Analysis

Let C_i^∗, S^∗, and O^∗ be the amount paid for cables of typei, for the core Steiner tree, and for opening facilities in the near-optimal solution, respectively. We defineCi to be the total cost paid for cables of typeiin the returned solution. LetDⁱ_j be the demand of nodejat stageiof the algorithm. LetT_i,P_i andN_i be cost incurred in the Aggregate-I step, the Aggregate-II step, and the both Consolidate steps of iteration i, respectively.

Also, let T_i^F and T_i^S denote the flow dependent and the setup cost components of the Aggregate-I step at iterationi. Analogously,P_i^F andP_i^S denote the flow dependent and

1The LBFL problem is a generalization of the facility location problem where each open facility is required to serve a certain minimum amount of demand (seeGuha et al.,2000;Svitkina,2010;Ahmadian and Swamy,2012).

Chapter 3. Connected Facility Location in Buy-at-Bulk Network Design 66

AlgDD– Approximation algorithm forDDConFL

1. Prune the set of cable types, as described in Theorem 3.6; letD₁ =D, and letDi be the set of demand points we have at the i-th stage;

guess a facilityr from the optimum solution.

2. For cable typei= 1,2, ..., K−1Do

- Aggregate I:Construct aρST-approximate Steiner treeTion demand pointsDi∪{r}

for edge costs σi per unit length. Install a cable of type i on each edge of this tree.

Root this tree at r. Send the demands from D_i upwards along the tree. Walking upwards along this tree, identify edges whose demand is larger than ^σ_rⁱ

i and cut the tree at these edges.

- Consolidate I: For every tree in the forest created in the preceding step, transfer the total demand at the root of the tree, which is at least ^σ_rⁱ

i, back to one of its sources using a shortest path of cable type i. Choose this source with a probability proportional to the demand at the source.

- Aggregate II: Solve the LBFL problem with (actual) demands D as clients, all nodes as facilities each having opening cost 0 and facility lower bound b_i, and edge costsr_i per unit length. The solution is a forest of shortest path trees. Then route the current demands (that is the demand aggregated on each node right after the Consolidate step) along these shortest path trees to their roots, by installing cables of typei.

- Consolidate II: For every tree in the forest created in the preceding step, transfer the total demand at the root of the tree, which is at leastbi, back to one of its sources with a probability proportional to the demand at that source using a shortest path with cables of typei. LetD_i+1 be the resulting set of demand locations.

3. For cables typeK Do

- Construct a ρ_ST-approximate Steiner tree T_K on demand points D_K∪ {r} for edge costsσ_K per unit length. Install cables of typeK on each edge of this tree. Root this tree at r. Send the demands from DK upwards along the tree. Walking along this tree, identify edges whose demand is larger than ^σ_r^K

K and cut the tree at these edges.

For every tree in the created forest, send the total demand in the root of tree back to one of its sources with a probability proportional to the demand at that source via a shortest path using cables of typeK.

- Solve the LBFL problem with (actual) clients D, facility set F, opening costs µi, facility lower bound b_K, and edge costs r_K per unit length. We obtain a forest of shortest path trees. Then route thecurrent demands along these trees to their roots, installing cables of typeK. Let F⁰ be the set of open facilities.

4. Compute aρ_ST-approximate Steiner treeT_core on nodesF⁰∪ {r}for edge costsM per unit length. Install the core cable on the edges ofTcore.

Chapter 3. Connected Facility Location in Buy-at-Bulk Network Design 67 the setup costs incurred in the Aggregate-II step. Recall that the set of discount cable types has been reduced depending on the constant parameter α∈(0,¹₂).

The following Lemma carries over from the uSSBB problem studied by Guha et al.

(2009) to our problem in a straightforward way.

Lemma 3.9.

(i) At the end of each consolidation step every node has E[D_jⁱ] =dj. (ii) E[Ni]≤Ti+Pi for each i.

(iii) P_i^S≤P_i^F and T_i^F ≤T_i^S for each i.

The following lemma bounds the setup costs of the cables installed in the Aggregate-I step of phase iof our algorithm.

Lemma 3.10. E[T_i^S]≤ρST Pi−1 j=1

1

β(2α)^i−jC_j^∗+PK

j=iα^j−iC_j^∗+¹₂α^K−iS^∗

Proof. We prove the claim by constructing a Steiner tree of cables of type i on nodes Di∪ {r} whose expected cost is at mostPi−1

j=1 1

β(2α)^i−jC_j^∗+PK

j=iα^j−iC_j^∗+¹₂α^K−iS^∗. Consider the near-optimum solution. We find an unsplittable flow sending the demand aggregated at points Di to facility r via this network solution. Then, out of edges of the near-optimum solution, we keep only those edges carrying positive flow and replace their (access and/or core) cables with access cables of type i. This obviously results a tree of cablesi on nodes Di∪ {r}. In the following we bound the expected cost of this network.

Note that, being in stage i, the expected demand of each point in Di is at least βbi. Hence the expected flow on each cablej < iis at leastβb_i as well. Therefore, by Lemma 3.7, the expected cost of all replacement cables for cables of type j < i is bounded by

1

β(2α)^i−jC_j^∗. Furthermore, the expected cost of the replacement cables for the cables j > i can be easily bounded byα^j−iC_j^∗, using the setup costs scale (see Theorem3.6).

Finally, with a similar argument, the cost of replacement for the core cables is bounded by ¹₂α^K−iS^∗; recall that 2σ_K < M.

Altogether, the expected setup cost of this Steiner tree using cables iis bounded by

i−1

X

j=1

1

β(2α)^i−jC_j^∗+

K

X

j=i

α^j−iC_j^∗+1

2α^K−iS^∗

As we use a ρST-approximation algorithm to solve this Steiner tree problem in our

algorithm, the claim follows.

Chapter 3. Connected Facility Location in Buy-at-Bulk Network Design 68

In the following lemma, we bound the flow dependent costs of the cables installed in the Aggregate-II step of phaseiof our algorithm.

Lemma 3.11. E[P_i^F]≤ν·ρF LPi

j=1α^i−j·C_j^∗

Proof. To prove this, similar to that of Lemma 3.10, we construct a feasible solution to the LBFL problem solved in this stage by making use of the near-optimum solution.

For this end, we only consider the forest defined by the edges with cable types 1 toiin the near-optimum solution and replace all cables of type less than iby cables of typei.

This results a feasible solution for this LBFL problem as the root node of each tree of the resulting forest has incoming demand of at leastb_i (recall property (ii) of Theorem 3.8). The flow dependent cost of the new solution on cables replaced by cables of type j < i is bounded by α^i−j ·C_j^∗ using the flow dependent costs scale. Hence, the cost of the resulting solution is at most Pi

j=1α^i−jC_j^∗. As our algorithm applies a bicriteria ν·ρ_{F L}-approximation algorithm to solve the lower bounded facility location problem in

this stage, the claim follows.

The opening costs and the flow dependent shortest path costs in the K-th stage of our algorithm can be bounded as follows.

Lemma 3.12. E[P_K^F +µ(F⁰)]≤ν·ρF L(PK

i=1α^K−i·C_i^∗+O^∗)

Proof. The proof is similar to the one for the previous lemma. We consider the access network of the near-optimum solution as a whole and replace all access cables (of type less thanK) by cables of typeK. This, combined with facilities opened in the near-optimum solution results a solution for this stage whose cost is at most PK

i=1α^K−iC_i^∗+O^∗. Let S be the cost of the core Steiner tree returned by our algorithm. This can be bounded as follows.

Lemma 3.13. E[S]≤ρST S^∗+_β²PK

j=1(C_j^∗+Cj)

Proof. Let F^∗,T_core^∗ and T_access^∗ be the set of open facilities, the tree connecting them, and the forest connecting clients to open facilities in the near-optimum solution, re-spectively. Let T_access be the forest connecting clients to open facilities in the solution returned by our algorithm. We prove the claim by constructing a feasible Steiner tree of core cables onF⁰∪ {r}, whose expected cost is at mostS^∗+_β²PK

j=1(C_j^∗+C_j).

Consider the near-optimum solution combined with the solution returned by the algo-rithm. In the algorithm’s solution, each facility l ∈ F⁰ serves at least a total demand

Chapter 3. Connected Facility Location in Buy-at-Bulk Network Design 69 of βbK. This demand, on the other hand, is served by the set of facilities in the near-optimum solution. Therefore, at least βb_K flow can be routed (possibly along more than one path) between each facility l ∈ F⁰ and the set of facilities in F^∗ using the access cables on edges of T_access^∗ ∪T_access (without violating the cables’ capacities). By contracting the facilitiesF^∗ into a single node ˜f, this guarantees that the access cables on edges of T_access^∗ ∪Taccess allow at least βbK flow crossing each cut that contains at least one facility inF⁰ and not containing ˜f. Using the fact that it is cheaper to use core cables rather than access cables when the demand is at least bK, we can deduce that the cost of a feasible (possibly fractional) solution to the core Steiner tree problem over F⁰∪ {f˜}is at most _β¹PK

j=1(C_j^∗+Cj). Since the integrality gap of the cut formulation of Steiner tree is at most 2, one can then use any (LP-rounding) approximation algorithm to construct a core Steiner tree onF⁰∪ {f}˜ with cost at most _β²PK

j=1(C_j^∗+C_j).

Finally, we uncontract node ˜f and augment the resulting forest by adding edges inT_core^∗ to get a core Steiner tree on F⁰∪ {r}. This completes the proof.

Together, Lemmas3.9–3.13 imply our main result.

Proof of Theorem 3.1.

Proof. By Lemmas 3.9–3.12, the total expected cost of access cables is bounded as follows:

K

X

i=1

Ci≤4

K

X

i=1

ν·ρF L i

X

j=1

α^i−jC_j^∗+ρST

X^K

j=i

α^j−iC_j^∗+

i−1

X

j=1

1

β(2α)^i−jC_j^∗+ 1

2α^K−iS^∗

≤4

ν·ρ_{F L}+ρ_ST

1−α + ρ_ST β(1−2α)

X^K

i=1

C_i^∗+ 2·ρ_ST 1−α S^∗

(3.1) Additionally, using Lemmas 3.12 and 3.13, the total cost of installing core cables and opening facilities is bounded by

ν·ρF L·O^∗+ρST·S^∗+ 2ρST

β X^K

i=1

C_i^∗+

K

X

i=1

Ci

.

Altogether, using Inequality (3.1) twice, we obtain a bound of

ν·ρ_{F L}·O^∗+h2ρ_ST

β + 4 1 +2ρ_ST β

ν·ρ_{F L}+ρ_ST

1−α + ρ_ST β(1−2α)

iX^K

i=1

C_i^∗ +h

1 +2ρ_ST β

2ρ_ST 1−α

+ρ_STi S^∗

Chapter 3. Connected Facility Location in Buy-at-Bulk Network Design 70 for the worst case ratio between the algorithm’s solution and a near optimal solution of DDConFL, according to Theorem 3.8.

With Theorems3.6and3.8, this yields a worst case approximation guarantee of _α¹(_α⁴+1) times the above ratio against an unrestricted optimal solution of the DDConFL problem.

Hence, the approximation guarantee of our algorithm can be bounded by 4 +α

α² ·max

ν.ρ_{F L},2ρ_ST

β + 4 1 +2ρ_ST β

ν·ρ_{F L}+ρ_ST

1−α + ρ_ST β(1−2α)

, ρ_ST 2

1−α+ 4ρ_ST

β(1−α) + 1

SettingρST =ln(4) due toByrka et al.(2010),ρF L = 1.488 due toLi(2013),αto some value in (0,¹₂), andβ to some value in (0,1) (and therebyν to its corresponding value) we obtain the first constant factor approximation algorithm for DDConFL.

Theorem3.1 together with Remark3.5implies Corollary 3.2.

One can observe that the approximation ratios obtained by this algorithm (even after setting parametersα and β to their best values) are larger than one thousand.

In the next section, we revise our general algorithmic approach to improve the factors.

Incorporating and exploiting random sampling techniques, we develop a new algorithm with an improved approximation factor for BBConFL, which, carries over to an improved algorithm for DDConFL as well.

Im Dokument Network design with facility location (Seite 67-72)