An Online Deterministic Algorithm for Non-Metric Facility Leasing

Christine Markarian and Claude Fachkha

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

Keywords:

Combinatorial Optimization, Online Algorithms, Competitive Analysis, Facility Location, Leasing.

Abstract:

Leasing has become one of the most widely spread business models in almost all markets. The online algo-

rithmic study of leasing was initiated in 2005. Unlike classical algorithms, online algorithms are not given

the entire input sequence at once. A portion of the input sequence is revealed in each step and the online

algorithm is required to react to each step while targeting the given optimization goal against the entire input

sequence. In a leasing setting, resources are leased and expire once their lease duration is over. Many well-

known optimization problems are deﬁned and studied in the leasing setting. In this paper, we continue the

online algorithmic study of leasing by addressing the so-called Online Non-metric Facility Leasing problem

(ONFL), the leasing variant of the non-metric Online Facility Location problem (non-metric OFL). Given a

collection of facility and client locations. Facilities can be leased using a ﬁxed number of lease types, each

characterized by length and price. Lease types respect the economy of scale, such that longer leases cost more

but are cheaper per unit of time. In each step, a client appears. The algorithm needs to promptly connect it

to a facility that is leased at the current time step. To this end, it needs to decide which facility locations to

lease, the start of their lease, and the lease duration. Connecting a client to a facility incurs a cost equal to the

distance between the facility and the client. The goal is to minimize the total connecting and facility leasing

costs. In this work, we develop the ﬁrst deterministic algorithm for ONFL and evaluate it using the notion

of competitive analysis, a worst-case performance analysis in which the solution of the online algorithm is

compared, over all instances of the given problem, to the optimal solution of the ofﬂine variant of the problem.

1 INTRODUCTION

Facility Location problems, which ask to place facil-

ities in the best possible way to satisfy a set of con-

straints, are one of the most well-studied optimiza-

tion problems in computer science and operations re-

search (Drezner and Hamacher, 2004; Mirchandani

and Francis, 1990). These problems appear in appli-

cations for warehouses, disaster management, health-

care, public transportation, power plants, among oth-

ers (Farahani and Hekmatfar, 2009; Adeleke and

Olukanni, 2020; Sundarakani et al., 2021; Farahani

et al., 2014; Laporte et al., 2019). Their NP-hardness,

network design structure, and combinatorial nature

have attracted researchers in a variety of ﬁelds, in-

cluding approximation algorithms (Vazirani, 2001),

online algorithms (Albers and Leonardi, 1999), and

combinatorial optimization (Korte et al., 2011).

The simplest form of a Facility Location prob-

lem (Shmoys et al., 1997) consists of a collection of

facility and client locations. Each facility is associ-

ated with an opening cost and each client-facility pair

is associated with a connecting cost, which is the dis-

tance between the client and the facility. To open a

facility, the corresponding facility cost needs to be

paid. To connect a client to a facility, the correspond-

ing connecting cost needs to be paid. The aim is to

open a set of facilities and connect each client to an

open facility, while minimizing the total opening and

connecting costs. There are two known variations

of Facility Location problems: the metric, in which

distances satisfy the triangle inequality, and the non-

metric, in which distances are arbitrary. Moreover,

Facility Location problems have been studied in the

ofﬂine and online settings (Shmoys et al., 1997; Jain

et al., 2002; Charikar et al., 2001; Meyerson, 2001;

Fotakis, 2008; Alon et al., 2006). In the ofﬂine set-

ting, the algorithm is given the entire input sequence,

to which it reacts once. In the online setting, the so-

called online algorithm is given a client in each step,

and needs to react to each step by connecting each

arriving client, as soon as it arrives, to an open facil-

ity. The Facility Location problem in the online set-

ting is known as the Online Facility Location problem

(OFL), ﬁrst introduced by (Meyerson, 2001) as a met-

ric variant. The goal in both settings is to minimize

Markarian, C. and Fachkha, C.

An Online Deterministic Algorithm for Non-Metric Facility Leasing.

DOI: 10.5220/0011958700003467

In Proceedings of the 25th International Conference on Enterprise Information Systems (ICEIS 2023) - Volume 1, pages 195-201

ISBN: 978-989-758-648-4; ISSN: 2184-4992

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

195

the total opening and connecting costs.

Online algorithms are evaluated using the notion

of competitive analysis (Borodin and El-Yaniv, 2005),

deﬁned as follows.

Deﬁnition 1. (Competitive Analysis). Let I be the

collection of all instances of a given problem P. We

designate by C(ALG,i) the cost of an online algorithm

ALG of P on instance i ∈ I and by C(OPT,i) the cost

of an optimal ofﬂine algorithm on instance i ∈ I . ALG

has competitive ratio r or is r-competitive if, for all

instances i ∈ I , C(ALG,i) ≤ r ·C(OPT,i)+c for some

constant c independent of i.

In this work, our focus will be the non-metric vari-

ant in the online setting. In particular, we address the

so-called Online Non-metric Facility Leasing prob-

lem (ONFL) (Markarian and auf der Heide, 2019), in

which facilities are leased rather than purchased.

Leasing is now widely used in almost all markets.

The ﬂexibility it offers makes it an indispensable busi-

ness model with numerous beneﬁts (Merrill, 2020).

Lease-based applications naturally call for online al-

gorithms, since decisions about leasing are often re-

quired to be made without knowing the future in ad-

vance. Consider, for instance, the cloud computing

market, in which a third-party company leases ma-

chines from the cloud to serve its clients. The com-

pany may decide to make a yearly lease contract for a

machine and then realize that the machine was not as

useful as thought throughout the year. Making wise

decisions in regards to when to lease which resource

and for how long is modeled as complex optimiza-

tion problems. An online algorithm will make such

leasing decisions with a provable guarantee (i.e., a

competitive ratio). Without knowing the future clients

in advance, it aims to achieve the given optimiza-

tion goal. This motivates the study of facility loca-

tion problems, which commonly appear in real-world

leasing applications, in an online leasing setting.

The Online Non-metric Facility Leasing problem

(ONFL) is deﬁned as follows.

Deﬁnition 2. (Online Non-metric Facility Leasing

or ONFL). Given a collection of facility and client

locations. Facilities can be leased using L different

lease types, each characterized by length and price.

Lease types respect the economy of scale, such that

longer leases cost more but are cheaper per unit of

time. In each step, a client appears. The algorithm

needs to immediately connect it to a facility that is

leased at the current time step. It needs to decide

which facility locations to lease, the start of their

lease, and the lease duration. Connecting a client to

a facility incurs a cost equal to the distance between

the facility and the client. The goal is to minimize the

total connecting and facility leasing costs.

ONFL is a generalization of non-metric OFL in

which there is one lease type that has inﬁnite length.

To the best of our knowlege, the only online algo-

rithm for ONFL in the literature is randomized, with

an O(log n log m + log L log n)-competitive ratio, due

to (Markarian and auf der Heide, 2019), where n is the

number of clients, m is the number of facilities, and L

is the number of lease types. The algorithm’s design

combines a randomized rounding strategy with a mul-

tiplicative incremental technique. We believe convert-

ing the latter into a deterministic algorithm does not

seem straightforward, if possible at all.

1.1 Our Contribution

In this paper, we design the ﬁrst online determin-

istic algorithm for the Online Non-metric Facility

Leasing problem (ONFL). We prove that the latter

has an O(log(m(L +

max

min

))(logl

max

+ log log(m(L +

max

min

))-competitive ratio, where:

– m is the number of facilities

– L is the number of lease types

– l

max

is the longest lease length

– l

min

is the shortest lease length

1.2 Roadmap

The rest of the paper is structured as follows. In

Section 2, we present the lower bounds associated

with the Online Non-metric Facility Leasing prob-

lem (ONFL). In Section 3, we describe a simpliﬁed

structure for the leases that helps ease the competi-

tive analysis. In Section 4, we present an overview of

works related to ONFL. In Section 5, we give a de-

scription of our online algorithm for ONFL and show

its competitive analysis in Section 6. In Section 7, we

conclude with a discussion about the results and open

problems that arise in the context of ONFL.

2 LOWER BOUNDS

The Online Non-metric Facility Leasing problem

(ONFL) generalizes the Parking Permit Problem due

to (Meyerson, 2005) and the Online Set Cover prob-

lem (OSC) due to (Alon et al., 2009). This enables

us to conclude the following lower bounds for ONFL.

Recall that L denotes the number of lease types, n the

number of clients, and m the number of facilities.

– No online deterministic algorithm for ONFL can

achieve a competitive ratio smaller than Ω(L),

ICEIS 2023 - 25th International Conference on Enterprise Information Systems

196

following the deterministic lower bound for the

Parking Permit Problem (Meyerson, 2005).

– No online deterministic algorithm for ONFL

can achieve a competitive ratio smaller than

Ω(

logm logn

loglog m+loglog n

), following the deterministic

lower bound for the Online Set Cover problem

(OSC) (Alon et al., 2009).

– A stricter bound of Ω(logmlogn) holds for

polynomial-time randomized algorithms for OSC,

due to (Korman, 2004), assuming BPP ̸= NP.

Hence, no online polynomial-time randomized al-

gorithm for ONFL can achieve a competitive ratio

smaller than Ω(logm log n), assuming BPP ̸= NP.

– No online randomized algorithm for ONFL can

achieve a competitive ratio smaller than Ω(logL),

following the randomized lower bound for the

Parking Permit Problem (Meyerson, 2005).

Using a simple observation, Bienkowski et al. (Bi-

enkowski et al., 2021) showed that any determinis-

tic algorithm for non-metric Online Facility Location

(non-metric OFL) that does not know the facility-

client graph with all the connections in advance can’t

achieve a competitive ratio better than m. Since

ONFL generalizes non-metric OFL, the same holds

for any deterministic algorithm for ONFL.

3 SIMPLIFIED LEASE

STRUCTURE

The ﬁrst leasing problem studied from the perspec-

tive of online algorithms was introduced by Meyer-

son (Meyerson, 2005), and was known as the Parking

Permit problem. The latter is a simple yet algorithmi-

cally rich problem, deﬁned as follows: On each day,

the algorithm is told if it is raining or not. If it is rain-

ing, the algorithm must provide a valid permit for the

day, selected from L different permit or lease types,

each with a duration and price. The algorithm only

knows whether it is raining or not on the same day. A

longer permit costs more but is less expensive per day.

For example, it would be cheaper to buy a weekly per-

mit for a rainy week than to buy seven daily permits,

one for each day. The online algorithm needs to min-

imize the total permit costs while covering all rainy

days.

In order to make the competitive analysis easier,

Meyerson (Meyerson, 2005) used a simpliﬁed form of

the lease structure known as the Interval model (The-

orem 2.2 in (Meyerson, 2005)), deﬁned as follows.

– Lengths of leases are powers of two.

– No leases of the same type overlap with one an-

other.

By using this structure, he showed that only a

factor of 4 is lost in the competitive ratio. For the

Parking Permit problem, he developed determinis-

tic and randomized algorithms, with competitive ra-

tios of O(L) and O(log L), respectively, where L is

the number of lease types. He also proved matching

lower bounds for both. This structure was used in the

majority of leasing problems that were studied after-

word (Abshoff et al., 2016; Li et al., 2018; Nagarajan

and Williamson, 2013). In this work, we assume the

same lease structure for the Online Non-metric Facil-

ity Leasing problem (ONFL).

4 RELATED WORK

Alon et al. (Alon et al., 2006) developed an online

randomized algorithm for the non-metric Online Fa-

cility Location problem (non-metric OFL), with com-

petitive ratio O(log m log n), where m is the number

of facilities and n is the number of clients. Their al-

gorithm is based on ﬁrst relaxing the problem to its

fractional variant and solving the latter using a multi-

plicative update approach, and then using randomized

rounding to compute a feasible integral solution.

A reduction from non-metric Facility Location

to Set Cover, that does not induce an exponential

increase in the input size, was given by Kolen et

al. (Kolen and Tamir, 1984). This reduction ac-

companied with doubling techniques and the deter-

ministic algorithm for the Online Set Cover prob-

lem due to (Alon et al., 2009), yields a deterministic

solution for non-metric OFL, with competitive ratio

O((logn + log m) · (log n + loglog m)).

Recently, Bienkowski et al. (Bienkowski et al.,

2021) improved this ratio by designing an online de-

terministic algorithm for non-metric OFL, with com-

petitive ratio O(logm·(log n+loglogm)). Their algo-

rithm is based on fractional relaxation of the problem

with clustered facilities and a combination of dual ﬁt-

ting and multiplicative weight-update approaches.

Other extensions of non-metric OFL were later

introduced in the context of service installation

costs (Markarian and Khallouf, 2021) and service

quality costs (Markarian, 2022).

Since the introduction of the leasing setting

by Meyerson (Meyerson, 2005), many optimization

problems (Bienkowski et al., 2017; De Lima et al.,

2018; Markarian and Kassar, 2022; Markarian, 2021;

Markarian and Khallouf, 2021; Markarian and Kas-

sar, 2020; Markarian, 2015), including Facility Lo-

cation problems (Nagarajan and Williamson, 2013;

An Online Deterministic Algorithm for Non-Metric Facility Leasing

197

Figure 1: Client-pair Formulation.

Kling et al., 2012; Markarian and auf der Heide, 2019;

Abshoff et al., 2016), were studied in this setting.

5 ONLINE ALGORITHM

Our online algorithm for ONFL makes use of the on-

line deterministic algorithm due to (Bienkowski et al.,

2021) for the non-metric Online Facility Location

problem. Their polynomial-time algorithm is based

on dual ﬁtting and multiplicative weight update ap-

proaches. It is up to log log-factor optimal and has an

O(logm · (log n + log log m))-competitive ratio, where

m is the number of facilities and n is the number of

clients.

Given an instance I of ONFL, we transform it into

an instance I

′

of non-metric OFL as follows.

1. The clients of I

′

will be formed as follows:

If the given instance I comprises of one lease type

L = 1 that has an inﬁnite length, then I would

be exactly an instance of non-metric OFL. Thus,

we just run the deterministic algorithm of (Bi-

enkowski et al., 2021) for non-metric OFL on I.

Otherwise, given client j of instance I. For each

time step t of instance I, we construct client ( j,t).

We do this for all clients of instance I. The clients

constructed will form the clients of instance I

′

We refer the reader to Figure 1 for an example.

We let N be the collection of all these clients.

2. The facilities of I

′

will be formed as follows:

Each facility of instance I can be leased with L

lease types. Given facility i of instance I. We de-

note facility i of lease length l and lease start t by

a triplet, (i,t,l). These triplets will form the facil-

ities of instance I

′

. We refer the reader to Figure 2

for an example. We let M be the collection of all

these facilities.

3. The facility-client connecting costs will be

formed as follows:

Given client ( j,t

′

) ∈ N of instance I

′

. For each

triplet (i,t, l) ∈ M of instance I

′

, we set the con-

Figure 2: Facility-triplet Formulation.

Figure 3: Connecting Costs.

necting cost between ( j,t

′

) and (i,t,l) to inﬁnity

if t

′

̸∈ [t + l] or otherwise, equal to the connecting

cost between j and i as per instance I. We refer

the reader to Figure 3 for an example.

The clients, facilities, and connecting costs of in-

stance I

′

will be formed before the arrival of the ﬁrst

client of instance I. Upon the arrival of a new client

j of instance I at time step t, the algorithm will con-

sider the client ( j,t) corresponding to j and time step

t as an input to the algorithm for non-metric OFL. All

the other pairs corresponding to j and the future time

steps will be ignored.

Running the online deterministic algorithm

ALG

OFL

of (Bienkowski et al., 2021) for non-metric

OFL on I

′

would yield to a feasible solution for I.

Whenever ALG

OFL

opens a facility (i,t,l) ∈ M ,

we immediately purchase the corresponding lease

for that facility. Whenever ALG

OFL

connects a pair

( j,t) ∈ N to a facility (i,t,l) ∈ M , we also do the

same and pay the corresponding connecting cost.

Notice that, for a client arriving at time step t, the

algorithm will never connect it to a facility whose

lease does not cover time step t, since the connecting

cost would be inﬁnity as per our construction.

ICEIS 2023 - 25th International Conference on Enterprise Information Systems

198

6 COMPETITIVE ANALYSIS

Following the simpliﬁed lease structure deﬁned ear-

lier, we divide the timeline into intervals of length

max

. We ﬁx any of these intervals, I , and evaluate

the performance of the algorithm over this interval.

As per the simpliﬁed lease structure, all leases on this

interval have a start time and end time within this in-

terval. This means that the optimal leases too. Hence,

by proving the competitive ratio of the algorithm over

this interval, we can conclude the competitive ratio of

the algorithm over the entire timeline.

Since in each time step a single client arrives, we

have in total l

max

clients of instance I appearing on the

interval I . According to the formation of our clients

of I

′

described earlier, we construct for each of these

max

clients, (l

max

)

clients of instance I

′

. Therefore,

we have that |N | = (l

max

)

As for the facilities, we show next an upper bound

on the cardinality of M following the formation

of facilities of I

′

described earlier. We order the

lease types in increasing order of length, denoted as

,...,l

}, such that l

j+1

> l

for 1 ≤ j ≤ L − 1.

We can now upper bound |M | as follows:

|M | ≤ m · (

∑

j=1

)

As per the simpliﬁed lease structure, lease lengths

s are increasing and powers of two. Hence, the sum

above can be upper bounded by the sum of a geomet-

ric series with ratio 1/2. Thus, we have that:

∑

j=1

≤ L + l



1−(1/2)

1−1/2

i

L + l



1 − (1/2)



Since L ≥ 1, we have:

L + l



1 − (1/2)



≤ L +

2 L

Therefore, |M | ≤ m·(L+

max

min

), where l

min

= l

is the

shortest lease length and l

max

= l

is the longest lease

length.

Since the algorithm for non-metric Online Facility

Location which we run on instance I

′

is O(log M ·

(logN + log logM ))-competitive, the theorem below

follows.

Theorem 1. There is an online O(log(m(L +

max

min

))(logl

max

+ loglog(m(L +

max

min

))-competitive de-

terministic algorithm for the Online Non-metric Fa-

cility Leasing problem, where m is the number of fa-

cilities, L is the number of lease types, l

max

is the

longest lease length, and l

min

is the shortest lease

length.

7 CONCLUSION

We have presented in this paper the ﬁrst determinis-

tic online algorithm for ONFL, with competitive ratio

depending on the parameters l

max

and l

min

. It would

be useful to achieve a competitive ratio that does not

depend on these parameters or prove a lower bound in

terms of these parameters.

We believe the design structure of our algorithm

for ONFL could be extended to other leasing prob-

lems for which there is no deterministic algorithm

in the literature. Even more interesting would be to

achieve a similar structure for a generalized transfor-

mation between the leasing and non-leasing variations

of any network design problem.

Considering other adverserial models, such as uni-

form distribution (as in (Meyerson, 2001; Kaplan

et al., 2023)), for the input sequence that would proba-

bly be less harsh on the algorithm is always worth the

investigation. Given that both leasing and Facility Lo-

cation scenarios appear as sub-problems in many real-

world applications, these investigations could play a

vital role in closing the gap between the theoretical

results and the practical world.

Finally, incremental (Arulselvan et al., 2015; Dai

and Zeng, 2010; Div

eki and Imreh, 2011; Fotakis,

2006; Fotakis, 2011). and dynamic algorithms (Cy-

gan et al., 2018; Feldkord and Meyer auf der Heide,

2018; Fotakis et al., 2021) have been used to address

many online Facility Location variants in the metric

setting. It would be interesting to design such algo-

rithms in the non-metric setting too.

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