Operations Research Letters 35 (2007) 707–712

OperationsResearchLetters

www.elsevier.com/locate/orl

A simpler and better derandomization of an approximationalgorithm for single source rent-or-buy

David P. Williamson, Anke van Zuylen∗

School of Operations Research and Industrial Engineering, 206 Rhodes Hall, Cornell University, Ithaca, NY 14853, USA

Received 12 December 2006; accepted 14 February 2007Available online 27 February 2007

Abstract

We present a very simple way of derandomizing the algorithm proposed by Gupta, Kumar and Roughgarden for singlesource rent-or-buy by using the method of conditional expectation. Using the improved analysis of Eisenbrand, Grandoniand Rothvoß, our derandomized algorithm has an approximation guarantee of 3.28.© 2007 Elsevier B.V. All rights reserved.

Keywords: Approximation algorithms; Derandomization; Network design; Rent-or-buy

1. IntroductionIn the multicommodity rent-or-buy problem, we are

given an undirected graph G=(V , E) where each edgee ∈ E has a nonnegative cost ce, a set of terminalpairs {(s1, t1), . . . , (sk, tk)} ⊆ V × V , and a parame-ter M > 1. A solution is a set of paths, one for eachterminal pair. For each edge in the solution, we caneither choose to rent the edge, incurring a cost ce foreach path that uses the edge e, or we can buy the edge,incurring a cost Mce. The special case when si = s

for every pair i = 1, . . . , k is called the single sourcerent-or-buy problem.

A related problem is the connected facility locationproblem. Here we are again given an undirected graph

∗ Corresponding author.E-mail address: avz2@cornell.edu (A. van Zuylen).

0167-6377/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.orl.2007.02.005

G = (V , E) with nonnegative edge costs ce, and aparameter M > 1, and we are given a set of demandpoints D ⊆ V . A solution is a set of facilities F ⊆ V ,a connected subgraph T of G that spans F , and anassignment of demand points to open facilities. Thecost of a solution is M times the cost of the edgesin T , plus for each demand point j ∈ D, the cost ofthe shortest path d(i, j) from j to the facility i ∈ F

that j is assigned to. Note that we can assume withoutloss of generality that T is a Steiner tree on F , andthat a “root” r ∈ V is given, such that r ∈ F mustbe satisfied for any solution. Single source rent-or-buycan then be formulated as a connected facility locationproblem with root r = s and D = {t1, . . . , tk}, since itis not hard to show that the edges bought in a solutionto single source rent-or-buy form a tree rooted at thesource.

708 D.P. Williamson, A. van Zuylen / Operations Research Letters 35 (2007) 707–712

Gupta et al. [8] gave the following approximationalgorithm SimpleCFL for connected facility locationand single source rent-or-buy, with q = 1:

1. Mark each demand j ∈ D independently withprobability q/M , and let D be the random set ofmarked demands.

2. Construct an �-approximate Steiner tree on D∪{r}and buy the edges of this tree.

3. Assign each demand to its closest facility in D∪{r}.

They show that the expected cost of this solution forq=1 is at most (2+�) times the cost of the optimal so-lution. Using the current best approximation algorithm[10] for the Steiner tree problem, SimpleCFL is an ex-pected 3.55-approximation algorithm. Eisenbrand etal. [2] recently improved the analysis of this algorithmand showed that the expected cost of the solution is atmost (2 + �q) times the cost of the optimal solution,if we choose q = 1

2 − 1/� + √�2 − 0.772� + 4/(2�).

Using the 1.55-algorithm from [10] in Step 2, thisimplies that for q = 0.591 we get an expected 2.92-approximation algorithm.

A similar type of algorithm was proposed in [6,7]for multicommodity rent-or-buy. In their boostedsampling algorithm for multicommodity rent-or-buy,we mark each terminal pair with probability 1/M ,and buy a Steiner forest F on the marked pairs. Wethen connect each remaining terminal pair (si, ti)

by renting the edges on the shortest si − ti pathin the graph G with the edges of F contracted. Itwas shown in [4] that using a primal–dual algorithm[1,5] to find F, this is an expected 5-approximationalgorithm.

It was shown in [9] that a variant of the SimpleCFLalgorithm for single source rent-or-buy yields approx-imately budget-balanced and group strategy proof costshares, and that it is possible to derandomize this al-gorithm and obtain a deterministic 4.2-approximationalgorithm for connected facility location and singlesource rent-or-buy. No derandomization of the boostedsampling algorithm for multicommodity rent-or-buyis known.

We will show that there is a very simple way of de-randomizing SimpleCFL by using the method of con-ditional expectation [3], with approximations of theconditional expected costs, instead of exact expres-sions. Using a 2-approximation algorithm in Step 2,

we obtain a deterministic 4-approximation algorithmfor q = 1 via the original analysis from [8]. The im-proved analysis from [2] shows that for q =0.636, ourderandomized algorithm has an approximation guar-antee of 3.28. Both of these results improve on the pre-viously best known deterministic approximation guar-antee of 4.2.

It remains an open question whether it is also pos-sible to use the ideas from this paper to derandomizethe boosted sampling algorithm for multicommodityrent-or-buy.

2. Derandomization of SimpleCFL

We start by repeating the lemmas from [8,2] fromwhich the expected performance guarantee of Sim-pleCFL follows. We also repeat the proof of Lemma2.1, because this will turn out to be helpful in explain-ing our approximation of the expected buying cost in-curred by the algorithm.

In the following, we denote by BOPT and ROPT

the buying and renting cost of an optimal solution tothe connected facility location problem.

Lemma 2.1 (Gupta et al. [8], Eisenbrand et al.[2]). The expected cost of Step 2 of SimpleCFL is atmost �[BOPT + qROPT ].

Proof. We will exhibit a Steiner tree T on D ∪ {r}for each realization of D, and we will showthat the expected cost of T is at most BOPT +qROPT . It then follows that the expected cost ofa minimum-cost Steiner tree is at most BOPT +qROPT .

Let F ∗ be the set of open facilities in an optimalsolution, and let T ∗ be the Steiner tree on F ∗ thatis bought in the optimal solution. For every j ∈ D,let i∗(j) ∈ F ∗ be the facility that j is assigned toin the optimal solution. We define the Steiner tree T

on D ∪ {r} as the union of the edges of T ∗ and theedges on the shortest j − i∗(j) path for each j ∈ D.Since each j ∈ D is included in D with probabilityq/M , we buy the edges on the shortest j − i∗(j) pathwith probability q/M , so the expected cost of T isat most

∑e∈T ∗Mce + ∑

j∈D(q/M)Md(i∗(j), j) =∑e∈T ∗Mce + q

∑j∈Dd(i∗(j), j) = BOPT +

qROPT . �

D.P. Williamson, A. van Zuylen / Operations Research Letters 35 (2007) 707–712 709

Lemma 2.2 (Gupta et al. [8]). For q=1, the expectedcost of Step 3 of SimpleCFL is at most 2(BOPT +ROPT ).

Lemma 2.3 (Eisenbrand et al. [2]). The expectedcost of Step 3 of SimpleCFL is at most (0.807/q)BOPT

+ 2ROPT .

Corollary 2.1 (Gupta et al. [8]). Using q = 1, andusing the 1.55-approximation algorithm from [10] inStep 2, SimpleCFL is an expected 3.55-approximationalgorithm.

Corollary 2.2 (Eisenbrand et al. [2]). Using q =0.591, and using the 1.55-approximation algorithmfrom [10] in Step 2, SimpleCFL is an expected 2.92-approximation algorithm.

In theory, the randomized algorithm SimpleCFL canbe derandomized by using the method of conditionalexpectation [3] as follows: take an arbitrary vertex j

in D. With probability q/M , this vertex will be in-cluded in D, and with probability 1 − q/M it is notincluded in D. Therefore, the expected cost incurredby SimpleCFL is equal to q/M times the expectedcost of SimpleCFL, conditioned on the fact that j ∈D plus 1 − q/M times the expected costs of Sim-pleCFL conditioned on j /∈ D. So it must be true thatone of the two conditional expected cost is not morethan the expected cost of SimpleCFL, and we choosewhether or not we will include j in D to minimizethe expected cost conditioned on this choice. We canrepeat this process until we have decided for all ver-tices in D whether or not to include the vertex in D.Since each step does not increase the expected cost ofthe algorithm, we find a deterministic set D such thatexecuting Steps 2 and 3 does not cost more than theexpected cost of SimpleCFL.

The problem with this approach for this particularproblem is that we cannot compute the (conditional)expected cost of SimpleCFL exactly. However, we cancompute the conditional expected assignment cost ofSimpleCFL, since for each j ∈ D the expected assign-ment cost is just the distance to the closest vertex thatis included in the random set D ∪ {r} (see Claim 2.1).

Furthermore, we will now show that there exists aneasily computable value cST (D∪{r}) that is not morethan twice the cost of the Steiner tree on D∪{r} found

by the primal–dual algorithm [1,5]. We will show inthe following lemma that given sets A, A ⊂ D, suchthat we have already decided to open facilities in A (soA ⊆ D), and not to open facilities in A (so A∩D=∅),that we can compute the conditional expectation ofcST (D ∪ {r}).

Lemma 2.4. For any subsets A, A ⊆ D, we can com-pute E[cST (D∪{r})|A ⊆ D, A∩D=∅] in polynomialtime such that

(i) E[cST (D ∪ {r})]�BOPT + qROPT ;(ii) The expected cost in Step 2 of SimpleCFL, condi-

tioned on the fact that A ⊆ D, A ∩ D = ∅, whenusing a primal–dual algorithm to find a Steinertree on D ∪ {r}, is at most 2E[cST (D ∪ {r})|A ⊆D, A ∩ D = ∅].

Proof. We use an idea similar to the one in the proofof Lemma 2.1: in Lemma 2.1 we used an optimalsolution to the connected facility location problem toexhibit a Steiner tree on D∪{r} that costs no more thanBOPT +qROPT in expectation. Here, we will use theoptimal solution to an LP relaxation of the connectedfacility location problem to demonstrate a fractionalSteiner tree on D ∪ {r}. The expected cost of thisfractional Steiner tree will be at most BOPT +qROPT ,and using a primal–dual algorithm we can compute aSteiner tree that costs no more than twice the cost ofthe fractional Steiner tree.

For a given set D, the cost of buying the Steiner treefound by the primal-dual algorithm on D ∪ {r} is atmost twice the objective value of any feasible solutionto the following LP [1,5]:

min∑e∈E

Mceye

(Buy(D)) s.t.∑

e∈�(S)

ye �1 for all S : D ∩ S = ∅, r /∈ S

ye �0.

We will define a feasible solution to Buy(D) for ev-ery possible realization of D with the help of the fol-lowing linear programming relaxation for connectedfacility location [11]. Let xij be a 0–1 variable that in-dicates whether demand vertex j is assigned to facilityi, and let ze = 1 if edge e is bought, and 0 otherwise.The first set of constraints ensures that every demandis assigned to a facility. For any subset S ⊂ V that

710 D.P. Williamson, A. van Zuylen / Operations Research Letters 35 (2007) 707–712

does not contain the root r , such that j is assigned tosome facility in S, any feasible solution must buy anedge that leaves S, which is enforced by the secondset of constraints.

min B + qR

s.t.∑i∈V

xij = 1 ∀j ∈ D,

(CFL)∑

e∈�(S)

ze �∑i∈S

xij ∀j∈D, S⊂V : r /∈S,

B = ∑e∈E

Mceze,

R = ∑i∈V

∑j∈D

d(i, j)xij ,

xij , ze, B, R�0.

Let (x∗, z∗, B∗, R∗) be an optimal solution to CFL.Note that B∗+qR∗ �BOPT +qROPT , because the op-timal solution to the connected facility location prob-lem gives a feasible solution to CFL with objectivevalue BOPT + qROPT .

Let �ije = 1 if e is on the shortest path from demand

j to node i (note that �ije is a constant, not a decision

variable). For a given set D let

ye(D) = z∗e +

∑j∈D

∑i∈V

�ije x∗

ij ∀e ∈ E.

We define cST (D ∪ {r}) to be the objective value of{ye(D)}e∈E for (Buy(D)), i.e.

∑e∈EMceye(D). Note

that after solving (CFL) and getting (x∗, z∗) we cancompute E[cST (D ∪ {r})|A ⊆ D, A ∩ D = ∅] as

E[cST (D ∪ {r})|A ⊆ D, A ∩ D = ∅]

=∑e∈E

Mce

⎛⎝z∗

e +∑j∈A

∑i∈V

�ije x∗

ij

+∑

j∈D\(A∪A)

∑i∈V

q

M�ije x∗

ij

⎞⎠ .

Then

E[cST (D∪{r})]=∑e∈E

Mce

⎛⎝z∗

e+∑j∈D

∑i∈V

q

M�ije x∗

ij

⎞⎠

=∑e∈E

Mcez∗e+q

∑j∈D

∑i∈V

(∑e∈E

�ije ce

)x∗ij

=∑e∈E

Mcez∗e + q

∑j∈D

∑i∈V

d(i, j)x∗ij

= B∗ + qR∗ �BOPT + qROPT .

It remains to show that {ye(D)}e∈E is a feasible solu-tion to Buy(D): Consider a set S such that r /∈ S, andthere exists j ′ ∈ D ∩ S. Then∑e∈�(S)

∑i∈V

�ij ′e x∗

ij ′ �∑

e∈�(S)

∑i /∈S

�ij ′e x∗

ij ′

=∑i /∈S

x∗ij ′

∑e∈�(S)

�ij ′e

�∑i /∈S

x∗ij ′ .

The last inequality follows from the fact that ifi /∈ S, j ′ ∈ S then the shortest i − j ′ path must go

from S to V \S, so∑

e∈�(S)�ij ′e �1. The feasibility of

(x∗, z∗) to (CFL) gives that∑e∈�(S)

z∗e �

∑i∈S

x∗ij ′ .

Therefore

∑e∈�(S)

ye(D) =∑

e∈�(S)

⎛⎝z∗

e +∑j∈D

∑i∈V

�ije x∗

ij

⎞⎠

�∑

e∈�(S)

(z∗e +

∑i∈V

�ij ′e x∗

ij ′

)

�∑i∈S

x∗ij ′ +

∑i /∈S

x∗ij ′ = 1.

So {ye(D)}e∈E is a feasible solution to Buy(D). �

Theorem 2.1. SimpleCFL can be derandomized toget a deterministic 4-approximation algorithm forconnected facility location using q = 1, and a 3.28-approximation algorithm using q = 0.636.

Proof. We show that for q �0.636, we can deran-domize SimpleCFL and get a (2 + 2q)-approximationalgorithm.

We begin by noting that, even though we are allowedto open a facility in any vertex in V , SimpleCFL onlyrandomly chooses vertices from D to open facilities. Inthe derandomization, we will therefore also only con-sider the vertices in D as possible sites for facilities.Suppose the vertices in D are numbered v1, . . . , v|D|.We will iterate through the vertices in D, and in iter-ation � we decide whether or not to open a facility inv�. We let A ⊆ D be the set of vertices for which we

D.P. Williamson, A. van Zuylen / Operations Research Letters 35 (2007) 707–712 711

have already decided to open a facility, and A ⊆ D isthe set of vertices for which we have already decidednot to open a facility. Initially, A, A = ∅.

Let E[cR(D ∪ {r})|A ⊆ D, A ∩ D = ∅] be the ex-pected assignment cost incurred by SimpleCFL, con-ditioned on the fact that A ⊆ D, A ∩ D = ∅.

Claim 2.1. For given subsets A, A ⊆ D, we can effi-ciently compute E[cR(D ∪ {r})|A ⊆ D, A ∩ D = ∅].

Proof of Claim 2.1. For a given demand vertex j ∈D\A, we order the nodes in (D\A)∪{r} according totheir nondecreasing shortest path distance from j . Letj (1), j (2), . . . , j (q) be the ordered set of nodes. Let� be the first index such that j (�) ∈ A ∪ {r}, i.e. � isthe first index for which we are sure to open a facilityat j (�), and for all earlier indices k we open a facilityat j (k) with probability q/M . Then the conditionalexpected assignment cost for j is

�−1∑k=1

(1− q

M

)k−1 q

Md(j (k), j)+

(1− q

M

)�−1d(j (�), j)

and we obtain the total expected assignment cost bysumming over all j ∈ D\A. �

Let E[cST (D∪{r})|A ⊆ D, A∩D=∅] be given byLemma 2.4. Let OPT = BOPT + ROPT be the costof the optimal solution. We maintain the invariant that

2E[cST (D ∪ {r})|A ⊆ D, A ∩ D = ∅]+ E[cR(D ∪ {r})|A ⊆ D, A ∩ D = ∅]

�(2 + 2q)OPT .

Note that initially, when A, A = ∅, the invariantholds for q = 1 by Lemma 2.4(i) and Lemma 2.2.Using the improved analysis from [2], we can makethe stronger statement that the invariant initially holdsfor any q �0.636, since by Lemma 2.4(i) and Lemma2.3 the left-hand side is at most (2+0.807/q)BOPT +(2q+2)ROPT , and 2+0.807/q �2q+2 for q �0.636.

Now, suppose we have already decided forv1, . . . , v�−1 whether or not to open a facility, andA ⊆ {v1, . . . , v�−1} contains the vertices at which wedecided to open a facility, and A={v1, . . . , v�−1}\A isthe set of vertices at which we will not open a facility.

Since

2E[cST (D ∪ {r})|A ⊆ D, A ∩ D = ∅]+ E[cR(D ∪ {r})|A ⊆ D, A ∩ D = ∅]

= q

M(2E[cST (D∪{r})|A∪{v�}⊆D, A∩D=∅]

+ E[cR(D ∪ {r})|A ∪ {v�} ⊆ D, A ∩ D = ∅])+(

1 − q

M

)(2E[cST (D ∪ {r})|A ⊆ D,

(A ∪ {v�}) ∩ D = ∅]+ E[cR(D ∪ {r})|A ⊆ D, (A ∪ {v�}) ∩ D = ∅]),

the smaller of (2E[cST (D ∪ {r})|A ∪ {v�} ⊆ D, A ∩D = ∅] + E[cR(D ∪ {r})|A ∪ {v�} ⊆ D, A ∩ D = ∅])and (2E[cST (D ∪ {r})|A ⊆ D, (A ∪ {v�}) ∩ D = ∅] +E[cR(D ∪ {r})|A ⊆ D, (A ∪ {v�}) ∩ D = ∅]) must beat most (2 + 2q)OPT and we choose to include v� inA or A accordingly.

At the end of the last iteration, we have obtainedtwo disjoint sets A, A such that A ∪ A = D. Notethat the conditional expectations are now constants.By Lemma 2.4(ii), buying the Steiner tree constructedby the primal–dual algorithm on A∪{r} costs at most2E[cST (D ∪ {r})|A ⊆ D, A ∩ D = ∅], the assignmentcost of the vertices in A to the closest facility in A ∪{r}is equal to E[cR(D ∪ {r})|A ⊆ D, A ∩ D = ∅], andby the invariant the total cost of this solution is at most(2 + 2q)OPT . �

Remark. The ideas from Lemma 2.4 can also be ap-plied to the boosted sampling algorithm for multicom-modity rent-or-buy. However, we do not have an easyway of computing the expected renting costs as wedid in Claim 2.1, since the renting cost will dependheavily on the forest that is bought.

Acknowledgment

This research was supported by NSF Grant CCF-0514628.

References

[1] A. Agrawal, P. Klein, R. Ravi, When trees collide: anapproximation algorithm for the generalized Steiner problemon networks, SIAM J. Comput. 24 (3) (1995) 440–456.

712 D.P. Williamson, A. van Zuylen / Operations Research Letters 35 (2007) 707–712

[2] F. Eisenbrand, F. Grandoni, T. Rothvoß, A tighter analysisof random sampling for connected facility location, 2007,submitted for publication.

[3] P. Erdos, J. Spencer, Probabilistic Methods in Combinatorics,Academic Press, New York, 1974.

[4] L. Fleischer, J. Könemann, S. Leonardi, G. Schäfer, Simplecost sharing schemes for multicommodity rent-or-buy andstochastic Steiner tree, in: STOC’06: Proceedings of theThirty-Eighth Annual ACM Symposium on Theory ofComputing, ACM Press, New York, 2006, pp. 663–670.

[5] M.X. Goemans, D.P. Williamson, A general approximationtechnique for constrained forest problems, SIAM J. Comput.24 (2) (1995) 296–317.

[6] A. Gupta, A. Kumar, M. Pál, T. Roughgarden, Approximationvia cost-sharing: a simple approximation algorithm forthe multicommodity rent-or-buy problem, in: FOCS’03:Proceedings of the 44th Annual IEEE Symposium onFoundations of Computer Science, IEEE Computer Society,Silverspring, MD, 2003, pp. 606–615.

[7] A. Gupta, A. Kumar, M. Pál, T. Roughgarden, Approximationvia cost-sharing: simpler and better approximation algorithmsfor network design, 2005, Submitted for publication.

[8] A. Gupta, A. Kumar, T. Roughgarden, Simpler and betterapproximation algorithms for network design, in: Proceedingsof the Thirty-Fifth Annual ACM Symposium on Theory ofComputing, ACM Press, New York, 2003, pp. 365–372.

[9] A. Gupta, A. Srinivasan, É. Tardos, Cost-sharing mechanismsfor network design, in: APPROX-RANDOM, 2004,pp. 139–150.

[10] G. Robins, A. Zelikovsky, Tighter bounds for graph Steinertree approximation, SIAM J. Discrete Math. 19 (1) (2005)122–134.

[11] C. Swamy, A. Kumar, Primal–dual algorithms for connectedfacility location problems, Algorithmica 40 (4) (2004)245–269.