TRANSFORMACIÓN DEL CASERÍO: DEL SIGLO XVI HASTA EL SIGLO

We start with an LP relaxation for ASYM-k-CENTERproblem by considering an unweighted directed graph on node set V . Specifically, for a parameter R_{≥ 0, we define the directed graph GR}= (V, E_R),

where E_R= {(u, v) : u, v ∈ V, d(u, v) ≤ R}. For a node v, letNbr−[v] = {u : (u, v) ∈ ER}S{v} denote

the in-neighbors, andNbr+[v] = {u : (v, u) ∈ ER}S{v} be the out-neighbors (including itself). Observe,

for any R_{≥ R}∗_d, there exists a set of k centers S_{⊆ V , such that S covers V in GR}, i.e.S

c∈SNbr+[c] = V .

Thus, given a parameter R, we can define the following LP relaxation on graph G_R. We use y_v as an indicator variable for open centers, and x_uv to denote if v is assigned to u.

(asym-kc-LP) X u∈V y_u≤ k xuv≤ yu ∀v ∈ V, u ∈ V X u∈Nbr−[v] x_{uv ≥ 1 ∀v ∈ V} y_v, x_{uv≥ 0}

Figure 4.3: LP relaxation for ASYM-k-CENTER

For ASYM-k-CENTER, Archer[13] showed that the integrality gap is atmost O(log∗k), Infact it is tight

within a constant factor[47]. The main result of the section is captured by the following theorem.

Theorem 4.3. LetI= (V, d, k) be a 2-perturbation resilient instance of ASYM-k-CENTERand let R∗_d be the cost of the optimal solution. Then, for any R< R∗_d,asym-kc-LPis infeasible.

4.4.1

Properties of 2-perturbation resilient A

-k-

instance

InSection 4.3we showed that the clusters in an optimal solution to a 2-PR k-center instance have a strong separation property: d(p, q) > R∗_d if p, q are in different clusters. For ASYM-k-CENTER the asymmetry in the distances does not permit a such a strong and simple separation property. However, we can show slightly weaker properties: (1) every optimal center is separated from any point in a different cluster by at least R∗_d; (2) points in a cluster which are far off from core points (these points have small distance "to" correponding cluster centers) in the cluster, are well-separated from core points of other clusters as well (SeeFigure 4.4a,Figure 4.4b). These properties suffice to prove our desired theorem. The rest of the section is dedicated to proving these properties.

Lemma 4.8. Consider a 2-perturbation resilient ASYM-k-CENTERinstanceI= (V, d, k). LetC= {C1, . . . Ck} be the unique optimal clustering, induced by a set of centers S= {c₁, . . . , c_{k}. Let the optimal radius be R}∗_d. Consider any center ci. Then for any point q in a different cluster Cj (i6= j), we have d(q, ci) > R∗_d.

Proof: Assume for the sake of contradiction, that the claim is false. That is, there exists a point q∈ Cj, such that d(q, ci) ≤ R∗_d. We construct a distance function d0, which is a metric 2-perturbation of d. And

> R∗ d > R∗ d > R∗ d ≤ R∗ d ≤ R∗ d ≤ R∗ d Ci Cj C` ci cj c` > R∗ d q

(a) Cluster centers are atleast R∗

daway from points in different

cluster C` ≤ R∗ d ≤ R∗ d ≤ R∗ d Ci Cj ci cj c` q ≤ R∗ d ≤ R∗ d ≤ R∗ d ≥ R∗ d > R∗ d p w > R∗ d (b) Cluster points R∗

daway from cluster core points, are R∗d

away from different cluster core points

Figure 4.4: Properties of a 2-perturbation resilient ASYM-k-CENTERinstance

show that in the instance thus constructed, the optimal clustering is not unique, which contradicts the definition of perturbation resilience.

We define d0as follows: consider the complete directed graph G on vertices V . Let E0= {(q, v) : v ∈ C_i}. The edge lengths in graph G are given by the function`, where for any edge (u, v),

`(u, v) =

  

mind(u, v), R∗_d (u, v) ∈ E0

d(u, v) otherwise

For any pair of points u, v, the distance d0(u, v) is the shortest path distance between u and v in graph G, using`.

Observation 4.3. d0is a metric2-perturbation of d.

Proof: For any v_{∈ Ci}, by triangle inequality, d(q, v) ≤ d(q, ci) + d(ci, v) ≤ 2 · R∗_d. Therefore, for any (u, v) ∈ E0_{,`(u, v) = min}

d(u, v), R∗_d ≥ d(u,v)₂ . For any other(u, v), by definition `(u, v) = d(u, v). That is, for any edge(u, v) we have, d(u,v)<sub>2</sub> ≤ `(u, v) ≤ d(u, v). The shortest path distance function d0, defined on such a graph G, can be easily shown is a metric 2-perturbation of d (details are chalked out in proof Lemma 4.2).

Consider the instance I0= (V, d0, k). Since,I0 is a 2-perturbed instance, the optimal clustering is given byC= {C1, . . . , Ck}. Let S0=



c₁0, . . . , c0_k be the optimal set of centers. Further, R∗_d0 denotes the cost of optimal solution. And we can show, R∗_d0 = R∗_d (follows fromLemma 4.3). Now there are two cases to consider.

• q 6= c0j. Consider the set of centers S00 = S0\ 

c_i0 S{q}. LetC00 be a Voronoi partition induced by

d0(q, u) ≤ R∗_d = R∗_d0. That is, for any point u∈ V , d0(S00_{, u) ≤ R}∗

d0. Thereforecostd0(C00, S00) ≤ R∗_d0. Now, in clusteringC00, the points q and c0_j are in different clusters, which is not true forC. Thus the optimal clustering is not unique, and this leads to contradiction.

• q = c0j. Consider the set of k− 1 centers S00= S0\ 

c0_i . LetC00 be a Voronoi partition induced by

S00. As in the previous case, we can show d(S00, u) ≤ R∗_d0, for any u∈ V , implyingcost_d0(C00, S00) ≤ R∗_d0. Therefore, we have a k_{− 1 clustering of}I0with cost at most the optimal cost. Then, byLemma 4.1, the optimal clustering ofI0is not unique. This contradicts the definition of perturbation resilience.

This completes the proof.

The next lemma formalizes the notion of core points and the property they enjoy.

Lemma 4.9. Consider a 2-perturbation resilient ASYM-k-CENTERinstanceI= (V, d, k). LetC= {C1, . . . Ck} be the unique optimal clustering induced by a set of centers S= {c1, . . . , ck}. Let the optimal radius is R∗_d. Suppose p_{∈ Ci} and q_{∈ Cj} where i_{6= j and d(p, ci}) ≤ R_d∗ and d(q, cj) ≤ R∗_d. Then for any w_{∈ Ci} such that d(p, w) ≥ R∗_d we have d(q, w) > R∗_d.

Proof: Consider a triplet of points p, w∈ Ci and q∈ Cj, where d(p, ci), d(q, cj) ≤ R∗d, and d(p, w) ≥ R∗d. Assume for the sake of contradiction, d(q, w) ≤ R∗_d. We construct a distance function d0, which is a metric 2-perturbation of d. Next we show that in the ASYM-k-CENTERinstance constructed using d0, the optimal clustering is not unique, which contradicts the definition of perturbation resilience.

We define d0as: consider the complete directed graph G on vertices V . Let E0= {(p, v) : v ∈ Ci} ∪ (q, v) : v ∈ Cj . The edge lengths in graph G are given by the function `, where for any edge (u, v),

`(u, v) =

  

mind(u, v), R∗_d (u, v) ∈ E0

d(u, v) otherwise

For any pair of points u, v, the distance d0(u, v) is the shortest path distance between u and v in graph G, using`.

Observation 4.4. d0is a metric2-perturbation of d.

Proof: For any v_{∈ Ci}, by triangle inequality, d(p, v) ≤ d(p, ci)+d(ci, v) ≤ 2·R∗_d. Similarly, for any v_{∈ Cj}, by triangle inequality, d(q, v) ≤ 2 · R∗_d Therefore, for any(u, v) ∈ E0,`(u, v) = mind(u, v), R∗<sub>d</sub> ≥ d(u,v)<sub>2</sub> . For any other(u, v), by definition `(u, v) = d(u, v). As we previously stated (inLemma 4.2), d0defined on graph G, satisfying the property d(u,v)₂ ≤ `(u, v) ≤ d(u, v) is a metric 2-perturbation of d.

Observation 4.5. For any point v_{∈ Ci}, we have d0(p, v) ≤ R∗_d. Similarly for any v_{∈ Cj}, d0(q, v) ≤ R∗_d. In particular, for point w, d0(p, w) = R∗_d and d0(q, w) ≤ R∗_d.

Proof: For any point v ∈ Ci, by definition,`(p, v) ≤ R∗<sub>d</sub>. Since, d0(p, v) ≤ `(p, v) the claim follows. Similarly, for any v_{∈ Cj}, it is easy to see d0(q, v) ≤ R∗_d.

Now, consider point w. Let P be any directed path from p   w in graph G, excluding the single edge path(p, w). If P includes an edge from E0, by triangle inequality,`(P) = P<sub>e∈P</sub>`(e) = P_e∈Pd(e) ≥ d(p, w) ≥ R∗_d. The last inequality follows from our choice of p, w at the outset. Otherwise, if P includes atleast one edge from E0, `(P0) = P<sub>e∈P</sub>0T E0`(e) + Pe∈P0_\E0`(e) ≥ Pe∈P0T E0min



d(e), R∗_d +

P

e∈P0_\E0d(e) ≥ min 

d(p, w), R∗_d ≥ R∗_d. Finally, by our definition`(p, w) = R∗_d. Therefore, d0(p, w) = R∗_d. The final observation, d0(q, w) ≤ R∗_d, follows from our assumption d(q, w) ≤ R∗_d.

Consider the instance I0= (V, d0, k). Since,I0 is a 2-perturbed instance, the optimal clustering is given byC= {C₁, . . . , C_{k}. Let S}0=c₁0, . . . , c0_k be the optimal set of centers. Further, R∗_d0 denotes the cost of optimal solution. We can show, R∗_d0= R∗_d.

Consider the set of centers S00=€S0\¦c0_i, c0_j©Š S_{{p, q}. Let}C00be a Voronoi partition induced by

S00. For any point u<sub>∈ C`</sub>, where` 6= i, j, d0(S00, u) ≤ d0(c_`0, u) ≤ R∗_d0. For any point u_{∈ Ci}, d0(S00, u) ≤

d0(p, u) ≤ R∗_d= R∗_d0. Similarly, for any point u_{∈ Cj}, d0(S00, u) ≤ d0(q, u) ≤ R∗_d= R∗_d0 That is, for any point

u∈ V , d0(S00, u) ≤ R∗_d0. Thereforecost_d0(C00, S00) ≤ R∗_d0. Recall byObservation 4.5, d0(p, w) = R∗_d, while d0(q, w) ≤ R∗_d. Therefore, we can assume without loss of generality, inC00, w and p are not in same cluster. This however is not true forC, implyingC00₆₌C. Thus the optimal clustering ofI0is not unique, and this contradicts the definition of perturbation resilience.

4.4.2

Proof ofTheorem 4.3

Let C₁, . . . C_k be the unique optimal clustering of instance I = (V, d, k), with optimal radius R∗_d. Consider an arbitrary R< R∗_d, and let G_Rdenote the corresponding threshold graph. Recall, graph G_Ris a directed graph defined over vertex set V , and the edge set E_R= {(u, v) : d(u, v) ≤ R}. Suppose, the ASYM-k-CENTERLP (asym-kc-LP) defined over graph G_R is feasible, and(x, y) is a feasible fractional solution.

FromLemma 4.8, in a 2-PR instance, we have the following: if q6∈ Ci then d(q, c_i) > R∗_d> R. This implies that, in the graph G_R, for any c_i, i_{∈ [k],}Nbr−[ci] ⊆ Ci. Let C_i0=u_∈Nbr−[ci] : yu> 0 . Since

(x, y) is a feasible solution, we must have P_u∈C0

i yu≥ 1. Since,

P

v∈V yv≤ k, and the clusters C1, . . . , Ck are disjoint, we haveP_u∈C

i yu= Pu∈Ci0yu= 1, for all i ∈ [k].

From the definition of R∗_dthere must be a cluster C_tsuch that min_c∈C

tmaxv∈Ctd(c, v) = R ∗

d. Consider its center c_tand let p∈ C_t0. Clearly d(p, ct) ≤ R < R∗d. Furthermore, since Ctis the largest radius cluster, there exists w_{∈ Ct}, such that d(p, w) ≥ R∗_d. Therefore in graph G_R, p/∈Nbr−[w]. For any other cluster Cj, byLemma 4.9, for any point q_{∈ C}0_j, we have d(q, w) > R∗_d. That is,Nbr−[w] T C0_j= ;, for any j 6= t. This

implies w can be covered only by points that belong to C_t0. ThereforeP_u∈Nbr−_[w]xuw≤ P

u∈C0t−pyu< 1