An Algorithmic Approach to Online Multi-Facility Location Problems

Christine Markarian

, Abdul-Nasser Kassar

and Manal Yunis

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

Department of Information Technology and Operations Management, Lebanese American University, Lebanon

Keywords:

Robustness, Multi-Facility Location, Online Algorithms, Competitive Analysis, Randomized Rounding.

Abstract:

Facility Location problems ask to optimally place facilities with respect to some objective so that all clients

requesting a facility service are served. These are one of the most well-studied optimization problems spanning

many research areas, such as operations research, computer science, and management science. Classical

algorithmic study of Facility Location problems is based on the assumption that clients need to be served

with one facility each. Nevertheless, in many real-world applications, facilities experience disruptions and

to overcome their failures, a robust service is desired. To obtain this, clients are served with more than one

facility, and this is commonly represented by an additional input parameter. The aim of the algorithm is then

to provide a robust service to all clients while minimizing costs. This is known as the Multi-Facility Location

problem (MFL), a well-known optimization problem, studied in the ofﬂine setting in which the entire input

sequence is known to the algorithm in advance. In this paper, we address MFL in the online setting, in which

client requests are not known in advance but revealed over time. We refer to it as the Online Multi-Facility

Location problem (OMFL) and study its metric and non-metric variants. We propose the ﬁrst online algorithms

for these variants and measure their performance using the standard notion of competitive analysis. The latter

is a worst-case analysis that compares the cost of the online algorithm to that of the optimal ofﬂine algorithm

that is assumed to know all demands in advance.

1 INTRODUCTION

Facility Location (FL) is a classical NP-hard opti-

mization problem widely spread in the ﬁelds of com-

puter science and operations research (Drezner, 1995;

Manne, 1964). In its simplest form, we are given a set

of facilities and a set of clients. Each facility has an

opening cost and each client i has a connecting cost to

each facility j, which is the distance between i and j.

The goal is to open a subset of the facilities and con-

nect the clients to open facilities so as to minimize

the sum of the facility costs and the connecting costs.

FL is known as two versions, metric and non-metric.

In the metric version, the distances are assumed to be

symmetric and satisfy the triangle inequality.

In this paper, we focus on the algorithmic view of

FL problems and consider an online setting in which

clients are not known in advance but revealed to the

algorithm over time. As soon as one arrives, it needs

to be connected. Many real-world applications, that

contain FL as a sub-problem, have this online na-

ture, in which one is expected to react to present de-

mands whenever they arrive, without knowing about

future demands. Maintaining a given optimization

goal becomes more challenging in the face of this un-

certainty. This encourages the study of online algo-

rithms (Fiat and Woeginger, 1998) for FL problems.

The standard framework to measure online algorithms

is competitive analysis, in which demands and their

arrival order are selected by an oblivious adversary,

that is unaware of the choices of the algorithm. An

online algorithm is c-competitive or has competitive

ratio c if for all sequences of demands, the cost in-

curred by the algorithm is at most c times the cost in-

curred by an optimal ofﬂine algorithm, which knows

the entire sequence of demands in advance.

The study of Facility Location (FL) in the on-

line setting was initiated by Meyerson (Meyer-

son, 2001), who introduced the metric Online Fa-

cility Location problem (OFL), and proposed an

O(logn)-competitive randomized algorithm, where n

is the number of clients. Alon et al. (Alon et al.,

2006) studied the non-metric version and proposed

an O(log n log m)-competitive randomized algorithm,

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

ber of facilities. Many other variations were known

for both metric and non-metric variants in the on-

line setting (Abshoff et al., 2016; Anagnostopoulos

et al., 2004; Div

eki and Imreh, 2011; Fotakis, 2007;

Fotakis, 2008; Markarian and Meyer auf der Heide,

2019).

Other variants include (Fotakis, 2011), in which

Markarian, C., Kassar, A. and Yunis, M.

An Algorithmic Approach to Online Multi-Facility Location Problems.

DOI: 10.5220/0010212200290035

In Proceedings of the 10th International Conference on Operations Research and Enterprise Systems (ICORES 2021), pages 29-35

ISBN: 978-989-758-485-5

the algorithm can merge existing facilities with each

other, and only the decision of assigning some de-

mands to the same facility is irrevocable. Another

variant by (Feldkord and Meyer auf der Heide, 2018)

allows the algorithm to adapt the position of the fa-

cilities for costs proportional to the distance by which

the position is changed.

All of these works assume that clients need to be

served with one facility each. In many real-world

applications, a robust service, in which a client is

served with more than one facility, is desirable (Sny-

der, 2006; Snyder and Daskin, 2006; Gerodimos,

1998). Facilities and/or connections to facilities may

be prone to failure and assigning clients to multi-

ple facilities would provide a fault-tolerant solution.

Such solutions are seeked in many applications, such

as providing replicated cash data in distributed net-

works (Byrka et al., 2010).

1.1 Our Contribution

In this paper, we explore a generalization of online

Facility Location (Alon et al., 2006; Fotakis, 2008), in

which we are additionally given a parameter k, which

is the number of facilities required to serve a client.

We refer to it as the Online Multi-Facility Location

problem, deﬁned as follows.

Deﬁnition 1. (Online Multi-Facility Location) We are

given a collection of m facilities, n clients, and a

positive integer k. Each facility has an opening cost

and each client has a connecting cost to each facility.

Clients arrive over time. As soon as one arrives, it

needs to be connected to at least k open facilities. To

open a facility, we pay its opening cost. To connect a

client to a facility, we pay the corresponding connect-

ing cost. The goal is to minimize the total opening

and connecting costs.

We address the metric and non-metric Online

Multi-Facility Location problems. As far as we are

aware of, there are no online algorithms in the liter-

ature that solve these variants or can be trivially ex-

tended to solve them.

Lower Bounds. For the non-metric version, there

is a lower bound of Ω(logmlogn), under the assump-

tion that NP 6⊆ BPP, where m is the number of facil-

ities and n is the number of clients, due to the lower

bound given for Online Set Cover (Korman, 2005).

As for the metric version, there is a lower bound

of Ω(

logn

loglog n

), where n is the number of clients, due

to the lower bound given for metric OFL (Fotakis,

2008).

Our results can be summarized as follows.

1. We refer to the non-metric version as Online Non-

metric Multi-Facility Location (ONMFL). We

propose an online O(log(kn) log m)-competitive

randomized algorithm for ONMFL, where m is

the number of facilities; n is the number of clients;

and k is the number of required connections.

The latter uses a randomized rounding approach

that ﬁrst constructs a fractional solution and then

rounds it into an integral one. Its competitive anal-

ysis is based on ﬁrst comparing the fractional so-

lution constructed by the algorithm to the optimal

ofﬂine solution and then measuring the fractional

solution in terms of the integral one.

2. We refer to the metric version as Online Met-

ric Multi-Facility Location (OMMFL). We pro-

pose an online O(max{

max

min

max

min

} · k ·

logn

loglog n

competitive deterministic algorithm for OMMFL,

where n is the number of clients; k is the num-

ber of required connections; c

max

and c

min

are

the maximum and minimum connecting costs, re-

spectively; f

max

and f

min

are the maximum and

minimum facility costs, respectively.

The idea of the algorithm is to ensure ﬁrst that

each client is connected to one facility by running

an algorithm for metric Online Facility Location

(OFL). Then, the k − 1 remaining connections are

made by choosing the cheapest possible facilities

so as not to worsen the competitive ratio by much.

Our approach can be seen as a general frame-

work that transforms any given online algorithm

for metric OFL into an algorithm for OMMFL, by

losing a bounded factor in the competitive ratio.

1.2 Outline

The remainder of this paper is structured as follows.

Section 2 gives an overview of literature related to

ONMFL and presents an algorithm for ONMFL along

with its competitive analyis. Section 3 gives an

overview of literature related to OMMFL and presents

an algorithm for OMMFL along with its competitive

analyis. Section 4 concludes the paper with some fu-

ture work.

2 ONLINE NON-METRIC

MULTI-FACILITY LOCATION

In this section, we start with an overview of works

related to ONMFL and some preliminaries. Then we

present an online randomized algorithm for ONMFL

and analyze its competitive ratio.

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

2.1 Preliminaries & Related Work

ONMFL is a generalization of the Online Non-metric

Facility Location (ONFL) (Alon et al., 2006) with

k = 1. Alon et al. (Alon et al., 2006) gave an

O(lognlogm)-competitive randomized algorithm for

ONFL, where n is the number of clients and m is the

number of facilities.

A closely related problem is the Online Set k-

Multicover (OSMC) (Berman and DasGupta, 2008).

Given a universe U of n elements, a family S of m

subsets of V , each associated with a cost, and a pos-

itive integer k. A subset D ⊆ U of elements arrives

over time. OSMC asks to select a collection C ⊆ S of

subsets, of minimum cost, such that each arriving el-

ement belongs to at least k subsets of C . Berman and

DasGubta (Berman and DasGupta, 2008) proposed an

O(logmlogd)-competitive randomized algorithm for

OSMC, where m is the number of subsets and d is the

maximum set size.

Transformations between OSMC and ONMFL in-

stances can be made in both directions. An instance

of ONMFL can be transformed into an instance of

OSMC as follows. We associate each facility with

each of the 2

− 1 possible groups of clients, and let

each facility/group be a subset, with cost equal to the

sum of the cost of the facility and the connecting costs

of the clients in the group to the facility. We let each

client be an element. Following this transformation,

the algorithm of Berman and DasGubta would im-

ply a feasible algorithm for ONMFL, but with com-

petitive ratio O(log(m(2

− 1)) log n), where n is the

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

An instance of OSMC can be transformed into an in-

stance of ONMFL as follows. We represent each sub-

set by a facility and let the opening cost be the sub-

set cost. We represent each element by a client and

set the connecting cost to 0 if the client belongs to

the subset/facility and inﬁnity otherwise.The ofﬂine

version of OSMC, in which all elements are given

in advance, has an O(log d)-approximation (Johnson,

1974; Virkumar, 2001), where d is the maximum set

size. Similar transformations can be made in the of-

ﬂine setting. This implies an O(log n)-approximation

for the ofﬂine version of ONMFL, where n is the num-

ber of clients. These are the best achievable unless

NP ⊆ DT IME(n

loglog n

) (Feige, 1998). The Online

Set Cover (OSC) (Alon et al., 2003) is a special case

of OSMC with k = 1. Korman’s lower bound on the

competitive ratio of OSC (Korman, 2005) implies an

Ω(logmlogn) lower bound on the competitive ratio of

ONMFL, under the assumption that NP 6⊆ BPP, where

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

clients.

For other online problems solved with similar

techniques as the ones used in this paper, see (Markar-

ian and Kassar, 2020; Hamann et al., 2018).

2.2 Online Algorithm

Our algorithm for ONMFL is based on constructing

a fractional solution ﬁrst and then rounding it online

into an integral solution. Unlike (Alon et al., 2003)

and (Alon et al., 2006) for Online Set Cover (OSC)

and Online Non-metric Facility Location (ONFL), re-

spectively, in which a potential function is used in the

competitive analysis of the algorithms, our analysis is

based on simple arguments, similar to those we gave

in our previous work for a related problem, the Online

Node-weighted Steiner Forest (Markarian, 2018).

We start with giving the following graph formu-

lation for ONMFL. Given a root node r, m facility

nodes, and n client nodes. We add an edge from r to

each facility node j, with cost equal to the opening

cost of facility j. We add an edge from each facility

node j to each client node i, with cost equal to the con-

necting cost of client i to facility j. As soon as a client

i arrives, we need to purchase k disjoint paths between

r and i. The goal is to minimize the total costs of the

paths purchased, where the cost of a path is the cost

of its edges. To output a solution for ONMFL, each

facility whose corresponding edge is purchased will

be opened, and each client whose corresponding edge

to an open facility is purchased will be connected to

that facility.

The algorithm initially knows n, the number of

clients; k, the number of required connections; and

the opening costs of facilities. A subset D of n

≤ n

clients arrives over time. As soon as a client i arrives,

the algorithm is given the connecting costs of i to each

facility, and is expected to react.

We consider the graph formulation described ear-

lier. Let r be a root node; each facility is represented

as a facility node, and has an edge to r, associated

with its opening cost. Upon the arrival of a new client

i, the algorithm creates a client node for it and adds an

edge from this node to each facility node, associated

with the given connecting cost. Let G = (V,E) be this

graph.

Each edge added to E is given a fraction, set inti-

tially to 0. The algorithm does not allow these frac-

tions to decrease over time. These form a fractional

solution for ONMFL. The maximum ﬂow between

node u and node v in G is the smallest total weight

of edges which if removed would disconnect u from

v. These edges form a minimum cut between u and v

in G. Let c

and f

be the cost and fraction of edge

e, respectively. A path is purchased if and only if its

An Algorithmic Approach to Online Multi-Facility Location Problems

edges are purchased.

Random Process. The algorithm makes its ran-

dom choices, based on α, the minimum among

log(kn + 1)

independently chosen random vari-

ables, distributed uniformly in the interval [0,1].

Next, we describe how the algorithm reacts upon the

arrival of a new client.

Input: G = (V,E) and client node i ∈ D

Output: Set of edges purchased

Make a copy G

of G;

As long as there are < k disjoint paths purchased

between r and i in G, do the following:

1. While the maximum ﬂow between r and i in G

less than 1, construct a minimum cut Q between r

and i in G

; for each edge e ∈ Q, make the follow-

ing fraction increase:

= f

· (1 + 1/c

) +

· c

2. Purchase each edge e with f

> α.

3. If there is no purchased path between r and i in

, ﬁnd a minimum-cost such path and purchase

it.

4. Refer to all facilities whose corresponding edges

were purchased as open; delete from G

the pur-

chased edges between i and each open facility.

2.3 Competitive Analysis

The algorithm buys edges in the second and third

steps. In the second step, its choices are made based

on the random process, whereas in the third step, its

choices are made to guarantee a feasible solution. We

measure the expected cost of each separately. Let Opt

be the cost of the optimal ofﬂine solution and let f rac

be the cost of the fractional solution constructed by

the algorithm in the ﬁrst step.

Choices Based on Random Process: Let S

be the

set of edges purchased in the second step of the al-

gorithm and let C

be its expected cost. These edges

are purchased by the algorithm based on the random

process described earlier. Let us ﬁx an l : 1 ≤ l ≤

log(kn + 1)

and an edge e. We denote by X

e,l

the

indicator variable of the event that e is chosen by the

algorithm based on the random choice of l.

∑

e∈S

log(kn+1)

∑

l=1

· Exp [X

e,l

]

= 2

log(kn + 1)

∑

e∈S

(1)

Notice that

∑

e∈S

is upper bounded by the cost

of the fractional solution. The latter can be measured

against the optimal ofﬂine solution, as follows. The

idea here is that every time the algorithm performs a

fraction increase, it does not exceed 2. Moreover, the

total number of fraction increases can be measured in

terms of the cost of the optimal ofﬂine solution.

The fraction increase contributed by each edge e

in a minimum cut Q is



·c



. The algorithm

would make a fraction increase only if the maximum

ﬂow is less than 1. This means we have that

∑

e∈Q

1 before a fraction increase. Therefore, each fraction

increase does not exceed:

∑

e∈Q



· c



< 2 (2)

As long as the algorithm hasn’t purchased at least

k disjoint paths between r and a given client i, it enters

the loop that starts by constructing a maximum ﬂow

on the graph G

. Notice how G

shrinks over time, as

the algorithm purchases the paths.

Lemma 1. Whenever the algorithm makes a fraction

increase, G

contains at least one path that is also in

the optimal ofﬂine solution.

Proof. To see why this holds, ﬁx any time t before a

fraction increase. Let s < k be the number of disjoint

paths purchased by the algorithm at time t. G

at time

t must contain at least one optimal path since G

constructed by removing s (less than k) feasible paths

from G and the optimal ofﬂine solution contains at

least k disjoint paths in G.

Finally, the algorithm would have an edge e from

the optimal ofﬂine solution in every minimum cut Q

of G

, since Q must contain an edge from each path,

by deﬁnition. Based on the equation for the fraction

increase, after O(log |Q|) fraction increases, the frac-

tion f

of e becomes 1, and no further increases can

be made, as e will not be in any future minimum cut.

The size of any minimum cut is upper bounded by

m, the number of facilities or the maximum available

paths between r and client i. We can now bound the

fractional solution:

f rac ≤ O(log m · Opt) (3)

Combining Equations 1, 2, and 3 imply an upper

bound on the expected cost C

of the edges bought in

the second step of the algorithm:

≤ O(log(kn)logm · Opt) (4)

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

Choices to Guarantee Feasible Solution: Let S

be the set of edges purchased in the third step of the al-

gorithm and let C

be its expected cost. These edges

are purchased by the algorithm only if a path has not

been bought by the random process in the second step.

Every time the algorithm purchases a path in this step,

its cost does not exceed Opt since the algorithm buys

the minimum-cost path in G

, and as we showed ear-

lier in Lemma 1, G

contains at least one path that is

also in the optimal solution.

• (one client, one path) We start with calculating the

expected cost incurred by a single client for pur-

chasing a single path. Fix a client i. Let Q

j+1

a minimum cut of G

constructed after the algo-

rithm has purchased j < k disjoint paths between

r and i and has completed the ﬁrst step. The prob-

ability of purchasing the ( j + 1)

path for a single

1 ≤ l ≤ 2

log(kn + 1)

is:

∏

e∈Q

j+1

(1 − f

) ≤ e

−

∑

e∈Q

j+1

≤ 1/e

Notice that the last inequality holds because the

algorithm ensures that

∑

e∈Q

j+1

≥ 1 at the end

of the ﬁrst step (Max-ﬂow min-cut theorem). The

expected cost of purchasing the ( j + 1)

path for

all 1 ≤ l ≤ 2

log(kn + 1)

, is less than 1/(kn)

Opt.

• (one client, k paths) The expected cost of purchas-

ing all k paths is the sum of the expected costs for

each path and is less than k · 1/(kn)

· Opt.

• (total cost of all clients) The total expected cost

incurred by all n

clients that arrived is less than:

· k · 1/(kn)

· Opt ≤ n · k · 1/(kn)

· Opt

= 1/kn · Opt

Therefore, the expected cost C

of the edges

bought in the third step of the algorithm is:

≤ 1/kn · Opt (5)

Equations 4 and 5 yield to the following theorem.

Theorem 1. There is an online O(log(kn)logm)-

competitive randomized algorithm for the Online

Non-metric Multi-Facility Location, where m is the

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

is the number of required connections.

3 ONLINE METRIC

MULTI-FACILITY LOCATION

In this section, we start with an overview of works

related to OMMFL and some preliminaries. Then we

present an online randomized algorithm for OMMFL

and analyze its competitive ratio.

While this problem has been intensively studied

in the ofﬂine setting (Byrka et al., 2010; Yan and

Chrobak, 2011), we are not aware of any online al-

gorithm for it.

3.1 Preliminaries & Related Work

OMMFL is a generalization of metric Online Facility

Location (OFL) (Fotakis, 2007; Fotakis, 2008; Mey-

erson, 2001) with k = 1. Meyerson (Meyerson, 2001)

introduced metric OFL and proposed an O(logn)-

competitive randomized algorithm, where n is the

number of clients. Fotakis (Fotakis, 2008) showed

that no randomized online algorithm can achieve

a competitive ratio better that Ω(

logn

loglog n

) against

an oblivious adversary and gave a deterministic al-

gorithm with asymptotically matching O(

logn

loglog n

competitive ratio. In another work later, he proposed

a primal-dual deterministic algorithm with O(log n)-

competitive ratio, that was simpler to formulate, ana-

lyze, and implement (Fotakis, 2007). The competitive

ratio we achieve for OMMFL is based on running the

deterministic algorithm of Fotakis (Fotakis, 2008) for

metric OFL.

The lower bound on the competitive ratio of

metric OFL by Fotakis (Fotakis, 2008) implies an

Ω(

logn

loglog n

) lower bound on the competitive ratio of

OMMFL.

3.2 Online Algorithm

Let A

OFL

be any online (deterministic or random-

ized) algorithm for metric Online Facility Location

(OFL), with competitive ratio r. Given an instance

I of OMMFL with positive integer k. Client i arrives.

Our algorithm needs to connect i to k different open

facilities.

1. The algorithm starts by running A

OFL

on instance

I, where k = 1. This results in opening some fa-

cilities and connecting i to one open facility.

2. If i is the ﬁrst client, we open the cheapest k −1 fa-

cilities other than the one i is connected to. From

this point on, there are at least k open facilities.

Until A

OFL

opens these facilities itself, these re-

main closed with respect to A

OFL

3. The algorithm will then connect i to any other k −

1 open facilities.

An Algorithmic Approach to Online Multi-Facility Location Problems

3.3 Competitive Analysis

Let I be an instance of OMMFL with positive integer

k. Let I

be the same instance as I except for k = 1.

Let Opt and Opt

be the cost of the optimal solution

for I and that for I

, respectively. Let C and C

be the

cost of our algorithm for I and that of A

OFL

for I

respectively. We denote by C

f ac

and C

con

the costs in-

curred by our algorithm to open facilities and to con-

nect clients, respectively. We denote by C

f ac

and C

con

the costs incurred by A

OFL

to open facilities and to

connect clients, respectively.

The algorithm opens the cheapest k − 1 facilities

other than the ones opened by A

OFL

. Let f

max

be the

maximum facility cost and f

min

the minimum facility

cost. Thus, we have that:

f ac

≤ C

f ac

+ f

max

· (k − 1) (6)

Given any client i. Apart from its connecting cost

incurred by A

OFL

, our algorithm connects i to k − 1

other facilities, each resulting in a connecting cost at

most

max

min

· c

, where c

max

and c

min

are the maximum

and minimum connecting costs, respectively. This

implies an overall connecting cost:

con

≤ C

con

· (1 +

max

min

(k − 1)) (7)

We now add the two equations above and do some

algebraic manipulations by using:

con

≤ C

f ac

≤ C

≥ c

min

≥ f

min

This yields to:

C ≤ C

· (2 +

max

min

(k − 1) +

max

min

(k − 1)) (8)

Recall that A

OFL

is r-competitive and so C

≤ r ·

Opt

. Since Opt

≤ Opt, the theorem below follows.

Theorem 2. Given an online (deterministic or ran-

domized) r-competitive algorithm for metric Online

Facility Location. Then there is an online

O(max{

max

min

max

min

} · k · r)-competitive algorithm for

the Online Metric Multi-Facility Location, where k is

the number of required connections; c

max

and c

min

are

the maximum and minimum connecting costs, respec-

tively; f

max

and f

min

are the maximum and minimum

facility costs, respectively.

Running the deterministic algorithm of Fo-

takis (Fotakis, 2008) for metric OFL, with O(

logn

loglog n

competitive ratio, results in the following.

Corollary 1. There is an online O(max{

max

min

max

min

} ·

k ·

logn

loglog n

)-competitive deterministic algorithm for the

Online Metric Multi-Facility Location, where n is the

number of clients; k is the number of required connec-

tions; c

max

and c

min

are the maximum and minimum

connecting costs, respectively; f

max

and f

min

are the

maximum and minimum facility costs, respectively.

4 CONCLUDING REMARKS &

FUTURE WORK

In this paper, we have assumed there is a unique pos-

itive integer k for all clients. In many application

scenarios, it is likely that clients have different num-

ber of required connections. A slight modiﬁcation

in our algorithms would yield to O(log(k

max

n)logm)

and O(max{

max

min

max

min

} · k

max

logn

loglog n

) competitive ra-

tios for ONMFL and OMMFL, respectively, where

max

is the maximum required connections. One re-

search direction is to target better competitive ratios

for these variants.

This brings us to the next question, for Online

Metric Multi-Facility Location (OMMFL), whether it

is possible to get rid of the parameters c

max

, c

min

, f

max

and f

min

from the competitive ratio or achieve lower

bounds in terms of these parameters. To achieve the

former, one may want to attempt a primal-dual ap-

proach, for instance, by trying to extend the algorithm

of Fotakis (Fotakis, 2007) for metric Online Facility

Location.

Finally, in our model here, demands and their ar-

rival order are given by an oblivious adversary. One

may want to consider other types of adversary for

these sequences, for instance, by exploring various

probability distributions.

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