Online Non-metric Facility Location with Service Installation Costs

Christine Markarian

Department of Engineering and Information Technology, University of Dubai, Dubai, U.A.E.

Keywords:

Facility Location Problem, Service-installation Costs, Online Algorithms, Competitive Analysis, Randomized

Rounding.

Abstract:

In this paper, we study the non-metric Online Facility Location with Service Installation Costs problem (OFL-

SIC), an extension of the well-known non-metric Online Facility Location problem. In OFL-SIC, we are

given a set of facilities, a set of services, and a set of requests arriving over time. Each request is composed

of a subset of the services. Facilities are enabled to offer a subset of the services when being opened and an

algorithm has to ensure that each arriving request is connected to a set of open facilities jointly offering the

requested services. Opening a facility incurs an opening cost and for each offered service, there is a service

installation cost that needs to be paid if the algorithm decides to install the service at the facility. Connecting a

request to an open facility incurs a connecting cost, which is equal to the distance between the request and the

facility. The goal is to minimize the total opening, service installation, and connecting costs. We propose the

ﬁrst online algorithm for non-metric OFL-SIC and show that it is asymptotically optimal under the standard

notion of competitive analysis which is used to evaluate the performance of online algorithms.

1 INTRODUCTION

With the rapid growth in urbanization, the demand for

different services in cities has been signiﬁcantly in-

creased. It is estimated that by the year 2050, about

66 percent of the world’s population will be living in

an urban environment (World Urbanization Prospects,

2014). With this trend, cities are expected to build fa-

cilities that offer various services not only to the peo-

ple currently residing in the city but also to the people

who are likely to move there in the future. Without ad-

equate information about the future, in regards to how

many people will move, when, and what services they

will request, decisions about where to locate facili-

ties and what services to assign to each facility, with

minimum possible costs, become more challenging.

At the heart of such decisions lie complex facility lo-

cation optimization problems, which we approach in

this paper from an algorithmic perspective. In partic-

ular, we are interested in developing algorithms that

can be proven to be optimal or best possible.

We consider the Facility Location with Ser-

vice Installation Costs problem (FL-SIC), which

is an extension of the well-studied Facility Lo-

cation problem (FL). FL-SIC was introduced by

Shmoys et al. (Shmoys et al., 2004) and is deﬁned as

follows. We are given a set of facilities, a set of ser-

vices, and a set of requests. Each request is composed

of a subset of services. Facilities are enabled to of-

fer a subset of the services when being opened and an

algorithm has to ensure that a request is connected to

a set of open facilities jointly offering the requested

services. Opening a facility incurs an opening cost

and for each offered service, there is a service instal-

lation cost that needs to be paid if the algorithm de-

cides to install the service at the facility. Connecting

a request to an open facility incurs a connecting cost,

which is equal to the distance between the request and

the facility. The goal is to minimize the total opening,

service installation, and connecting costs.

Shmoys et al. (Shmoys et al., 2004) studied the

metric version of FL-SIC in which facilities and re-

quests are assumed to reside in a metric space. A

more general variant of FL-SIC, known as the Multi-

Commodity Facility Location problem (MCFL), has

been studied for both the metric (Poplawski and

Rajaraman, 2011; Ravi and Sinha, 2010; Shmoys

et al., 2004; Svitkina and Tardos, 2010) and non-

metric (Fleischer et al., 2006) versions. In MCFL,

one facility cost is given for opening a facility and

installing its services. In FL-SIC, facility costs are

split, so as we pay a ﬁxed cost for opening a facility

and then for each service installed at the facility, we

pay a service cost associated with it. FL-SIC is also

known as MCFL with linear costs. FL-SIC becomes

a special case of MCFL if we set each facility cost (in

MCFL) to the sum of opening and service installation

costs (in FL-SIC).

In many real-world scenarios, predicting future

events is difﬁcult if not impossible, and yet, we are

Markarian, C.

Online Non-metric Facility Location with Service Installation Costs.

DOI: 10.5220/0010469207370743

In Proceedings of the 23rd International Conference on Enterprise Information Systems (ICEIS 2021) - Volume 1, pages 737-743

ISBN: 978-989-758-509-8

737

expected to make immediate wise decisions with as

few regrets as possible. From an algorithmic perspec-

tive, these scenarios are modeled as online problems

and are solved with online algorithms. Motivated by

such scenarios, we study the online variant of FL-SIC

which we refer to as the non-metric Online Facil-

ity Location with Service Installation Costs problem

(OFL-SIC), deﬁned as follows.

Deﬁnition 1. (Non-metric Online Facility Location

with Service Installation Costs) We are given a set

of facilities, a set of services, and a set of requests

arriving over time. Each request is composed of a

subset of the services. Facilities are enabled to of-

fer a subset of the services when being opened and

an algorithm has to ensure that each arriving request

is connected to a set of open facilities jointly offering

the requested services. All decisions are to be made

irrevocably. Opening a facility incurs an opening cost

and for each offered service, there is a service instal-

lation cost that needs to be paid if the algorithm de-

cides to install the service at the facility. Connecting

a request to an open facility incurs a connecting cost,

which is equal to the distance between the request and

the facility. The goal is to minimize the total opening,

service installation, and connecting costs.

To the best of our knowledge, no online algorithm ex-

ists for this variant.

2 OUR CONTRIBUTION

We propose the ﬁrst online algorithm for non-metric

OFL-SIC and analyze it under the standard notion

of competitive ratio. An online algorithm is c-

competitive or has competitive ratio c if for all se-

quences of demands, the cost incurred by the algo-

rithm is at most c times the cost incurred by an op-

timal ofﬂine algorithm, which knows the entire se-

quence of demands in advance.

Our algorithm is based on randomized rounding,

a technique commonly used in the design and analy-

sis of online algorithms. It has an O(log(nk) logm)-

competitive ratio, where m is the number of facilities,

n is the number of requests, and k is the number of

services. The competitive analysis of our algorithm is

based on ideas taken from the competitive analysis of

Meyerson for the Parking Permit problem (Meyerson,

2005).

We also show that the competitive ratio of our al-

gorithm is asymptotically optimal by giving a lower

bound of Ω(log(nk)logm) on the competitive ratio of

any polynomial-time randomized algorithm for non-

metric OFL-SIC, under the assumption that NP 6⊆

BPP. The latter is the result of two reductions from the

Online Set Cover problem (OSC), which was intro-

duced by Alon et al. (Alon et al., 2009). Korman (Ko-

rman, 2005) gave an Ω(logmlog n) lower bound on

the competitive ratio of any polynomial-time random-

ized algorithm for OSC, under the assumption that NP

6⊆ BPP.

Outline. The rest of the paper is structured as fol-

lows. In Section 3, we give an overview of related

works. In Section 4, we present a lower bound on

the competitive ratio of any online polynomial-time

randomized algorithm for non-metric OFL-SIC. In

Section 5, we describe our online algorithm for non-

metric OFL-SIC and give its competitive analysis in

Section 6. We conclude with some open problems in

Section 7.

3 RELATED WORK

In the ofﬂine setting, Fleischer et al. (Fleischer et al.,

2006) studied the non-metric Multi-Commodity Fa-

cility Location problem (MCFL) and gave an ap-

proximation ratio logarithmic in the number of re-

quests, facility locations, and services. As for met-

ric MCFL, there were many works, mostly varying

in the facility cost function (Poplawski and Rajara-

man, 2011; Ravi and Sinha, 2010; Shmoys et al.,

2004; Svitkina and Tardos, 2010), including that of

Shmoys et al. (Shmoys et al., 2004) for the Facil-

ity Location with Service Installation Costs problem

(FL-SIC).

In the online setting, only the metric version of

the Online Multi-Commodity Facility Location prob-

lem (OMCFL), the online variant of MCFL, has been

studied, by Castenow et al. (Castenow et al., 2020).

Castenow et al. (Castenow et al., 2020) gave deter-

ministic and randomized online algorithms along with

a lower bound on the competitive ratio of any random-

ized online algorithm for metric OMCFL.

A closely related online problem is the non-metric

Online Facility Location problem (OFL), the online

variant of non-metric FL, due to Alon et al. (Alon

et al., 2006). Non-metric OFL is a special case of

non-metric OFL-SIC in which the number of ser-

vices is 1. Alon et al. (Alon et al., 2006) proposed

a randomized algorithm for non-metric OFL, with

asymptotically optimal O(log m logn)-competitive ra-

tio, where m is the number of facilities and n is

the number of clients. The metric variant of On-

line Facility Location has been intensively studied.

Meyerson (Meyerson, 2001) introduced a random-

ized algorithm with O (log n)-competitive ratio. Fo-

takis (Fotakis, 2003) later proved that the algorithm

ICEIS 2021 - 23rd International Conference on Enterprise Information Systems

738

is O(

logn

loglog n

)-competitive and showed that this is the

best possible competitive ratio for any online algo-

rithm. He also gave a deterministic algorithm with the

same competitive ratio. Fotakis (Fotakis, 2007) also

provided a simpler online algorithm with an O(log n)-

competitive ratio.

4 LOWER BOUND

In this section, we give an Ω(log(nk)log m) lower

bound on the competitive ratio of any online

polynomial-time randomized algorithm for non-

metric OFL-SIC, under the assumption that NP 6⊆

BPP.

The lower bound is achieved by two reductions

from the Online Set Cover problem (OSC), intro-

duced by Alon et al. (Alon et al., 2009), and deﬁned

as follows.

Deﬁnition 2. (Online Set Cover). We are given a

universe of n elements and m subsets of this universe,

each associated with a cost. Elements are revealed to

the algorithm over time and as soon as one arrives,

the online algorithm needs to make sure that there is

at least one subset purchased that contains the ele-

ment. The goal is to minimize the total cost of subsets

purchased.

Korman (Korman, 2005) gave an Ω(log m log n)

lower bound on the competitive ratio of any

polynomial-time randomized algorithm for OSC,

where m is the number of subsets and n is the number

of elements, under the assumption that NP 6⊆ BPP.

Our lower bound is based on the following two

observations.

Observation 1. No randomized online polynomial-

time algorithm for the non-metric Online Facility Lo-

cation with Service Installation Costs problem (OFL-

SIC) can achieve a competitive ratio better than

Ω(logm logn), under the assumption that NP 6⊆ BPP,

where m is the number of facilities and n is the num-

ber of requests.

Proof. Let I be an instance of Online Set Cover. We

transform I into an instance I

of non-metric OFL-SIC

as follows. We set the number of services to 1. We

represent each subset of I as a facility offering the

service and set its opening cost to the corresponding

subset cost. Service installation costs are all set to 0.

We represent each element as a request with the (one)

service. For each request, we let the distance from

the request to the facility be 0 if the corresponding el-

ement belongs to the subset, and inﬁnity otherwise.

Now, every solution to I

corresponds to a solution to

I of the same cost, and vice versa. Due to the lower

bound of Korman (Korman, 2005), we can imply the

result, where n is the number of requests (elements)

and m is the number of facilities (subsets).

Observation 2. No randomized polynomial-time on-

line algorithm for the non-metric Online Facility Lo-

cation with Service Installation Costs problem (OFL-

SIC) can achieve a competitive ratio better than

Ω(logm logk), under the assumption that NP 6⊆ BPP,

where m is the number of facilities and k is the num-

ber of services.

Proof. Let I be an instance of Online Set Cover. We

transform I into an instance I

of non-metric OFL-SIC

as follows. Each element is represented as a service.

Each subset is represented as a facility with opening

cost equal to the corresponding subset cost and offer-

ing the services of the corresponding elements. Ser-

vice installation costs are all set to 0. In each step,

a request with the (one) service arrives. In the I

in-

stance, the algorithm is allowed to open any facility

and install at it any subset of the services. The OSC

instance I need not include all such combinations and

so for each facility with a subset of services that is

not yet created, we create it and set its opening cost to

inﬁnity. Now, every solution to I

corresponds to a so-

lution to I of the same cost, and vice versa. Due to the

lower bound of Korman (Korman, 2005), we can im-

ply the result, where k is the number of services (ele-

ments) and m is the number of facilities (subsets).

By combining the two observations, we conclude

the following.

Theorem 1. (Lower Bound). No online polynomial-

time randomized algorithm for the non-metric On-

line Facility Location with Service Installation Costs

problem (OFL-SIC) can achieve a competitive ratio

better than Ω(log(nk) log m), where m is the number

of facilities, n is the number of requests, and k is the

number of services, under the assumption that NP 6⊆

BPP.

5 ONLINE ALGORITHM

In this section, we present an online randomized al-

gorithm for non-metric OFL-SIC.

We formulate non-metric OFL-SIC as a directed

edge-weighted graph, as follows. We generate a node

for each of the k services, called service nodes. For

each facility, we generate a node, called facility nodes.

We add an edge from each facility node to each ser-

vice node. Each edge from a facility node to a ser-

vice node has weight equal to the corresponding ser-

Online Non-metric Facility Location with Service Installation Costs

739

vice installation cost. We make a copy of each facil-

ity node, called duplicate facility nodes. Each facil-

ity node will be connected to its duplicate through an

edge of weight equal to the opening cost of the facil-

ity.

Since requests are not given all at once, a request

node will be created as soon as a request arrives. We

add an edge from the request node to each facility du-

plicate node and set its weight to the distance between

the request and the facility (the connecting cost).

Figure 1: Graph formulation of Non-metric OFL-SIC.

Figure 1 gives an example of two requests, three fa-

cilities, and three services. Assume request r arrives

and asks for the services 1 and 3. To serve r, the algo-

rithm needs to ﬁnd two paths: a path from the request

node r to the service node 1 and another path from the

request node r to the service node 3. It is easy to see

that the edges on these paths represent a feasible so-

lution for r. Let p be a solution path. p would contain

an edge that corresponds to some duplicate facility -

the algorithm will connect the client to that facility

and open the facility, if it is not already open. More-

over, the algorithm will install a service at a facility if

the outgoing edge from the facility node to the service

node is on p and if it is not already installed.

Remark. It is worth noting that the solution paths

can be found by running any online algorithm for

the well-known Online Steiner Forest problem (OSF).

In OSF, we are given an edge-weighted graph and

pairs of nodes arriving over time. As soon as a

pair arrives, the online algorithm needs to ensure that

there is at least one path, whose edges are purchased

by the algorithm, that connects the pair. The goal

is to minimize the total weights of the edges pur-

chased. The best competitive ratio achievable for OSF

is O(log

n), where n is the number of nodes in the

graph. There exists an online algorithm in the liter-

ature with this ratio, due to Awerbuch et al. (Awer-

buch et al., 2004). Nevertheless, this algorithm will

not imply a desirable asymptotically optimal compet-

itive ratio for our problem, without further analysis.

This is because the number of nodes in our formu-

lated graph can get as many as n + k + mk, implying

a competitive ratio of O(log

(km +n)) for non-metric

OFL-SIC. Hence, a different approach is required.

Random Variable α. Before any request arrives,

the algorithm chooses α to be the minimum among

log(kn)

independently chosen random variables,

distributed uniformly in the interval [0,1].

The algorithm ﬁnds the solution paths as follows.

It initially knows k, the number of services, and n, the

number of requests. Let G be the formulated graph,

growing over time as new requests arrive. We assign

to each edge e in G of weight w

, a fraction f

. We

set the values of all fractions to 0 initially. The online

algorithm will increase these fractions throughout its

execution. We deﬁne the max-ﬂow from node u to

node v in G to be the smallest total fractions of edges

which if removed would disconnect u from v. These

edges form a min-cut from u to v in G.

Whenever a new request arrives, a request node

and its outgoing edges are added to G. A fraction set

initially to 0 is assigned to each new edge. The weight

of a path is the sum of the weights of its edges.

For each service requested, the algorithm executes

the following. Let r be the request node and s the

service node requested.

Online Algorithm for Non-metric OFL-SIC.

1. While the max-ﬂow from the request node r to the

service node s in G is less than 1, construct a min-

cut K from r to s in G; for each edge e ∈ K, make

the following increment:

= f

· (1 + 1/w

) +

· w

2. Purchase each edge e with f

> α.

3. If there is no purchased path from r to s in G, ﬁnd

a minimum-weight such path and purchase it.

Notice that the last step guarantees that the algorithm

achieves a feasible solution.

6 COMPETITIVE ANALYSIS

In this section, we show that our algorithm has an

O(log(nk)logm)-competitive ratio, where m is the

number of facilities, n is the number of requests, and

k is the number of services.

The algorithm makes purchases in Step 2 and Step

3. We will measure the cost of the algorithm in each

step separately. Note that the cost of the algorithm is

ICEIS 2021 - 23rd International Conference on Enterprise Information Systems

740

equivalent to the total weight of the edges purchased

in the formulated graph.

Algorithm’s Cost in Step 2. Let E

be the set of

the edges purchased in the second step of the algo-

rithm and let Cost

be the expected cost. Recall that

an edge is purchased if its fraction exceeds α. We ﬁx

any i : 1 ≤ i ≤ 2

log(kn + 1)

and an edge e. We de-

note by X

e,i

the indicator variable of the event that e is

purchased by the algorithm. Let w

be the weight of

edge e and f

its fraction. We have that:

Cost

∑

e∈E

log(kn+1)

∑

i=1

· Exp[X

e,i

] = 2

log(kn + 1)

∑

e∈E

(1)

Next, we give an upper bound for

∑

e∈E

in terms

of the optimal ofﬂine solution. Let E be the set of all

edges in the graph.

∑

e∈E

(i.e., with all edges in-

cluded) is the cost of the so-called fractional solution

of the algorithm. In a fractional solution, the algo-

rithm is allowed, for each edge, to buy a fraction of

it and pay the corresponding fraction of its cost. The

cost of a fractional solution is called the fractional

cost of the algorithm. When fractions are either 0

or 1, then the solution is called an integral solution.

The algorithm’s goal is to ultimately ﬁnd a feasible

integral solution. To achieve that, the algorithm pro-

duces, in the ﬁrst step, a fractional solution, which is

rounded into an integral solution in the second step.

The integral solution in the second step is not neces-

sarily feasible. In the third step, the algorithm ensures

that the ﬁnal integral solution is feasible.

Now, we compare the cost of the fractional solu-

tion to the cost of the optimal integral solution. The

following lemma will be used in the comparison.

Lemma 1. Every min-cut constructed in Step 1 con-

tains at least one edge of the optimal integral solution.

Proof. Assume there is no such edge. Given a pair

that needs to be connected. The optimal solution

needs to connect this pair through at least one path,

p. By the deﬁnition of a cut, every cut should contain

at least one of the edges of p.

We call it a min-cut construction every time the al-

gorithm constructs a min-cut. Observe that, each op-

timal edge, i.e., an edge in the optimal solution, can

appear in zero, or more min-cut constructions, which

do not have to be consecutive. We look into all the

optimal edges that appeared in at least one min-cut

construction. We calculate the costs paid by the frac-

tional solution during all the min-cut constructions in

which each of these edges appeared. By doing so, we

would have measured the total cost of the fractional

solution. This is true because an increment is only

made after a min-cut construction and due to Lemma

Let e be an edge that appeared in more than one

min-cut construction. The optimal algorithm pays w

We calculate now what the online algorithm pays dur-

ing the min-cut constructions in which e appeared.

Lemma 2. Each increment increases the fractional

cost by at most 2.

Fix any min-cut K constructed. Each edge e in K in-

creases the cost by w



·w



. Before an in-

crement, the max-ﬂow is less than 1 (or

∑

e∈K

< 1).

Hence, adding up over all |K| edges, we get for each

increment a total cost of:

∑

e∈K



· w



< 2

The fraction f

of e will become 1 after a ﬁnite num-

ber of min-cut constructions and this number can be

upper bounded as follows. Based on the increment

equation of the algorithm, after O(w

log|K|) min-cut

constructions, f

becomes 1 and e cannot appear in

any further min-cut construction. Due to Lemma 2

and since each min-cut construction is accompanied

with only one increment, we imply that the algorithm

pays at most O(w

log|K|) during the min-cut con-

structions in which e appeared. This is O(log|K|)

times what the optimal has paid. Moreover, we have

that |K| is at most m, which is the number of paths be-

tween any pair, each containing one facility node. The

same analysis holds for every optimal edge appearing

in at least one min-cut construction. By summing up

over all these edges, we achieve an upper bound for

the fractional cost of the algorithm:

∑

e∈E

≤ O(logm · Opt) (2)

Since

∑

e∈E

≤

∑

e∈E

and from Equations 1

and 2, we conclude that:

Cost

≤ O(log(nk)log m · Opt) (3)

Algorithm’s Cost in Step 3. Now, we measure the

cost of the algorithm in the third step. Let Cost

the expected cost of this step.

We ﬁx a pair (request node r, service node s) and

1 ≤ i ≤ 2

log(kn + 1)

accompanied with the random

variable α chosen before the arrival of requests. We

record the time at which the algorithm has already

completed the second step. We ﬁx any min-cut K

from r to s at this point. The probability that the al-

gorithm did not purchase a path from r to s in the

second step is equal to the probability that it did not

Online Non-metric Facility Location with Service Installation Costs

741

purchase any of the edges in K. To see why this holds,

we let e be an edge in K. All the edges in every path

containing e must have a fraction equal to at least the

fraction of e (Max ﬂow - Min Cut theorem). So if e

is purchased, i.e., e > α, then all the other edges on

the paths containing e are purchased too, since they

have a higher fraction than e. Hence, the probability

is equal to:

∏

e∈K

(1 − f

) ≤ e

−

∑

e∈K

≤ 1/e

The second inequality holds since the algorithm en-

sures that

∑

e∈K

≥ 1 at this point. Thereby, the

expected cost of purchasing a path in the third step,

for all 1 ≤ i ≤ 2

log(kn + 1)

, would be less than

1/(kn)

· Opt, where the optimal solution cost Opt

can be used as an upper bound for the minimum-

weight path constructed by the algorithm in the third

step.

The total number of pairs the algorithm receives

is at most kn, since each of the n requests can ask for

at most k services. Summing up over all these pairs,

we conclude the expected cost of the algorithm in the

third step:

Cost

≤ 1/(kn) · Opt (4)

By combining Equations 3 and 4, we conclude the

following.

Theorem 2. (Upper Bound). There is an online ran-

domized algorithm for the non-metric Online Facility

Location with Service Installation Costs (OFL-SIC),

that has an asymptotically optimal competitive ratio

of O (log(nk)log m), where m is the number of facili-

ties, n is the number of requests, and k is the number

of services.

7 CONCLUDING REMARKS &

FUTURE WORK

In this paper, we have studied the non-metric On-

line Facility Location with Service Installation Costs

problem (OFL-SIC), which could also be called the

non-metric Online Multi-Commodity Facility Loca-

tion with linear costs problem (non-metric OMCFL

with linear costs). A next step would be to consider

the non-metric Online Multi-Commodity Facility Lo-

cation problem (OMCFL) for other facility cost func-

tions, such as the cost functions deﬁned for the metric

case in (Castenow et al., 2020). It seems like other

techniques than the ones used in this paper would be

needed to achieve results for these cost functions.

Moreover, unlike in the ofﬂine setting, for the gen-

eral facility cost function, there are no online algo-

rithms in the literature for both the metric and non-

metric cases. So there is a lot to investigate in this

direction.

Another direction is to assume facilities with ca-

pacities, for both the metric and non-metric variants.

This would reﬂect a more natural real-world facil-

ity location problem, in which the number of clients

served by each facility is limited by the resources

available at the facility (Cygan et al., 2018).

REFERENCES

Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., and

Naor, J. S. (2006). A general approach to online net-

work optimization problems. ACM Trans. Algorithms,

2(4):640–660.

Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., and

Naor, J. S. (2009). The online set cover problem.

SIAM J. Comput., 39(2):361–370.

Awerbuch, B., Azar, Y., and Bartal, Y. (2004). Online

generalized steiner problem. Theoretical Computer

Science, 324(2):313 – 324. Online Algorithms: In

Memoriam, Steve Seiden.

Castenow, J., Feldkord, B., Knollmann, T., Malatyali, M.,

and Meyer Auf der Heide, F. (2020). The online multi-

commodity facility location problem. In Proceedings

of the 32nd ACM Symposium on Parallelism in Algo-

rithms and Architectures, SPAA ’20, page 129–139,

New York, NY, USA. Association for Computing Ma-

chinery.

Cygan, M., Czumaj, A., Mucha, M., and Sankowski, P.

(2018). Online facility location with deletions. ArXiv,

abs/1807.03839.

Fleischer, R., Li, J., Tian, S., and Zhu, H. (2006). Non-

metric multicommodity and multilevel facility loca-

tion. In Cheng, S.-W. and Poon, C. K., editors, Al-

gorithmic Aspects in Information and Management,

pages 138–148, Berlin, Heidelberg. Springer Berlin

Heidelberg.

Fotakis, D. (2003). On the competitive ratio for online fa-

cility location. In Baeten, J. C. M., Lenstra, J. K.,

Parrow, J., and Woeginger, G. J., editors, Automata,

Languages and Programming, pages 637–652, Berlin,

Heidelberg. Springer Berlin Heidelberg.

Fotakis, D. (2007). A primal-dual algorithm for online non-

uniform facility location. Journal of Discrete Algo-

rithms, 5(1):141 – 148.

Korman, S. (2005). On the use of randomization in the on-

line set cover problem. Master’s thesis, Weizmann In-

stitute of Science, Israel.

Meyerson, A. (2001). Online facility location. In Proceed-

ings 42nd IEEE Symposium on Foundations of Com-

puter Science, pages 426–431.

Meyerson, A. M. (2005). The parking permit problem. In

Proceedings of the 46th Annual IEEE Symposium on

ICEIS 2021 - 23rd International Conference on Enterprise Information Systems

742

Foundations of Computer Science, FOCS ’05, page

274–284, USA. IEEE Computer Society.

Poplawski, L. J. and Rajaraman, R. (2011). Multicommod-

ity facility location under group steiner access cost. In

SODA ’11.

Ravi, R. and Sinha, A. (2010). Approximation algorithms

for multicommodity facility location problems. SIAM

J. Discret. Math., 24(2):538–551.

Shmoys, D. B., Swamy, C., and Levi, R. (2004). Facility

location with service installation costs. In Proceed-

ings of the Fifteenth Annual ACM-SIAM Symposium

on Discrete Algorithms, SODA ’04, page 1088–1097,

USA. Society for Industrial and Applied Mathematics.

Svitkina, Z. and Tardos, E. (2010). Facility location with

hierarchical facility costs. ACM Trans. Algorithms,

6(2).

Online Non-metric Facility Location with Service Installation Costs

743