Algorithmic View of Online Prize-collecting Optimization Problems

Christine Markarian

1

and Abdul Nasser El-Kassar

2

1

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

2

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

Keywords:

Prize-collecting, Facility Location Planning, Decision Making, Online Algorithms, Competitive Analysis.

Abstract:

Online algorithms have been a cornerstone of research in network design problems. Unlike in classical ofﬂine

algorithms, the input to an online algorithm is revealed in portions over time and the online algorithm reacts

to each portion while targeting a given optimization goal. Online algorithms are deployed in real-world opti-

mization problems in which provably good decisions are expected in the present without knowing the future.

In this paper, we consider a well-established branch of online optimization problems, known as online prize-

collecting problems, in which the online algorithm may reject some input portions at the cost of paying an

associated penalty. These appear in business applications in which a company decides to lose some customers

by paying an associated penalty. Particularly, we study the online prize-collecting variants of three well-known

optimization problems: Connected Dominating Set, Vertex Cover, and Non-metric Facility Location, namely,

Online Prize-collecting Connected Dominating Set (OPC-CDS), Online Prize-collecting Vertex Cover (OPC-

VC), and Online Prize-collecting Non-metric Facility Location (OPC-NFL), respectively. We propose the ﬁrst

online algorithms for these variants and evaluate them using competitive analysis, the standard framework to

measure online algorithms, in which the online algorithm is measured against the optimal ofﬂine algorithm

that knows the entire input sequence in advance and is optimal.

1 INTRODUCTION

Many real-world optimization problems are online

in nature, requiring smart decisions to be made im-

mediately, even when future information is not fully

known. The challenge is then to make decisions that

will cause as few regrets as possible in the future.

Classically, most optimization problems have been

modeled in an ofﬂine setting, in which the entire in-

put sequence is assumed to be known to the decision

maker in advance. In many real-world scenarios, hav-

ing access to future input sequences is most of the

time hard or even impossible. Over the past decades,

online algorithms have gained a lot of popularity in

such scenarios, since they can provide performance

guarantee to the decision maker. That is, decisions

made by online algorithms are measured against op-

timal decisions that are made by the assumption of

knowing the entire input sequence in advance. Hence,

a decision maker would know how good or bad the

decision is at the time it is made. Moreover, it would

allow decision makers to know and try to achieve the

best provably possible decision achievable by any de-

cision maker. Unlike in classical ofﬂine models, the

input to an online algorithm is revealed in portions

over time and the online algorithm reacts to each por-

tion, as it targets a given optimization goal against the

entire input sequence.

We adopt the competitive analysis framework to

evaluate the online algorithms (Sleator and Tarjan,

1985). In this framework, the performance of the

online algorithm is measured against the optimal of-

ﬂine algorithm, that knows the entire input sequence

in advance and is optimal, in the worst case. Given

an input sequence σ, let C

A

(σ) and C

OPT

(σ) de-

note the cost incurred by an online algorithm A, pos-

sibly randomized, and an optimal ofﬂine algorithm

OPT , respectively. A has competitive ratio c or is

c-competitive if there exists a constant α such that

C

A

(σ) ≤ c · C

OPT

(σ) + α for all input sequences σ.

We assume the oblivious adversarial model, in which

the adversary speciﬁes all of the input at the begin-

ning and does not know the random outcomes of the

algorithm.

In this paper, we consider a well-established

branch of online optimization problems known as on-

line prize-collecting problems, in which the online al-

gorithm may reject some input portions at the cost of

paying an associated penalty. These appear in net-

work planning applications for service providers, in

744

Markarian, C. and El-Kassar, A.

Algorithmic View of Online Prize-collecting Optimization Problems.

DOI: 10.5220/0010471507440751

In Proceedings of the 23rd International Conference on Enterprise Information Systems (ICEIS 2021) - Volume 1, pages 744-751

ISBN: 978-989-758-509-8

Copyright

c

2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

which a company may decide to lose some customers

by paying a corresponding penalty in the revenue.

Since (Qian and Williamson, 2011) introduced the

online prize collecting framework, many graph opti-

mization problems, such as variants of Steiner prob-

lems and metric Facility Location, were deﬁned in

this model (see (Felice et al., 2015; Hajiaghayi et al.,

2013; Markarian, 2018)). It is worth noting that the

online prize-collecting model is a generalization of

the online model in which all penalties are set to in-

ﬁnity.

2 OUR CONTRIBUTION

In this paper, we study the online prize-collecting

variants of three classical optimization problems:

Connected Dominating Set, Vertex Cover, and Non-

metric Facility Location.

2.1 Online Prize-collecting Connected

Dominating Set

Consider an advertising company trying to reach po-

tential customers in a social network for the purpose

of advertising a certain product or service. The com-

pany assumes a customer is reached either if he has

direct access to the ad or one of his friends has. Ev-

ery now and then, a number of potential customers are

announced. These are people who would likely be in-

terested in the ad. The point is to send the ad to as few

people as possible so as to reduce costs, while reach-

ing as many people as possible. To achieve this, the

company wishes to ﬁnd a group of connected people

who can spread the message to each other and to the

rest of the potential customers through the word-of-

mouth effect. The company may ignore some of the

potential customers, by paying an associated penalty.

The goal is to minimize the total costs of sending ads

and penalties.

From an algorithmic perspective, this scenario can

be formalized as the Online Prize-collecting Con-

nected Dominating Set problem (OPC-CDS). OPC-

CDS is the online prize-collecting variant of Con-

nected Dominating Set. It generalizes the Online

Set Cover problem (OSC) introduced by (Alon et al.,

2003), deﬁned as follows.

Deﬁnition 1. (OSC) Given a universe U of elements

and a collection S of subsets of U, each associated

with a cost. A subset D ⊆ U of elements arrives over

time and OSC asks to ﬁnd a minimum cost of subsets

C ⊆ S that cover all elements in D.

(Korman, 2005) gave a lower bound of Ω(log mlogn)

on the competitive ratio of any online polynomial-

time randomized algorithm for OSC, under the as-

sumption that NP 6⊆ BPP, where m is the number of

subsets and n is the number of elements. This implies

a lower bound of Ω(log

2

n) on the competitive ratio of

any randomized polynomial-time algorithm for OPC-

CDS, where n is the number of nodes. OPC-CDS is

deﬁned as follows.

Deﬁnition 2. (OPC-CDS) Given an undirected con-

nected graph G = (V, E) with |V | = n, node-weight

function w : V → R

+

, and penalty-cost function p :

V → R

+

. A sequence of disjoint subsets of V arrives

over time. A subset S ⊆ V serves as a connected dom-

inating set of a given subset D ⊆ V if every node in

D is either in S or has an adjacent node in S, and the

subgraph induced by S is connected in G. In each step

t, a subset D

t

⊆ V arrives; for each u ∈ D

t

, OPC-CDS

asks to either pay the penalty p

u

of u or add u to a sub-

set D

0

t

⊆ D

t

that is served by a connected dominating

set, at time t. The goal is to minimize the total weight

of the connected dominating set constructed and the

total penalties paid.

To the best of our knowledge, no online algo-

rithm with non-trivial competitive ratio exists for

OPC-CDS. A special case of OPC-CDS, in the online

model, namely, the Online Connected Dominating Set

problem (OCDS), was introduced by (Hamann et al.,

2018), in the context of modern robotic warehouses.

Unlike in this paper, (Hamann et al., 2018) studied a

special case in which all node-weights are uniform.

Hence, their approach cannot be applied to our prob-

lem.

In this paper, we propose the ﬁrst online algorithm

for OPC-CDS, with O(

w

max

w

min

log

2

n)-competitive ratio,

where n is the number of nodes, w

max

is the maximum

node weight, and w

min

is the minimum node weight.

Our algorithm is randomized and makes use of the

deterministic algorithm of (Alon et al., 2003) for the

Online Set Cover problem (OSC), deﬁned earlier, and

the randomized algorithm of (Hajiaghayi et al., 2014)

for the Online Node-weighted Steiner Tree problem

(OPC-NWST).

2.2 Online Prize-collecting Vertex

Cover

Consider the advertising company described earlier.

For some ads, the company wishes to assure that the

ad is reached to the customers without relying on the

word-of-mouth effect. This means, it assumes a cus-

tomer is inﬂuenced only through close friends. Every

now and then, a number of friendships are announced.

Algorithmic View of Online Prize-collecting Optimization Problems

745

These are pairs of people who would likely inﬂuence

each other. If one of them receives the ad, the other

is assumed to have been reached. Moreover, the com-

pany may wish, as before, to ignore some friendships,

at the cost of paying a penalty. The goal is to mini-

mize the total costs of sending ads and penalties.

This scenario can be modeled as the Online Prize-

collecting Vertex Cover problem (OPC-VC). OPC-

VC is the online prize-collecting variant of Vertex

Cover and is deﬁned as follows.

Deﬁnition 3. (OPC-VC) Given an undirected graph

G = (V, E) with |V | = n and node-weight function w :

V → R

+

. A sequence of edges, each associated with

a penalty, arrives over time. In each step, an edge

arrives; OPC-VC asks to output a set S of nodes, such

that each edge has at least one of its endpoints in S or

its penalty is paid, at the current step. The goal is to

minimize the total weight of S and the total penalties

paid.

To the best of our knowledge, no online algorithm

with non-trivial competitive ratio exists for OPC-VC.

A special case of OPC-VC is the Online Vertex Cover

problem (OVC). For the unweighted variant of OVC,

in which all node weights are uniform, there is a sim-

ple greedy algorithm with 2-competitive ratio. As

soon as an edge arrives, if it is not covered, this al-

gorithm adds both of its endpoints to the solution.

In this paper, we propose the ﬁrst online algorithm

for OPC-VC, with 3-competitive ratio. Our algorithm

is deterministic and is based on a simple classical

primal-dual approach.

2.3 Online Prize-collecting Non-metric

Facility Location

Assuming the word-of-mouth effect, the advertising

company now wishes to send the ad to very few peo-

ple so as every potential customer announced is not

very far from one of these people. As before, some

potential customers may be ignored at the cost of

paying a penalty. The goal is to minimize the total

costs of sending ads, penalties, and the total distances

from the potential customers to the nearest people

who have received the ad.

This scenario can be formulated as the Online

Prize-collecting Non-metric Facility Location prob-

lem (OPC-NFL). OPC-NFL is the online prize-

collecting variant of Non-metric Facility Location and

is deﬁned as follows.

Deﬁnition 4. (OPC-NFL) Given a complete bipartite

graph G = ((F ∪D), E), where F is the set of facilities

that may be opened and D is the set of clients arriv-

ing over time, an edge-weight function w : E → R

+

,

a facility-opening-cost function f : F → R

+

, and a

penalty-cost function p : D → R

+

. To connect client

i ∈ D to facility j, the weight w

(i, j)

of edge (i, j) is to

be paid, and to open facility j ∈ F, the opening facil-

ity cost f

j

is to be paid. In each step t, a client i ∈ D

arrives; OPC-NFL asks to either pay the penalty as-

sociated with i or connect i to an open facility. The

goal is to minimize the total penalties, the total facil-

ity opening costs, and the total connecting costs paid.

To the best of our knowledge, no online algorithm

with non-trivial competitive ratio exists for OPC-

NFL. There only is an online algorithm for the metric

version, in which facilities and clients reside in a met-

ric space and all distances respect the triangle inequal-

ity, as in (Felice et al., 2015). The latter is essential

to prove the competitive ratio of the algorithm and so

the result does not carry over to OPC-NFL.

OPC-NFL generalizes the Online Set Cover prob-

lem (OSC), due to (Alon et al., 2003), and this im-

plies an Ω(logm logn) lower bound on the competi-

tive ratio of any online randomized polynomial-time

algorithm for OPC-NFL, where m is the number of

facilities and n is the number of clients.

In this paper, we propose the ﬁrst online al-

gorithm for OPC-NFL, with asymptotically optimal

O(logm logn)-competitive ratio, where m is the num-

ber of facilities and n is the number of clients. Our al-

gorithm is randomized and is based on reducing OPC-

NFL to the Online Non-metric Facility Location prob-

lem (ONFL), due to (Alon et al., 2006).

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

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

related to our problems. In Sections 4, 5, and 6,

we present our results for OPC-CDS, OPC-VC, and

OPC-NFL, respectively. We conclude with a discus-

sion of our results and open problems in Section 7.

3 STATE-OF-THE-ART

(Qian and Williamson, 2011) initiated the study of

online prize-collecting Steiner problems by provid-

ing an O(logn)-competitive algorithm for the On-

line Prize-collecting Steiner Tree problem (OPC-ST).

(Hajiaghayi et al., 2014) proposed an online algorithm

with the same competitive ratio but gave a simpler

analysis. (Hajiaghayi et al., 2014) proposed a generic

technique that reduces online prize-collecting Steiner

problems to their corresponding fractional non-prize-

collecting variants, by losing logarithmic factor in

the competitive ratio. This has implied O(log

3

n)-

competitive and O(log

4

n)-competitive randomized

ICEIS 2021 - 23rd International Conference on Enterprise Information Systems

746

algorithms for the Online Prize-collecting Node-

weighted Steiner Tree problem (OPC-NWST) and the

Online Prize-collecting Node-weighted Steiner For-

est problem (OPC-NWSF), respectively. (Hajiaghayi

et al., 2013) gave a primal-dual algorithm with opti-

mal O(log n)-competitive ratio for ONWST in graphs

excluding a ﬁxed graph as minor (which include pla-

nar graphs). (Hajiaghayi et al., 2014) extended this

result to OPC-NWST in graphs excluding a ﬁxed

graph as minor, for which they gave an O(log

2

n)-

competitive algorithm. Other well-known optimiza-

tion problems were also studied in the online prize-

collecting model, such as (Ausiello et al., 2008).

(Hamann et al., 2018) proposed an online ran-

domized algorithm for a special case of OPC-CDS

in which all penalties are set to inﬁnity, the Online

Connected Dominating Set problem (OCDS). They

showed that their algorithm has asymptotically opti-

mal O(log

2

n)-competitive ratio, where n is the num-

ber of nodes. (Markarian and Kassar, 2020) later pro-

posed an online deterministic algorithm for the prob-

lem with the same O(log

2

n)-competitive ratio.

(JunFeng and JianHua, 2014) gave an optimal 2-

approximation algorithm for the prize-collecting vari-

ant of Vertex Cover (in the ofﬂine setting). (Demange

and Paschos, 2005) studied an online model of Vertex

Cover, that is substantially different than the one in

this paper, providing competitive ratios characterized

by the maximum degree of the graph.

(Alon et al., 2006) proposed an online randomized

algorithm for a special case of OPC-NFL in which

all penalties are set to inﬁnity, the Online Non-metric

Facility Location problem (ONFL). They showed that

their algorithm has O(log mlog n)-competitive ratio,

where m is the number of facilities and n is the

number of clients. The latter is asymptotically op-

timal since ONFL generalizes OSC. A related prob-

lem to ONFL is the metric variant, known as the On-

line Facility Location problem (OFL), in which fa-

cilities and clients reside in a metric space and all

distances respect the triangle inequality. The latter

is commonly used in competitive analysis. (Meyer-

son, 2001) gave an O(log n)-competitive randomized

algorithm for OFL, where n is the number of clients.

This was improved to an O(

logn

loglog n

)-competitive al-

gorithm by (Fotakis, 2008), who showed that this is

the best competitive ratio achievable for the problem.

Another paper of (Fotakis, 2007) gave a simple deter-

ministic primal-dual O(log n)-competitive algorithm

for the problem. The prize-collecting metric variant,

the Online Prize-collecting Facility Location prob-

lem was studied by (Felice et al., 2015), who pro-

posed an O(log n)-competitive algorithm for the prob-

lem. Their algorithm is based on previous algorithms

of (Fotakis, 2008) and (Nagarajan and Williamson,

2013).

4 ONLINE PRIZE-COLLECTING

CONNECTED DOMINATING

SET (OPC-CDS)

In this section, we present a randomized online algo-

rithm for OPC-CDS and analyze its competitive ratio.

4.1 Online Algorithm

The algorithm has two phases. In the ﬁrst phase, we

transform the given instance I into an OSC instance I

0

as follows.

Given an instance I of OPC-CDS containing a

connected graph G = (V, E), a penalty cost function

p : V → R

+

, and a sequence of disjoint subsets of V

arriving over time. We construct an instance I

0

of OSC

as follows. The elements of I

0

are the nodes of V .

Each node u ∈ V is represented by two sets:

− a set containing u and all nodes adjacent to u, with

cost w

u

, the weight associated with u

− a set containing u, with cost p

u

, the penalty asso-

ciated with u

When a subset D

t

⊆ V arrives at step t, the algorithm

returns the set P

t

containing the nodes of D

t

whose

penalties are paid for and the set CDS

t

that contains

a connected dominating set of the remaining nodes

D

t

\ P

t

. Now, the algorithm runs the algorithm for

OSC, due to (Alon et al., 2003), on I

0

and adds the

corresponding nodes to the sets P

t

and CDS

t

based on

the sets returned by the algorithm. Note that a node

may end up covered by more than one set, meaning

that its penalty might be paid for in addition to being

dominated.

Note that, any OSC solution for I

0

of cost c is a

solution of the same cost c for Phase 1 of the algo-

rithm. Moreover, an O(logm logn)-competitive algo-

rithm for OSC implies an O(log

2

n)-competitive al-

gorithm for Phase 1 of the algorithm, since the num-

ber of sets in I

0

is double the number of nodes in I

(m = 2n).

In the second phase, we connect the dominating

set nodes constructed in the ﬁrst phase directly. We

run the randomized algorithm for the Online Node-

weighted Steiner Tree problem (OPC-NWST) due to

(Hajiaghayi et al., 2014) on these nodes. The two

phases of the algorithm are depicted below.

Algorithmic View of Online Prize-collecting Optimization Problems

747

Online Algorithm for OPC-CDS.

Input: G = (V, E) and subset D

t

⊆ V

Output: P

t

∪CDS

t

1. Run the OSC algorithm on I

0

. Add the nodes

whose penalties are paid for to P

t

and the domi-

nating set nodes to a set S

t

. If t = 1, assign any of

the nodes in S

t

as a root node r. Add all the nodes

in S

t

to CDS

t

.

2. Run the OPC-NWST algorithm to construct a tree

that connects all the nodes in S

t

to r. Add all the

nodes of this tree, that are not already in CDS

t

, to

CDS

t

.

4.2 Competitive Analysis

We denote by Opt the cost of an optimal solution Opt

I

for an instance I of OPC-CDS and by C

1

and C

2

the

cost of the two phases of the algorithm, respectively.

Phase 1. The cost C

1

of Phase 1 of the algorithm

can be bounded as follows.

C

1

≤ O(log

2

n) · Opt

Phase 2. The cost C

2

of Phase 2 of the algorithm

is the cost of the Steiner tree nodes connecting the

dominating set nodes constructed in the ﬁrst phase.

Let Opt

St

be the cost of a minimum Steiner tree of

these nodes. Since the algorithm for OPC-NWST,

due to (Hajiaghayi et al., 2014), has an O(log

2

n)-

competitive ratio, we have that C

2

≤ O(log

2

n)·Opt

St

.

It remains to compare Opt

St

to the cost of the opti-

mal solution Opt. The latter is not a feasible solution

for Phase 2 of the algorithm. This would have been

the case in the ofﬂine setting.

Remark. In the ofﬂine setting, algorithms that ﬁrst

ﬁnd a dominating set and then run a Steiner tree algo-

rithm to connect the dominating set, have a straight-

forward approximation analysis, depending on the

analysis of the Steiner tree and Dominating Set al-

gorithms themselves. This means that the approxi-

mation bounds attained, in such algorithms, depend

on the approximation bounds for Dominating Set/Set

Cover and Steiner Tree (see (Guha and Khuller,

1998)).

To compare Opt

St

to Opt, we construct a Steiner

tree S for the nodes of Phase 1 of the algorithm, as fol-

lows. S will contain the nodes in the optimal solution

and some additional nodes. We add to S:

− all the nodes in the optimal solution

min

∑

i∈V

x

i

w

i

+

∑

e∈E

p

e

y

e

Subj to: ∀e = (i, j) ∈ E : x

i

+ x

j

+ y

e

≥ 1

∀i ∈ V, e ∈ E : x

i

, y

e

≥ 0 max

∑

e∈E

z

e

Subj to: ∀i ∈ V :

∑

e∈γ(i)

z

e

≤ w

i

∀e ∈ E : z

e

≤ p

e

∀e ∈ E : z

e

≥ 0

Figure 1: LP Formulation of OPC-VC.

− all the nodes added in Phase 2 of the algorithm, in

addition to the nodes added in Phase 1 (these are

the terminals and so have weight 0 each)

− one additional node from the demand set D

t

, for

any t (this has weight at most w

max

)

The cost of S is upper bounded by: Opt +C

2

+ w

max

.

Thus, Opt

St

is at most Opt +C

2

+ w

max

. Therefore,

C

2

≤ O(logn) · (Opt +C

2

+ w

max

).

Applying asymptotic notation with simple algebra

and using the fact that Opt is at least w

min

, we con-

clude that

C

2

≤ O(logn) ·

w

min

w

max

By adding the two costs C

1

and C

2

of the algorithm,

we conclude the following theorem.

Theorem 1. There is an online O(

w

max

w

min

log

2

n)-

competitive randomized algorithm for the Online

Prize-collecting Connected Dominating Set problem

(OPC-CDS), where n is the number of nodes, w

max

is

the maximum node weight, and w

min

is the minimum

node weight.

5 ONLINE PRIZE-COLLECTING

VERTEX COVER (OPC-VC)

In this section, we present a deterministic online algo-

rithm for OPC-VC and analyze its competitive ratio.

5.1 Online Algorithm

The algorithm is a classical primal-dual algorithm.

The LP formulation of OPC-VC is depicted in Fig-

ure 1. x

i

is the indicator variable set to 1 if node i of

weight w

i

belongs to the solution, and set to 0 other-

wise. y

e

is the indicator variable set to 1 if the penalty

p

e

of edge e is paid, and set to 0 otherwise. γ(i) is the

set of edges incident to node i.

The primal-dual algorithm is depicted below. Let

S be the set of all edges whose penalties are paid for

and all nodes that are purchased by the algorithm.

ICEIS 2021 - 23rd International Conference on Enterprise Information Systems

748

Online Algorithm for OPC-VC.

Input: G = (V, E) and e ∈ E

Output: S

1. Increase the dual variable z

e

of e until one of the

dual constraints becomes tight.

2. Set the primal variable corresponding to each tight

constraint to 1.

3. Purchase each node corresponding to a tight con-

straint and pay each penalty corresponding to a

tight constraint.

5.2 Competitive Analysis

Let S be the primal solution constructed by the al-

gorithm. Recall that S is the set of all edges whose

penalties are paid for and all nodes that are purchased

by the algorithm. We have that the dual constraint cor-

responding to each node i ∈ S is tight: w

i

=

∑

e∈γ(i)

z

e

.

Moreover, the dual constraint corresponding to each

e ∈ S is tight: p

e

= z

e

. Thus,

∑

e∈S

p

e

y

e

≤

∑

e∈E

z

e

and

∑

i∈S

x

i

w

i

=

∑

i∈S

∑

e∈γ(i)

z

e

≤ 2 ·

∑

e∈E

z

e

.

By the Weak Duality theorem, we have that

∑

e∈E

z

e

≤ Opt, where Opt is the cost of the optimal

solution, and hence the theorem follows.

Theorem 2. There is an online 3-competitive de-

terministic algorithm for the Online Prize-collecting

Vertex Cover problem (OPC-VC).

6 ONLINE PRIZE-COLLECTING

NON-METRIC FACILITY

LOCATION (OPC-NFL)

In this section, we present a randomized online algo-

rithm for OPC-NFL and analyze its competitive ratio.

6.1 Online Algorithm

Given an instance I of OPC-NFL that contains a

complete bipartite graph G = ((F ∪ D), E), an edge-

weight function w : E → R

+

, a facility-opening-cost

function f : F → R

+

, and a penalty-cost function

p : D → R

+

. The algorithm is based on transforming

I into an instance I

0

of the Online Non-metric Facility

Locatiom problem (ONFL), as follows.

− We add to the set F, a facility j and set its opening

cost to 0.

− For each client i ∈ D that arrives, we add an edge

from i to j and set its weight to the penalty cost of

i.

The algorithm is depicted below.

Online Algorithm for OPC-NFL.

Input: G = ((F ∪ D), E) and instance I of OPC-NFL

Output: Set of penalties, facility costs, and connect-

ing costs paid

1. Transform I into I

0

, as described earlier.

2. Run the algorithm for ONFL on I

0

.

3. Purchase all facilities and edges outputted by the

ONFL algorithm. For each arriving client i, pay

its associated penalty if the corresponding edge in

I

0

is purchased by the ONFL algorithm.

6.2 Competitive Analysis

Let I be the original instance of OPC-NFL. Let Opt

be an optimal solution of I and let C

Opt

be its cost.

Let I

0

be the new instance of ONFL generated from I

as above. Let Opt’ be an optimal solution of I

0

and let

C

Opt

0

be its cost.

We need to show that Opt is a feasible solution

of I

0

: Given a client i, whenever its penalty is pur-

chased in Opt, we purchase the corresponding edge

in I

0

; whenever a facility is opened in Opt, we open

it too in I

0

and whenever an edge is paid for, we pay

for it too in I

0

. This means that every time a client

arrives, it is connected to at least one facility and the

connecting edge is paid for by the solution Opt. Thus,

Opt is a feasible solution of I

0

. Hence, C

Opt

0

≤ C

Opt

,

since every feasible solution is lower bounded by the

cost of the optimal solution.

The algorithm for ONFL has an O(logm

0

logn

0

)-

competitive ratio, where m

0

is the number of facilities

and n

0

is the number of clients. According to our re-

duction, m

0

= m+1 and n

0

= n, where m is the number

of facilities and n is the number of clients in the orig-

inal instance I.

Let C be the cost of our solution for I. Our solution

is constructed by running the algorithm for ONFL and

thus C ≤ O(log m log n) ·C

Opt

0

≤ O(log m log n) ·C

Opt

and the theorem below follows.

Theorem 3. There is an online asymptotically opti-

mal O(log m log n)-competitive randomized algorithm

for the Online Prize-collecting Non-metric Facility

Location problem, where m is the number of facilities

and n is the number of clients.

Algorithmic View of Online Prize-collecting Optimization Problems

749

7 DISCUSSION AND OPEN

PROBLEMS

In this paper, we have studied OPC-CDS in general

graphs. Connected Dominating Set problems have

been extensively studied in the ofﬂine setting. These

were motivated by real-world network applications in

which geometric graph models were used (see (Mah-

dian and Yan, 2011; Amb

¨

uhl et al., 2006)). One re-

search direction is to extend this study to the online

setting by considering such graphs for OPC-CDS and

its variants. This might yield to competitive ratios

dependent on the properties of the geometric graphs

rather than the number of nodes.

Another set of open problems would generate by

considering other online models. We have made our

study based on the oblivious adversary model. It is

interesting to consider other weaker adversary models

such as stochastic as in (Manshadi et al., 2010), or

random as in (Mahdian and Yan, 2011).

Competitive analysis offers a worst case perfor-

mance evaluation of online algorithms. It would be

interesting to also consider other models. (Cheung,

2016) performed a computational study of various on-

line algorithms for Steiner problems to reveal their

performance in average. It would be interesting to do

the same for our proposed algorithms, as in (Hamann

et al., 2018)), for the Online Connected Dominating

Set problem. Another study by (Angelopoulos, 2019)

based the analysis of online algorithms on additional

parameters of the problem, known as parameterized

analysis of online algorithms. For the Online Node-

weighted Steiner Tree problem, he showed a tight

competitive ratio that depends on the maximum node

weight, minimum node weight, and the number of ter-

minals. Investigating parameterized analysis for our

problems would initiate an interesting study.

REFERENCES

Alon, N., Awerbuch, B., and Azar, Y. (2003). The on-

line set cover problem. In Proceedings of the Thirty-

ﬁfth Annual ACM Symposium on Theory of Comput-

ing, STOC ’03, pages 100–105, New York, NY, USA.

ACM.

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.

Amb

¨

uhl, C., Erlebach, T., Mihal

´

ak, M., and Nunkesser, M.

(2006). Constant-factor approximation for minimum-

weight (connected) dominating sets in unit disk

graphs. In Approximation, Randomization, and Com-

binatorial Optimization. Algorithms and Techniques,

pages 3–14. Springer.

Angelopoulos, S. (2019). Parameterized analysis of the on-

line priority and node-weighted steiner tree problems.

Theory Comput. Syst., 63(6):1413–1447.

Ausiello, G., Bonifaci, V., and Laura, L. (2008). The online

prize-collecting traveling salesman problem. Informa-

tion Processing Letters, 107(6):199 – 204.

Cheung, S. S. (2016). Ofﬂine and online facility location

and network design. PhD thesis, Operations Research

and Information Engineering, Cornell University.

Demange, M. and Paschos, V. T. (2005). Online vertex-

covering. Theoretical Computer Science, 332(1):83 –

108.

Felice, M. C. S., Cheung, S.-S., Lee, O., and Williamson,

D. P. (2015). The online prize-collecting facility loca-

tion problem. Electronic Notes in Discrete Mathemat-

ics, 50:151 – 156. LAGOS’15 – VIII Latin-American

Algorithms, Graphs and Optimization Symposium.

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

uniform facility location. Journal of Discrete Algo-

rithms, 5(1):141 – 148.

Fotakis, D. (2008). On the competitive ratio for online fa-

cility location. Algorithmica, 50(1):1–57.

Guha, S. and Khuller, S. (1998). Approximation algo-

rithms for connected dominating sets. Algorithmica,

20(4):374–387.

Hajiaghayi, M., Liaghat, V., and Panigrahi, D. (2014). Near-

optimal online algorithms for prize-collecting Steiner

problems. In Esparza, J., Fraigniaud, P., Husfeldt, T.,

and Koutsoupias, E., editors, Automata, Languages,

and Programming, pages 576–587, Berlin, Heidel-

berg. Springer Berlin Heidelberg.

Hajiaghayi, M. T., Liaghat, V., and Panigrahi, D. (2013).

Online node-weighted Steiner forest and extensions

via disk paintings. In 54th Annual IEEE Symposium

on Foundations of Computer Science, FOCS 2013, 26-

29 October, 2013, Berkeley, CA, USA, pages 558–567.

Hamann, H., Markarian, C., Meyer auf der Heide, F., and

Wahby, M. (2018). Pick, pack, & survive: Charging

robots in a modern warehouse based on online con-

nected dominating sets. In 9th International Confer-

ence on Fun with Algorithms, FUN 2018, June 13-15,

2018, La Maddalena, Italy, pages 22:1–22:13.

JunFeng, D. and JianHua, T. (2014). A factor 2-

approximation algorithm for the prize-collecting ver-

tex cover problem. Journal of Beijing University

of Chemical Technology (Natural Science Edition),

41(2):120.

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

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

stitute of Science, Israel.

Mahdian, M. and Yan, Q. (2011). Online bipartite matching

with random arrivals: An approach based on strongly

factor-revealing lps. In Proceedings of the Forty-Third

Annual ACM Symposium on Theory of Computing,

STOC ?11, page 597?606, New York, NY, USA. As-

sociation for Computing Machinery.

Manshadi, V. H., Gharan, S. O., and Saberi, A. (2010). On-

line stochastic matching: online actions based on of-

ﬂine statistics. Math. Oper. Res., 37:559–573.

Markarian, C. (2018). An optimal algorithm for online

prize-collecting node-weighted steiner forest. In Com-

ICEIS 2021 - 23rd International Conference on Enterprise Information Systems

750

binatorial Algorithms - 29th International Workshop,

IWOCA 2018, Singapore, July 16-19, 2018, Proceed-

ings, pages 214–223.

Markarian, C. and Kassar, A. (2020). Online deterministic

algorithms for connected dominating set & set cover

leasing problems. In Parlier, G. H., Liberatore, F., and

Demange, M., editors, Proceedings of the 9th Interna-

tional Conference on Operations Research and Enter-

prise Systems, ICORES 2020, Valletta, Malta, Febru-

ary 22-24, 2020, pages 121–128. SCITEPRESS.

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

ceedings of the 42Nd IEEE Symposium on Founda-

tions of Computer Science, FOCS ’01, pages 426 –

431, Washington, DC, USA. IEEE Computer Society.

Nagarajan, C. and Williamson, D. P. (2013). Ofﬂine and on-

line facility leasing. Discrete Optimization, 10(4):361

– 370.

Qian, J. and Williamson, D. P. (2011). An O(logn)-

competitive algorithm for online constrained forest

problems. In Automata, Languages and Program-

ming - 38th International Colloquium, ICALP 2011,

Zurich, Switzerland, July 4-8, 2011, Proceedings, Part

I, pages 37–48.

Sleator, D. D. and Tarjan, R. E. (1985). Amortized efﬁ-

ciency of list update and paging rules. Commun. ACM,

28(2):202–208.

Algorithmic View of Online Prize-collecting Optimization Problems

751