABZ18]) . Any basic feasible solution { y v : i ∈ { , } , v ∈ L i } of the LP polytope has at most 2 loose variables. A variable y v is loose if 0 < y v < y p ( v ) = 0in case v ∈ L .In particular, if y ( L ) ≤ k −
2, then the above lemma along with Proposition 1 implies there existsan integral solution with value ≥ m . That is, the instance is valuable. Conversely, the fact7hat the instance is not valuable asserts that y ( L ) > k − cov ( L ) > k − L .This is where we exploit the ideas in [BIPV19, JSS20, AAKZ20]. By choosing α > L are “well-separated”. More precisely,we can ensure for any two u, v ∈ L we have d ( u, v ) > α r (from Fact 2). The well-separatedcondition implies that the same center cannot be fractionally covering two different points in L .Therefore, cov ( L ) > k − cov , cov ) ∈ P I cov is in the integer hull, then there must exist aninteger solution which opens at most not cover points in L . For the time beingassume in fact no such center exists and cov ( L ) = k . Indeed, the integrality gap example inFigure 2 satisfies this equality.Our last piece of the puzzle is that if the cov ’s are concentrated on separated points, thenindeed we can apply the round-or-cut framework to obtain an approximation algorithm. To thisend, we make the following definition, and assert the following theorem. Definition 4 ( Well-Separated Robust -NU k C ) . The input is the same as
Robust -NU k C , alongwith Y ⊆ X where d ( u, v ) > r for all pairs u, v ∈ Y , and the algorithm is allowed to open theradius r -centers only on points in Y . Theorem 2.
Given a
Well-Separated Robust -NU k C instance there is a polynomial time algorithmusing the ellipsoid method that either gives a -approximate solution, or proves that the instance isinfeasible. We remark the natural (
Robust -NU k C LP) relaxation still has a bad integrality gap, and we needthe round-or-cut approach. Formally, given fractional coverages ( cov , cov ) we run Algorithm 2(with α = α = 2) to get a instance. If the instance is valuable, we are done by Lemma 1.Otherwise, we prove that ( cov , cov ) / ∈ P I cov by exhibiting a separating hyperplane. This cruciallyuses the well-separated-ness of the instance and indeed, the bad example shown in Figure 2 is not well-separated. This implies Theorem 2 using the ellipsoid method.In summary, to prove Theorem 1, we start with ( cov , cov ) purported to be in P I cov . Our goalis to either get a constant approximation, or separate ( cov , cov ) from P I cov . We first run theCGK Algorithm 2 with α = 8 and α = 2. If cov ( L ) ≤ k −
2, we can assert that the instance is valuable and get a 10-approximation. Otherwise, cov ( L ) > k −
2, and we guess the O ( n ) many possible centers “far away” from L , and obtain that many well-separated instances.We run the algorithm promised by Theorem 2 on each of them. If any one of them gives a 4-approximate solution, then we immediately get an 8-approximate solution to the original instance.If all of them fail, then we can assert cov ( L ) ≤ k − valid inequality for P I cov , and thusobtain a hyperplane separating ( cov , cov ) from P I cov . The polynomial running time is impliedby the ellipsoid algorithm. Note that there are two nested runs of the ellipsoid method in thealgorithm. Figure 3 below shows an illustration of the ideas. Before we move to describing algorithms proving Theorem 2 and Theorem 1, let us point out whythe above set of ideas does not suffice to prove the full CGK conjecture, that is, give an O (1)-approximation for NU k C with constant many type of radii. Given fractional coverages, the CGK The factor doubles as we need to double the radius, but that is a technicality. start Robust 2-NU 𝑘 C instance I = ((𝑋, 𝑑), (𝑟 ! , 𝑟 " ), (𝑘 ! , 𝑘 " ),m) fail Ellipsoid on 𝒫 cov I co-v = (co-v ! , co-v " ) CGK (Algorithm2) with 𝛼 ! = 8 𝛼 " = 2 Yes No No Yes 2-FF instance T = ((𝐿 ! , 𝐿 " ,Leaf, w), (𝑘 ! , 𝑘 " )) co-v & (𝐿 & ) ≤ 𝑘 & 𝑖 ∈ {1,2} co-v ! (𝐿 ! )≤ 𝑘 ! − 2 Yes Solve T (Proposition 5 and Lemma 1) I end Proposition 6 Reduce I to Well-Separated Robust 2-NU 𝑘 C Well-Separated Robust 2-NU 𝑘 C instances I ’ ∀𝑞 ∈ 𝑄 ∪ ∅, for 𝑄 ⊆ 𝑋 ∃ feasible I ’ 𝑞 ∈ 𝑄 ∪ ∅ Ellipsoid (Theorem 2) 𝑘 C 8-approximate solution for I Yes
Proposition 6 asserts cov ! (𝐿 ! ) ≤ 𝑘 ! − 2 No No co-v(X) ≥ m
Figure 3: Our framework for approximating
Robust -NU k C . The three black arrows each representseparating hyperplanes we feed to the outer ellipsoid. The box in the bottom row stating “4-approximation for well-separated Robust -NU k C ” runs the inner ellipsoid method.algorithm now returns a t -layered firefighter instance and again if such an instance is valuable(which can be checked in n O ( t ) time), we get an O (1)-approximation. As above, the main challengeis when the firefighter instance is not valuable. Theorem 2, in fact, does generalize if all layers areseparated. Formally, if there are t types of radii, and there are t sets Y , . . . , Y t such that (a) any twopoints p, q ∈ Y i are well-separated, that is, d ( p, q ) > r i , and (b) the r i -radii centers are only allowedto be opened in Y i , then in fact there is an O (1)-approximation for such instances. Furthermore,if we had fractional coverages ( cov , cov , . . . , cov t ) such that in the t -layered firefighter instancereturned, all layers have “slack”, that is cov i ( L i ) ≤ k i − t , then one can repeatedly use Lemma 2to show that the tree instance is indeed valuable.The issue we do not know how to circumvent is when some layers have slack and some layersdo not. In particular, even with 3 kinds of radii, we do not know how to handle the case whenthe first layer L is well-separated and cov ( L ) = k , but the second layer has slack cov ( L ) ≤ k −
3. Lemma 2 does not help since all the loose vertices may be in L , but they cannot all bepicked without violating the budget. At the same time, we do not know how to separate such cov ’s,or whether such a situation arises when cov ’s are in the integer hull. We believe one needs moreideas to resolve the CGK conjecture. Well-Separated Robust -NU k C In this section we prove Theorem 2 stated in Section 2.2. As mentioned there, the main idea is torun the round-or-cut method, and in particular use ideas from a previous paper [CN19] of ours.The main technical lemma is the following. 9 emma 3.
Given
Well-Separated Robust -NU k C instance I and fractional coverages ( ˆ cov ( v ) , ˆ cov ( v )),if the output of the CGK Algorithm 2 is not valuable, there is a hyperplane separating ( ˆ cov ( v ) , ˆ cov ( v ))from P I cov . Furthermore, the coefficients of this hyperplane are bounded in value by | X | . Remark 2.
We need to be careful in one place. Recall that HS is used in the CGK Algorithm 2. Weneed to assert in HS , that points u with d ( u, Y ) ≤ r are prioritized over points v with d ( v, Y ) > r to be taken in L . This is w.l.o.g. since cov ( v ) = 0 if d ( v, Y ) > r by definition of Well-SeparatedRobust -NU k C .Using the ellipsoid method, the above lemma implies Theorem 2. Proof of Theorem 2.
The goal is to either prove P I cov is empty, or give a 4-approximate solution.To do so, we run the ellipsoid algorithm. Each time the ellipsoid algorithm provides a purportedfractional point ( ˆ cov ( v ) , ˆ cov ( v )) ∈ P I cov and asks for a separating hyperplane. Given such asolution, we first check if ˆ cov ( v ) = 0 for all v with d ( v, Y ) > r . By the well-separatednessproperty of I , this must be a valid equality and we can force the ellipsoid method to run overthese equalities. Then we run CGK Algorithm 2 with this ( ˆ cov , ˆ cov ) and α = α = 2. If theresulting instance is valuable, we get a 4-approximate solution by Lemma 1. If not, Lemma 3provides a separating hyperplane to feed to ellipsoid. Since our hyperplanes can be described inpolynomial size, ellipsoid terminates in polynomial time, either giving us some ˆ cov leading to a4-approximation along the way, or prompts that P I cov is empty thereby proving I is infeasible.The rest of this section is dedicated to proving Lemma 3. Fix a well-separated Robust -NU k C instance I . Recall that Y ⊆ X is a subset of points, and the radius r centers are only allowed tobe opened at Y . Let T be the instance output by Algorithm 2 on I and cov with α = α = 2.Recall, T = (( L , L , Leaf , w ) , k , k ). The key part of the proof is the following valid inequality incase T is not valuable. Lemma 4. If T is not valuable (cid:80) v ∈ L w ( v ) cov ( v ) ≤ m − cov ( v ) ∈ P I cov .Before we prove Lemma 4, let us show how it proves Lemma 3. Given ( ˆ cov , ˆ cov ) we first check that (cid:80) u ∈ X ˆ cov ( u ) ≥ m , or otherwise that would be the hyperplane separating it from P I cov . Nowrecall that in Algorithm 2, for v ∈ L , w ( v ) = | Child ( v ) | which is the number of points assigned to v by HS (see Line 1 of Algorithm 2). By definition of w and then parts d) and c) of Fact 2, (cid:88) v ∈ L w ( v ) ˆ cov ( v ) = (cid:88) v ∈ L (cid:88) u ∈ Child ( v ) ˆ cov ( v ) ≥ (cid:88) v ∈ L (cid:88) u ∈ Child ( v ) ˆ cov ( u ) = (cid:88) u ∈ X ˆ cov ( u ) ≥ m . That is, ( ˆ cov , ˆ cov ) violates the valid inequality asserted in Lemma 4, and this would complete theproof of Lemma 3. All that remains is to prove the valid inequality lemma above. of Lemma 4. Fix a solution cov ∈ P I cov and note that this is a convex combination of coveragesinduced by integral feasible solutions in F . The main idea of the proof is to use the solutions in F to construct solutions to the tree instance T . Since T is not valuable, each of these solutions willhave “small” value, and then we use this to prove the lemma. To this end, fix S = ( S , S ) ∈ F where | S i | ≤ k i for i ∈ { , } . The corresponding solution T = ( T , T ) for T is defined as follows:For i ∈ { , } and any u ∈ L i , u is in T i iff S i ∈ F i ( u ). That is, d ( u, S i ) ≤ r i . recall, ˆ cov ( v ) = ˆ cov ( v ) + ˆ cov ( v ). roposition 2. T satisfies the budget constraints | T i | ≤ k i for i ∈ { , } . Proof.
For i ∈ { , } and two different u, v ∈ T i by Fact 2 and our choice of α i = 2, d ( u, v ) > r i .By the triangle inequality, a facility in S i cannot cover both u and v meaning | T i | ≤ | S i | ≤ k i .The next claim is the only place where we need the well-separated-ness of I . Basically, we willargue that the leaves covered by T capture all the points covered by S . Proposition 3. If u ∈ L but u / ∈ T then no v ∈ Leaf ( u ) can be covered by a ball of radius r in S . Proof.
We will prove the contrapositive by showing that if u = p ( v ) and v is covered through f ∈ S ,then u as well must be covered by the same f and therefore, u ∈ T . Consider the following twocases: either d ( u, Y ) > r in which case, by our assumption on HS , v is prioritized over u to be chosenin L so this cannot happen. Thus, we must have d ( u, Y ) ≤ r which means there is f u ∈ Y with d ( u, f u ) ≤ r . This f u has to be equal to f otherwise, by definition of Y we must have d ( f, f u ) > r that contradicts the following: d ( f u , f ) ≤ d ( f u , u ) + d ( u, v ) + d ( v, f ) ≤ r + α r + r = 4 r .Next, we can prove that overall, the leaves covered by T capture the whole set of points coveredby S . Recall that C ( T ) = { v ∈ L : v ∈ T ∨ p ( v ) ∈ T } is the set of leaves covered by T . For v ∈ X let F ( v ) := F ( v ) ∪ F ( v ) be the set of solutions that cover v . Proposition 4.
Take solution T corresponding to Well-Separated Robust -NU k C solution S as described earlier. We have: (cid:88) v ∈ L : S ∈ F ( v ) w ( v ) ≤ w ( C ( T )) . That is, the total w of the points covered by S is at most w ( C ( T )). Proof.
The leaves covered by T are covered either by T or T . Thus, we get w ( C ( T )) = (cid:88) u ∈ T (cid:88) v ∈ Leaf ( u ) w ( v ) + (cid:88) u/ ∈ T (cid:88) v ∈ Leaf ( u ): v ∈ T w ( v ) . (1)The first of these terms can be lower-bounded as (cid:88) u ∈ T (cid:88) v ∈ Leaf ( u ) w ( v ) ≥ (cid:88) u ∈ T (cid:88) v ∈ Leaf ( u ): S ∈ F ( v ) w ( v ) . that is, we only consider the leaves v of u ∈ T which are covered by the Robust -NU k C solution S . The second term, by definition of T is (cid:88) u/ ∈ T (cid:88) v ∈ Leaf ( u ): v ∈ T w ( v ) = (cid:88) u/ ∈ T (cid:88) v ∈ Leaf ( u ): S ∈ F ( v ) w ( v ) = (cid:88) u/ ∈ T (cid:88) v ∈ Leaf ( u ): S ∈ F ( v ) w ( v ) . where the last equality uses Proposition 3 which implies for u / ∈ T and v ∈ Leaf ( u ), d ( v, S ) > r .Thus, the solution S covers v iff S covers v . Plugging back in (1), we complete the proof.11he proof of Lemma 4 now follows from the fact that T is not valuable thus w ( C ( T )) ≤ m − S ∈ F we have (cid:80) v ∈ L : S ∈ F ( v ) w ( v ) ≤ m −
1. So we have: (cid:88) v ∈ L w ( v ) cov ( v ) = ( P I cov .1) (cid:88) v ∈ L w ( v ) (cid:88) S ∈ F ( v ) z S = (cid:88) S ∈ F z S (cid:88) v ∈ L : S ∈ F ( v ) w ( v ) ≤ ( m − (cid:88) S ∈ F z S = ( P I cov .2) m − . As mentioned in Section 2, we focus on the feasibility version of the problem: given an instance I of Robust -NU k C we either want to prove it is infeasible, that is, there are no subsets S , S ⊆ X with (a) | S i | ≤ k i and (b) | (cid:83) i (cid:83) u ∈ S i B ( u, r i ) | ≥ m , or give a 10-approximation that is, open subsets S , S that satisfy (a) and | (cid:83) i (cid:83) u ∈ S i B ( u, r i ) | ≥ m . To this end, we apply the round-or-cutmethodology on P I cov . Given a purported ˆ cov := ( ˆ cov ( v ) , ˆ cov ( v ) : v ∈ X ) we want to either useit to get a 10-approximate solution, or find a hyperplane separating it from P I cov . Furthermore, wewant the coefficients in the hyperplane to be poly-bounded. Using the ellipsoid method we indeedget a polynomial time algorithm thereby proving Theorem 1.Upon receiving ˆ cov , we first check whether ˆ cov ( X ) ≥ m or not, and if not that will be theseparating hyperplane. Henceforth, we assume this holds. Then, we run CGK Algorithm 2 with α = 8 and α = 2 to get instance T = (( L , L , Leaf , w ) , ( k , k )). Let { y v : v ∈ L ∪ L } bethe solution described in Proposition 1. Next, we check if ˆ cov i ( L i ) = y ( L i ) ≤ k i for both i ∈ { , } ;if not, by Proposition 1 that hyperplane would separate ˆ cov from P I cov (and even Robust -NU k C LP in fact). The algorithm then branches into two cases.
Case I: y ( L ) ≤ k − . In this case, we assert that T is valuable, and therefore by Lemma 1 weget an α + α = 10-approximate solution for I via Lemma 1, and we are done. Proposition 5. If y ( L ) ≤ k −
2, then there is an integral solution T for T with w ( C ( T )) ≥ m . Proof.
Since y ( L ) ≤ k −
2, we see that there is a feasible solution to the slightly revised LP below.max (cid:88) v ∈ L w ( v ) Y ( v ) : (cid:88) u ∈ L y u ≤ k − , (cid:88) u ∈ L y u ≤ k ,Y ( v ) := y p ( v ) + y v ≤ , ∀ v ∈ L Consider a basic feasible solution { y (cid:48) v : v ∈ L ∪ L } for this LP, and let T := { v ∈ L : y (cid:48) v > } .By definition y (cid:48) ( T ) = y (cid:48) ( L ) ≤ k −
2. According to Lemma 2, there are at most 2 loose variablesin y (cid:48) . So there are at most 2 fractional vertices in T . This implies | T | ≤ k . Let U be the set ofleaves that are not covered by T , that is, U := { v ∈ L : p ( v ) / ∈ T } . Let T be the top k membersof U according to decreasing w order. We return T = ( T , T ).We claim T has value at least m , that is, w ( C ( T )) ≥ m . Note that w ( C ( T )) = w ( T ) + (cid:80) u ∈ T w ( Leaf ( u )). By the greedy choice of T , w ( T ) ≥ (cid:80) v ∈ U w ( v ) y (cid:48) v . Since y (cid:48) p ( v ) = 0 forany v ∈ U , we have w ( T ) ≥ (cid:80) v ∈ U w ( v ) y (cid:48) v = (cid:80) v ∈ U w ( v ) Y (cid:48) ( v ). Furthermore, by definition,12 u ∈ T w ( Leaf ( u )) = (cid:80) v ∈ L \ U w ( v ) which in turn is at least (cid:80) v ∈ L \ U w ( v ) Y (cid:48) ( v ). Adding up provesthe claim as the objective value is at least m . w ( C ( T )) ≥ (cid:88) v ∈ U w ( v ) Y (cid:48) ( v ) + (cid:88) v ∈ L \ U w ( v ) Y (cid:48) v = (cid:88) v ∈ L w ( v ) Y (cid:48) ( v ) ≥ m . Case II, y ( L ) > k − . In this case, we either get an 8-approximation or prove that the followingis a valid inequality which will serve as the separating hyperplane (recall ˆ cov ( L ) = y ( L )). cov ( L ) ≤ k − . (2)To do so, we need the following proposition which formalizes the idea stated in Section 2.2 that incase II, we can enumerate over O ( | X | ) many well-separated instances . Proposition 6.
Let ( cov , cov ) ∈ P I cov be fractional coverages and suppose there is a subset Y ⊆ X with d ( u, v ) > r for all u, v ∈ Y . Then either cov ( Y ) ≤ k −
2, or at least one of thefollowing
Well-Separated Robust -NU k C instances are feasible I ∅ := (( X, d ) , (2 r , r ) , ( k , k ) , Y, m ) I q := (( X \ B ( q, r ) , d ) , (2 r , r ) , ( k − , k ) , Y, m − | B ( q, r ) | ) ∀ q ∈ X : d ( q, Y ) > r . Before proving the above proposition, let us use it to complete the proof of Theorem 1. Welet Y := L , and obtain the instances I ∅ and I q ’s as mentioned in the proposition. We applythe algorithm in Theorem 2 on each of them. If any of them returns a solution, then we have an8-approximation. More precisely, if I ∅ is feasible, Theorem 2 gives a 4-approximation for it whichis indeed an 8-approximation for I (the extra factor 2 is because I ∅ uses 2 r as its largest radius).If I q is feasible for some q ∈ X and Theorem 2 gives us a 4-approximate solution S (cid:48) = ( S (cid:48) , S (cid:48) ) forit and S = ( S (cid:48) ∪ { q } , S (cid:48) ) is an 8-approximation for I . If none of them are feasible, then we seethat cov ( L ) ≤ k − cov and P I cov . This endsthe proof of Theorem 1. Proof of Proposition 6.
Let us assume cov ( Y ) > k −
2, and prove that one of the proposed
Well-Separated Robust -NU k C instances are feasible. First of all, note that the described Well-SeparatedRobust -NU k C instances indeed satisfy the definition: Y is separated enough for radius 2 r andby definition of q , Y ⊆ ( X \ B ( q, r )).Suppose, for the sake of contradiction, none of the described Well-Separated Robust -NU k C instances are feasible. Since cov ( Y ) > k − cov ∈ P I cov , there has to be some S = ( S , S ) ∈ F such that S covers strictly more than k − Y . Take any such S . There are two typesof centers in S , the ones that do contribute to cov ( Y ), and the ones that do not. The former is A := { f ∈ S : d ( f, Y ) ≤ r } and the latter is B := { f ∈ S : d ( f, Y ) > r } . By our assumption of cov ( Y ) > k − Y points are more than 2 r apart, | A | > k −
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