Online Deterministic Algorithms for Connected Dominating Set & Set

Cover Leasing Problems

Christine Markarian

1

and Abdul-Nasser Kassar

2

1

Department of Mathematical Sciences, Haigazian University, Beirut, Lebanon

2

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

Keywords:

Connected Dominating Sets, Set Cover, Leasing, Online Algorithms, Competitive Analysis.

Abstract:

Connected Dominating Set (CDS) and Set Cover (SC) are classical optimization problems that have been

widely studied in both theory and practice, as many variants and in different settings, motivated by appli-

cations in wireless and social networks. In this paper, we consider the online setting, in which the input

sequence arrives in portions over time and the so-called online algorithm needs to react to each portion. On-

line algorithms are measured using the notion of competitive analysis. An online algorithm A is said to have

competitive ratio r, where r is the worst-case ratio, over all possible instances of a given minimization prob-

lem, of the solution constructed by A to the solution constructed by an ofﬂine optimal algorithm that knows

the entire input sequence in advance. Online Connected Dominating Set (OCDS) (Hamann et al., 2018) is an

online variant of CDS that is currently solved by a randomized online algorithm with optimal competitive ra-

tio. We present in this paper the ﬁrst deterministic online algorithm for OCDS, with optimal competitive ratio.

We further introduce generalizations of OCDS, in the leasing model (Meyerson, 2005) and in the multiple hop

model (Coelho et al., 2017), and design deterministic online algorithms for each of these generalizations. We

also propose the ﬁrst deterministic online algorithm for the leasing variant of SC (Abshoff et al., 2016), that is

currently solved by a randomized online algorithm.

1 INTRODUCTION

Dominating Set problems, where the goal is to ﬁnd

a minimum subgraph of a given (undirected) graph

such that each node is either in the subgraph or has

an adjacent node in it, form a fundamental class of

optimization problems that have received signiﬁcant

attention in the last decades. The Connected Dom-

inating Set problem (CDS) - which asks for a min-

imum such subgraph that is connected - is one of

the most well-studied problems in this class (Du and

Wan, 2013) with a wide range of applications in

wireless networks (Yu et al., 2013) and social net-

works (Daliri Khomami et al., 2018; Barman et al.,

2018; Halawi et al., 2018; Wagner et al., 2017).

CDS is known to be N P -complete even in planar

graphs (Garey and Johnson, 1979) and admits an

O(ln∆)-approximation for general graphs, where ∆ is

the maximum node degree of the input graph (Guha

and Khuller, 1998). The latter is the best possible

unless N P ⊆ DT IME(n

loglog n

) (Feige, 1998; Lund

and Yannakakis, 1994). Motivated by applications

in modern robotic warehouses (D’Andrea, 2012), an

online variant of CDS, the Online Connected Domi-

nating Set problem (OCDS), has been introduced by

Hamann et al. (Hamann et al., 2018) - the input to the

so-called online algorithm is an undirected connected

graph G = (V,E), and a sequence of subsets of V ar-

riving over time. OCDS asks to construct a subset S

of V inducing a connected subgraph in G, such that

for each subset D

t

of V arriving at time t, each node

of D

t

must be either in S or have an adjacent node in

S at time t. The goal is to minimize the cardinality of

S. Online algorithms are evaluated using the notion of

competitive analysis, in which the performance of the

online algorithm is measured against the optimal of-

ﬂine solution. Given an input sequence σ - let C

A

(σ)

and C

OPT

(σ) be the cost of an online algorithm A and

an optimal ofﬂine algorithm, respectively. A is said to

be c-competitive (or have competitive ratio c) if there

exists a constant α such that C

A

(σ) ≤ c · C

OPT

(σ) + α

for all input sequences σ. Hamann et al. (Hamann

et al., 2018) proposed an online randomized algorithm

for OCDS, with an asymptotically optimal O(log

2

n)-

competitive ratio, where n is the number of nodes.

In this paper, we give the ﬁrst deterministic

Markarian, C. and Kassar, A.

Online Deterministic Algorithms for Connected Dominating Set Set Cover Leasing Problems.

DOI: 10.5220/0008866701210128

In Proceedings of the 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020), pages 121-128

ISBN: 978-989-758-396-4; ISSN: 2184-4372

Copyright

c

2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

121

algorithm for OCDS, with asymptotically optimal

O(log

2

n)-competitive ratio. Moreover, motivated

by inﬂuence spreading applications in social net-

works (Berman and Coulston, 1997; Daliri Khomami

et al., 2018), in which a small group of people (in-

ﬂuential people) is selected to spread information to

the rest of the group (dominated people), we study

the online variant of the r-hop Connected Dominat-

ing Set problem, where r is a positive integer that

denotes the maximum allowable distance (number of

edges or hops) between the inﬂuential node and the

dominated node. Only ofﬂine model for r-hop con-

nected dominating sets has been known (Coelho et al.,

2017). In our online model, groups of people to be

dominated are revealed over time and need to be inﬂu-

enced, rather than all at once as in the ofﬂine model.

Many classical optimization problems, includ-

ing Set Cover (Abshoff et al., 2016), Facility Lo-

cation (Nagarajan and Williamson, 2013; Markar-

ian and Meyer auf der Heide, 2019), and Steiner

Tree (Meyerson, 2005; Bienkowski et al., 2017), have

been studied in the online leasing model (Meyerson,

2005) and its extensions (Feldkord et al., 2017), in

which rather than being bought, resources are leased

for different time duration with costs respecting econ-

omy of scale, where a long expensive lease costs less

per unit time. In this paper, we give the ﬁrst determin-

istic online algorithm for the Online Set Cover Leas-

ing problem (OCSL), the leasing variant of Set Cover

(SC). Given a universe U and a collection S of sub-

sets of U, SC asks to ﬁnd a minimum number of sub-

sets C ⊆ S whose union is U. Abshoff et al. (Abshoff

et al., 2016) gave the ﬁrst online algorithm for OCSL,

which was randomized. Furthermore, we introduce

the leasing variants of Connected Dominating Set and

r-hop Connected Dominating Set and give a deter-

ministic algorithm for each. All of our algorithms in

this paper are online, deterministic, and evaluated us-

ing the standard competitive analysis. Our results are

summarized as follows.

• We propose the ﬁrst deterministic algorithm for

the Online Connected Dominating Set problem

(OCDS), with asymptotically optimal competi-

tive ratio of O(log

2

n), where n is the number

of nodes (Section 3). The currently best result

for OCDS is a randomized algorithm by Hamann

et al. (Hamann et al., 2018), with asymptotically

optimal O(log

2

n)-competitive ratio.

• We introduce the Online r-hop Connected Domi-

nating Set problem (r-hop OCDS), and give a de-

terministic O(2r · log

3

n)-competitive algorithm,

where n is the number of nodes (Section 4). r-

hop OCDS has been studied in the ofﬂine setting -

Coelho et al. (Coelho et al., 2017) gave inapprox-

imability results for the problem in some special

graph classes.

• We propose the ﬁrst deterministic algorithm for

the Online Set Cover Leasing problem (OSCL),

with O(log σ log(mL + 2m

σ

l

1

))-competitive ratio,

where m is the number of subsets, L is the num-

ber of lease types, σ is the longest lease length,

and l

1

is the shortest lease length (Section 5). The

currently best result for OSCL is a randomized al-

gorithm by Abshoff et al. (Abshoff et al., 2016),

with O(logσ log(mL))-competitive ratio.

• We introduce the Online Connected Dominating

Set Leasing problem (OCDSL), and give a de-

terministic O

(σ + 1) · logσlog(nL + 2n

σ

l

1

) + L ·

logn

-competitive algorithm, where n is the num-

ber of nodes, L is the number of lease types, σ is

the longest lease length, and l

1

is the shortest lease

length (Section 6).

• We introduce the Online r-hop Connected Dom-

inating Set Leasing problem (r-hop OCDSL),

and give a deterministic O

L(1 + σ(2r −

1))logσ log(nL + 2n

σ

l

1

)logn

-competitive algo-

rithm, where n is the number of nodes, L is

the number of lease types, σ is the longest lease

length, and l

1

is the shortest lease length (Section

7).

2 RELATED WORK

Online Connected Dominating Sets and Online

Set Cover. While there are many works that ad-

dress Connected Dominating Set problems and other

related problems in the ofﬂine setting (Guha and

Khuller, 1998; Yu et al., 2013; Haraty et al., 2015;

Haraty et al., 2016), only few consider the online set-

ting. Boyar et al. (Boyar et al., 2016) studied an on-

line variant of the Connected Dominating Set prob-

lem (CDS), in which the input graph is unknown in

advance, and restricted to a tree, a unit disk graph, or

a bounded degree graph. Each step a node is either

inserted or deleted and the goal is to maintain a con-

nected dominating set of minimum cardinality. Boyar

et al. showed that a simple greedy approach attains a

(1 +

1

OP T

)-competitive ratio in trees - where OP T is

the cost of the optimal ofﬂine solution, an (8 + ε)-

competitive ratio in unit disk graphs - for arbitrary

small ε > 0, and b-competitive ratio in b-bounded de-

gree graphs. Recently, Hamann et al. (Hamann et al.,

2018) introduced the Online Connected Dominating

Set problem (OCDS), an online variant of CDS, in

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

122

which the graph is known in advance, and proposed

an O(log

2

n)-competitive randomized algorithm for

OCDS in general graphs, where n is the number of

nodes. Their work was motivated by applications

in modern robotic warehouses, in which geometric

graphs were used to model the topology of a ware-

house. Alon et al. (Alon et al., 2003) gave a deter-

ministic O(logmlogn)-competitive algorithm and an

Ω(logmlogn/(loglogm + log logn)) lower bound for

the online variant of the Set Cover problem, where

m is the number of sets and n is the number of ele-

ments. Korman (Korman, 2005) then improved the

lower bound to Ω(logmlogn). For the unweighted

case where costs are uniform, Alon et al. (Alon et al.,

2003) gave an O(log n log d) competitive ratio, which

was later improved by Buchbinder et al. (Buchbinder

and Naor, 2005) to O(log(n/Opt)logd), where Opt

is the optimal ofﬂine solution and d is the maximum

number of sets an element belongs to.

Leasing Variants. Meyerson (Meyerson, 2005)

gave deterministic O(L)-competitive and randomized

O(logL)-competitive algorithms along with match-

ing lower bounds for the Parking Permit problem.

He also introduced the leasing variant of the Steiner

Forest problem, for which he proposed a random-

ized O(log n log L) competitive algorithm, where n is

the number of nodes, and L is the number of lease

types. Nagarajan and Williamson (Nagarajan and

Williamson, 2013) gave an O(L · logn)-competitive

algorithm for the leasing variant of the Facility Lo-

cation problem, where n is the number of clients.

Abshoff et al. (Abshoff et al., 2016) gave an online

randomized algorithm for the leasing variant of Set

Cover, with O(log(mL) log σ)-competitive ratio and

improved previous results for other online variants

of Set Cover. Bienkowski et al. (Bienkowski et al.,

2017) proposed a deterministic algorithm that has an

O(L log k)-competitive ratio for the leasing variant of

Steiner Tree, where k is the number of terminals.

3 ONLINE CONNECTED

DOMINATING SET (OCDS)

Deﬁnition. Given a connected graph G = (V,E) and

a sequence of disjoint subsets of V arriving over time.

A subset S of V serves as a connected dominating set

of a given subset D of 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. Each step, a subset

of V arrives and needs to be served by a connected

dominating set of G. OCDS asks to grow a connected

dominating set of minimum number of nodes.

Preliminaries. A dominating set of a subset D is a

subset DS of nodes such that each node in D is either

in DS or has an adjacent node in DS. DS is minimal

if no proper subset of DS is a dominating set of D. A

minimal dominating set can be constructed online us-

ing the online deterministic algorithm by Alon et al.

for the Online Set Cover problem (OSC) (Alon et al.,

2003), the online variant of the classical Set Cover

problem. A Set Cover instance is formed by making

each node an element, and corresponding each node

to a set that contains the node itself, along with its

adjacent nodes. Alon et al. (Alon et al., 2003) gave

a deterministic O(log m log n)-competitive algorithm

for OSC, where m is the number of sets and n is the

number of elements.

A Steiner tree of a subset D is a tree con-

necting each node in D to a given root s. A

Steiner tree can be constructed online using the

online deterministic O(logn)-competitive algo-

rithm by Berman et al. (Berman and Coulston,

1997). The Steiner tree problem studied by

Berman et al. (Berman and Coulston, 1997) is for

edge-weighted graphs and the algorithmic cost is

measured by adding the costs of all edges outputted

by the online algorithm. Our model in this paper

assumes no weights on the nodes, and hence the

competitive ratio given by Berman et al. for edge-

weighted graphs carries over to our graph model in

this paper. To see this, assume we are given a graph

G with a weight of 1 on all edges and all nodes, and a

set of terminals that need to be connected. Let Opt

e

be the cost of an optimal Steiner tree T measured

by counting the edges in T . Let Opt

n

be the cost of

an optimal Steiner tree T

0

measured by counting the

nodes in T

0

. We have that Opt

e

= Opt

n

+ 1. The

proof is straightforward, by contradiction. Assume

Opt

e

> Opt

n

+ 1. We can construct a tree which

has an edge cost lower than that of T : the tree T

0

with edge cost Opt

n

+ 1, and this contradicts the

fact that T is an optimal Steiner tree. Now assume

Opt

e

< Opt

n

+1. We can construct a tree which has a

node cost lower than that of T

0

: the tree T with node

cost Opt

e

− 1, and this contradicts the fact that T

0

is

an optimal Steiner tree. This would not have been

the case had there been non-uniform weights on the

nodes since the node-weighted variant of the Steiner

tree problem generalizes the edge-weighted variant

by replacing each edge by a node with the corre-

sponding edge cost. Moreover, the node-weighted

variant of the Steiner tree problem generalizes the

Online Set Cover problem which has a lower bound

of Ω(log m log n) (Korman, 2005) on its competitive

ratio.

Online Deterministic Algorithms for Connected Dominating Set Set Cover Leasing Problems

123

Algorithm. The algorithm assigns, at the ﬁrst time

step, any of the nodes purchased by the algorithm, as

a root node s. At time step t:

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

t

of V

Output: A connected dominating set CDS

t

of D

t

1. Find a minimal dominating set DS

t

of D

t

.

2. Assign to each node in DS

t

a connecting node,

that is any adjacent node from the set D

t

. If t = 1,

assign any of the nodes in DS

t

as a root node s.

3. Find a Steiner tree that connects all connecting

nodes to s. Add all the nodes in this tree includ-

ing the nodes in DS

t

and their connecting nodes to

CDS

t

.

Competitive Analysis. OCDS has a lower bound of

Ω(log

2

n), where n is the number of nodes, result-

ing from Korman’s lower bound of Ω(log m log n) for

OSC (Korman, 2005), where m is the number of sub-

sets and n is the number of elements.

Let Opt be the cost of an optimal solution Opt

I

of

an instance I of OCDS. Let C1, C2, and C3 be the cost

of the algorithm in the three steps, respectively. The

ﬁrst step of the algorithm constructs online a minimal

dominating set. Let Opt

DS

be the cost of a minimum

dominating set of I. Note that Opt

I

is a dominating

set of I. Hence, Alon et al.’s (Alon et al., 2003) de-

terministic algorithm yields: C1 ≤ log

2

n · Opt

DS

≤

log

2

n · Opt. The second step adds at most one node

for each node bought in the ﬁrst step. Hence we have

that: C2 ≤ C1. As for the third step, Opt

I

is a Steiner

tree for the connecting nodes bought in the second

step, since all connecting nodes belong to the set of

nodes that need to be served and Opt

I

serves as a con-

nected dominating set of these nodes. Let Opt

St

be

the cost of a minimum Steiner tree of these connect-

ing nodes. Since Berman et al.’s (Berman and Coul-

ston, 1997) algorithm has an O(logn)-competitive ra-

tio, we conclude that C3 ≤ log n · Opt

St

≤ log n · Opt.

The total cost of the algorithm is then upper bounded

by: C1 + C2 +C3 = (2 · log

2

n + logn) · Opt and the

theorem below follows.

Theorem 1. There is an asymptotically optimal

O(log

2

n)-competitive deterministic algorithm for the

Online Connected Dominating Set problem, where n

is the number of nodes.

4 ONLINE r-hop CONNECTED

DOMINATING SET (r-hop

OCDS)

Deﬁnition. Given a connected graph G = (V,E), a

positive integer r, and a sequence of disjoint subsets

of V arriving over time. A subset S of V serves as

an r-hop connected dominating set of a given subset

D of V if for every node v in D, there is a vertex

u in S such that there are at most r hops (edges)

between v and u in G, and the subgraph induced

by S is connected in G. Each step, a subset of V

arrives and needs to be served by an r-hop connected

dominating set of G. r-hop OCDS asks to grow an

r-hop connected dominating set of minimum number

of nodes.

OCDS is equivalent to r-hop OCDS with r = 1.

Preliminaries. Given a graph G = (V,E) and a pos-

itive integer r. A subset DS of V is an r-hop domi-

nating set of a given subset D of V if for every node

v in D, there is a vertex u in DS such that there are

at most r hops between v and u in G. DS is mini-

mal if no proper subset of DS is an r-hop dominat-

ing set of D. We can transform an r-hop dominat-

ing set instance into a Set Cover instance by making

each node an element, and corresponding each node

to a set that contains the node itself, along with all

nodes that are at most r hops away from it. Hence, we

can construct a minimal r-hop dominating set by run-

ning the online deterministic algorithm by Alon et al.

for the Online Set Cover problem (OSC) (Alon et al.,

2003). A Steiner tree can be constructed online, as

in Section 3, using the online deterministic O(log n)-

competitive algorithm by Berman et al. (Berman and

Coulston, 1997).

Algorithm. The algorithm assigns, at the ﬁrst time

step, any of the nodes purchased by the algorithm, as

a root node s. At time step t:

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

t

of V

Output: An r-hop connected dominating set rCDS

t

of D

t

1. Find a minimal r-hop dominating set rDS

t

of D

t

.

If t = 1, assign any of the nodes in rDS

t

as a root

node s.

2. Find a Steiner tree that connects all nodes in rDS

t

to s. Add all the nodes in this tree including the

nodes in rDS

t

to rCDS

t

.

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

124

Competitive Analysis. The only lower bound for r-

hop OCDS is the one for OCDS, Ω(log

2

n), where n

is the number of nodes. The proof of the competitive

analysis is ommited due to lack of space.

Theorem 2. There is a deterministic O(2r · log

3

n)-

competitive algorithm for the Online r-hop Connected

Dominating Set problem, where n is the number of

nodes.

5 ONLINE SET COVER LEASING

(OSCL)

Deﬁnition. Given a universe U of elements (|U| =

n), a collection S of subsets of U (|S| = m), and a

set of L different lease types, each characterized by a

duration and cost. A subset can be leased using lease

type l for cost c

l

and remains active for d

l

time steps.

Each time step t, an element e ∈ U arrives and there

needs to be a subset S ∈ S active at time t such that

e ∈ S. OSCL asks to minimize the total leasing costs.

We assume the following conﬁguration on the leases.

Deﬁnition 1. (Lease Conﬁguration) Leases of type l

only start at times t with t ≡ 0 mod d

l

, where d

l

is

the length of lease type l. Moreover, all lease lengths

are power of two.

This conﬁguration has been similarly deﬁned by

Meyerson for the Parking Permit problem (Meyerson,

2005), who showed that by assuming this conﬁgura-

tion, one loses only a constant factor in the compet-

itive ratio. A similar argument can be easily made

for OSCL, as was the case for all generalizations of

the Parking Permit problem (Abshoff et al., 2016;

Bienkowski et al., 2017; Nagarajan and Williamson,

2013).

Preliminaries. Our algorithm for OSCL is based

on running Alon et al.’s (Alon et al., 2003) deter-

ministic algorithm for the Online Set Cover prob-

lem (the weighted case), which constructs a frac-

tional solution that is rounded online into an inte-

gral deterministic solution. Alon et al.’s algorithm

has an O(logmlogn)-competitive ratio and requires

the knowledge of the set cover instance to make it de-

terministic. What is unknown to the algorithm is the

order and subset of arriving elements. We will trans-

form an instance α of OSCL into an instance α

0

of the

Online Set Cover problem and run Alon et al.’s deter-

ministic algorithm on α

0

. An instance of the Online

Set Cover problem consists of a universe of elements

and a collection of subsets of the universe - an element

of the universe arrives in each step. The algorithm

needs to purchase subsets such that each arriving ele-

ment is covered, upon its arrival, by one of these sub-

sets, while minimizing the total costs of subsets. The

algorithm may end up covering elements that never

arrive.

Algorithm. Suppose the algorithm is given a uni-

verse U of elements and a collection S of subsets of

U. If there is one lease type, of inﬁnite lease length

(L = 1), we have exactly an instance of the Online

Set Cover problem and so Alon et al.’s (Alon et al.,

2003) deterministic algorithm would solve it. Other-

wise, we do the following - we represent each ele-

ment e ∈ U by n pairs, one for each of the at most

n potential time steps at which e can arrive. We let

pair (e,t) represent element e at time step t. We de-

note by N the collection of all these pairs. A subset

S ∈ S can be leased using lease type l for cost c

l

and

remains active for d

l

time steps. We represent subset

S of lease type l at time t as a triplet (S, l,t). We de-

note by M the collection of all these triplets. We now

construct an instance of the Online Set Cover prob-

lem with N and M being the collection of elements

and of subsets, respectively. Pair (e,t) can be covered

by triplet (S, l,t

0

) if e ∈ S and t ∈ [t

0

,t

0

+ d

l

]. When

an element arrives at time t, pair (e,t) is given as in-

put to the Online Set Cover instance for step t. Note

that each element e ∈ U arrives only once. An al-

gorithm for the Online Set Cover problem will ensure

that e

0

s corresponding pair at the time it arrives is cov-

ered. Moreover it will ignore (not necessarily cover)

all other pairs corresponding to the other time steps

and this is equivalent to having elements that never

arrive in an Online Set Cover instance. Hence, run-

ning Alon et al.’s (Alon et al., 2003) algorithm will

yield a feasible deterministic solution for OSCL.

Competitive Analysis. OSCL has a lower bound of

Ω(logmlogn + L) resulting from the Ω(logm logn)

lower bound for OSC (Korman, 2005), where m is

the number of subsets and n is the number of ele-

ments, and the Ω(L) lower bound for the Parking Per-

mit problem (Meyerson, 2005), where L is the number

of lease types.

We ﬁx any interval I of length σ and show that

the algorithm would be O(log σ log(mL + 2m

σ

l

1

))-

competitive if this interval were the entire input,

where l

1

is the length of the shortest lease, σ is the

length of the longest lease, L is the number of lease

types, and m is the number of subsets. Since all

leases including the ones in the optimal solution end

at the end of I due to the lease conﬁguration deﬁned

earlier, this would imply that the algorithm has an

O(logσ log(mL +2m

σ

l

1

))-competitive ratio. Note that

Online Deterministic Algorithms for Connected Dominating Set Set Cover Leasing Problems

125

there are at most σ elements over I, since at most

one element arrives in each time step. The competi-

tive ratio O(log |M | log |N |) of the algorithm follows

directly by setting the number of elements and sub-

sets to |N | and |M |, respectively. Now, we have that

|N | = σ

2

since there are σ

2

pairs in total. Next, we

give an upper bound to |M | over I.

|M | ≤ m · (

L

∑

j=1

l

σ

l

j

m

)

Since l

j

s are increasing and powers of two, we con-

clude that the sum above can be upper bounded by the

sum of a geometric series with a ratio of 1/2.

L

∑

j=1

l

σ

l

j

m

≤ L + σ

h

1

l

1

1−(1/2)

σ

1−1/2

i

=

L + σ

h

2

l

1

1 − (1/2)

L

i

Since L ≥ 1, we have:

L + σ

h

2

l

1

1 − (1/2)

L

i

≤ L +

2σ

l

1

.

Therefore, |M | ≤ m·(L +

2σ

l

1

), and the theorem below

follows.

Theorem 3. There is a deterministic

O(logσ log(mL + 2m

σ

l

1

))-competitive algorithm

for the Online Set Cover Leasing problem, where m is

the number of subsets, L is the number of lease types,

σ is the longest lease length, and l

1

is the shortest

lease length.

6 ONLINE CONNECTED

DOMINATING SET LEASING

(OCDSL)

Deﬁnition. Given a connected graph G = (V,E), a

sequence of disjoint subsets of V arriving over time,

and a set of L different lease types, each character-

ized by a duration and cost. A node can be leased

using lease type l for cost c

l

and remains active for

d

l

time steps. A subset S of nodes of V serves as a

connected dominating set of a given subset D of 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. Each time step t, a subset of V arrives and needs

to be served by a connected dominating set of nodes

active at time t. OCDSL asks to grow a connected

dominating set with mininum leasing costs. OCDS is

equivalent to OCDSL with one lease type (L = 1) of

inﬁnite length. Note that in both OCDS and OCDSL,

the algorithm ends up purchasing (leasing) nodes that

form one connected subgraph - the difference is that

in OCDSL, at a certain time step t, only the currently

active nodes needed to serve the nodes given at time

t, are connected by nodes active at time t, to at least

one of the previously leased nodes, thus maintaining

one single connected subgraph.

We assume the lease conﬁguration introduced ear-

lier in Deﬁnition 1.

Algorithm. The algorithm assigns, at the ﬁrst time

step, any of the nodes leased by the algorithm, as a

root node s. At time step t:

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

t

of V

Output: A set of leased nodes that form a connected

dominating set of D

t

1. Lease a set DS

t

of nodes that form a minimal dom-

inating set of D

t

.

2. Assign to each node in DS

t

a connecting node,

that is any adjacent node from the set D

t

. Buy the

cheapest lease for each of these connecting nodes.

If t = 1, assign any of the nodes in DS

t

as a root

node s.

3. Lease a set of nodes that connect all connecting

nodes to s.

Algorithm Description. To ﬁnd a set of leased

nodes that form a minimal dominating set of a subset

D

t

, we run our deterministic algorithm for Online Set

Cover Leasing presented in Section 5. An Online

Set Cover Leasing instance is formed by making

each node an element, and corresponding each node

to a set that contains the node itself, along with its

adjacent nodes - sets are leased with L different lease

types. Our algorithm for Online Set Cover Leasing

has an O(logσlog(mL + 2m

σ

l

1

))-competitive ratio,

where m is the number of subsets, L is the number of

lease types, σ is the longest lease length, and l

1

is the

shortest lease length.

To ﬁnd a set of leased nodes that connect a sub-

set of nodes to s, we run the deterministic algorithm

for Online Steiner Tree Leasing problem (OSTL) by

Bienkowski et al. (Bienkowski et al., 2017), deﬁned

as follows. Given a connected graph G = (V,E),

a root node s, a sequence of nodes of V (called

terminals) arriving over time, and a set of L different

lease types, each characterized by a duration and

cost. An edge can be leased using lease type l for cost

c

l

and remains active for d

l

time steps. Each step t, a

node arrives and needs to be connected to s through a

path of edges active at time t. OSTL asks to minimize

the total leasing costs. The algorithm by Bienkowski

et al. (Bienkowski et al., 2017) has an O(L log k)-

competitive ratio, where k is the number of terminals.

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

126

The Online Steiner Tree Leasing problem studied

by Bienkowski et al. (Bienkowski et al., 2017) is

for edge-weighted graphs and the algorithmic cost

is measured by adding the leasing costs of the edges

and not the nodes. Our model in this paper assumes

no weights on the nodes, and hence the competitive

ratio given by Bienkowski et al. (Bienkowski et al.,

2017) for edge-weighted graphs carries over to our

graph model in this paper. This would not have

been the case had there been non-uniform weights

on the nodes since the node-weighted variant of the

Online Steiner Tree Leasing problem generalizes

the edge-weighted variant. Hence, whenever the

algorithm for Online Steiner Tree Leasing leases an

edge (u, v) at time t with lease type l, we lease both

u and v at the same time t with the same lease type l

and hence the cost will only double.

Competitive Analysis. Since OCDSL generalizes

OSCL, Ω(log

2

n + L) is a lower bound for OCDSL,

where n is the number of nodes and L is the number

of lease types. The proof of the competitive analysis

is ommited due to lack of space.

Theorem 4. There is a deterministic O

(σ + 1) ·

logσ log(nL + 2n

σ

l

1

) + L · log n

competitive algo-

rithm for the Online Connected Dominating Set Leas-

ing problem, where n is the number of nodes, L is the

number of lease types, σ is the longest lease length,

and l

1

is the shortest lease length.

7 ONLINE r-hop CONNECTED

DOMINATING SET LEASING

(r-hop OCDSL)

Deﬁnition. Given a connected graph G = (V,E), a

positive integer r, a sequence of disjoint subsets of

V arriving over time, and a set of L different lease

types, each characterized by a duration and cost. A

node can be leased using lease type l for cost c

l

and

remains active for d

l

time steps. A subset S of nodes

of V serves as an r-hop connected dominating set of

a given subset D of V if for every node v in D, there

is a vertex u in S such that there are at most r hops

between v and u in G, and the subgraph induced by

S is connected in G. Each time step t, a subset of V

arrives and needs to be served by an r-hop connected

dominating set of nodes active at time t. r-hop

OCDSL asks to grow an r-hop connected dominating

set with minimum leasing costs.

OCDSL is equivalent to r-hop OCDSL for r = 1. We

assume the lease conﬁguration introduced earlier in

Deﬁnition 1.

Algorithm. The algorithm assigns, at the ﬁrst time

step, any of the nodes leased by the algorithm, as a

root node s. At time step t:

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

t

of V

Output: A set of leased nodes that form r-hop

connected dominating set of D

t

1. Lease a set rDS

t

of nodes that form a minimal r-

hop dominating set of D

t

. If t = 1, assign any of

the nodes in rDS

t

as a root node s.

2. Lease a set of nodes that connect all nodes in rDS

t

to s.

Algorithm Description. To ﬁnd a set of leased

nodes that form a minimal r-hop dominating set of a

subset D

t

, we run our deterministic algorithm for On-

line Set Cover Leasing presented in Section 5. An On-

line Set Cover Leasing instance is formed by making

each node an element, and corresponding each node

to a set that contains the node itself, along with all

nodes that are at most r hops away from it - sets are

leased with L different lease types. Our algorithm for

Online Set Cover Leasing has an O(logσ log(mL +

2m

σ

l

1

))-competitive ratio, where m is the number of

subsets, L is the number of lease types, σ is the

longest lease length, and l

1

is the shortest lease length.

To ﬁnd a set of leased nodes that connect a subset

of nodes to s, we run the deterministic O(L logk)-

competitive algorithm for Online Steiner Tree Leasing

problem (OSTL) by Bienkowski et al. (Bienkowski

et al., 2017), deﬁned earlier.

Competitive Analysis. The only lower bound for r-

hop OCDSL is the one for OCDSL, Ω(log

2

n + L),

where n is the number of nodes and L is the number

of lease types. The proof of the competitive analysis

is ommited due to lack of space.

Theorem 5. There is a deterministic O

L(1+σ(2r −

1))logσ log(nL + 2n

σ

l

1

)logn

-competitive algorithm

for the Online r-hop Connected Dominating Set Leas-

ing problem, where n is the number of nodes, L is the

number of lease types, σ is the longest lease length,

and l

1

is the shortest lease length.

Online Deterministic Algorithms for Connected Dominating Set Set Cover Leasing Problems

127

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