A Vertex Degree-Based GRASP Approach for the Minimum

Independent Dominating Set Problem

André Eduardo Alessi

, Dalcimar Casanova

, Lucas Caldeira de Oliveira

Marco Antonio de Castro Barbosa

, Marcelo Teixeira

, Ives Rene Venturini Pola

and Fernanda Paula Barbosa Pola

Federal University of Technology - Paraná, Pato Branco, Brazil

Keywords: Minimum Independent Dominating Set Problem, GRASP, Vertex Degree.

Abstract: The Minimum Independent Dominating Set (MIDS) problem is a classical graph theory problem, with

applications in sensors networks and database management systems. As a NP-hard problem, there is no exact

solution that can be processed in polynomial time, unless 𝑃 = 𝑁𝑃. Some metaheuristic approaches have been

proposed to tackle the problem in polynomial time. In this paper, we develop our own metaheuristic approach

called GRASP+VD, a GRASP that uses vertex degree as its greedy function. We show that GRASP+VD

outperforms the state-of-the-art approach drMIDS in all BHOSLIB dataset and in most of the DIMACS

dataset.

1 INTRODUCTION

The Minimum Independent Dominating Set (MIDS)

problem is a classical graph theory issue which the

solution has shown to be a promising alternative to

search for energy efficient wireless sensor network

topologies (Santos et al., 2009); eliminate redundancy

in wireless sensor and actor network (Akyildiz et al.,

2004); among others (Alowa et al., 2022). Also, it is

used on a new concept of sets used as a

straightforward way to handle similarity in database

management systems, from complex datasets. Some

results towards this direction have been presented in

Pola et al. (2015).

Given a Graph 𝐺=

(

𝑉, 𝐸

)

, a dominating set 𝐷 of

𝐺 is a set of vertices such that every vertex in 𝑉  𝐷

is adjacent to a vertex in 𝐷. Furthermore, a set 𝐼 is

independent if there is no pair of adjacent vertices in

𝐼. Thus, an independent dominating set (𝐼𝐷𝑆) of 𝐺is

a set that is both dominating and independent. The

https://orcid.org/0000-0003-0268-9801

https://orcid.org/0000-0002-1905-4602

https://orcid.org/0000-0002-7881-8414

https://orcid.org/0000-0001-9674-2348

https://orcid.org/0000-0002-1008-7838

https://orcid.org/0000-0001-7300-7535

https://orcid.org/0000-0002-0449-0954

MIDS problem focus, therefore, to point out the

smallest 𝐼𝐷𝑆 in a graph.

Some variations of the MIDS problem have been

studied in recent literature. For example, the

independent perfect domination sets in Cayley graph

(Wang et al., 2021), the open-independent, open-

locating-dominating sets in complementary prism

graphs (Capelle et al., 2022) and the IDS with

obligation in graphs (Laforest and Martinod, 2022),

to name a few.

Some exact algorithms have been exploited to

resolve the MIDS problem (Johnson et al. 1988) (Liu

and Song, 2006) (Gaspers and Liedloff, 2006)

(Laforest and Phan, 2013) (Burgois et al., 2013).

However, they are limited by the fact that its

processing has proven to be 𝑁𝑃-Hard (Irving, 1991),

meaning that an exact solution cannot be processed in

polynomial time lest 𝑃=𝑁𝑃 (Halldórsson, 1993).

Even though these solutions guarantee optimal

results, they are limited by the exponential processing

time.

318

Alessi, A., Casanova, D., Caldeira de Oliveira, L., Barbosa, M., Teixeira, M., Pola, I. and Pola, F.

A Vertex Degree-Based GRASP Approach for the Minimum Independent Dominating Set Problem.

DOI: 10.5220/0011924000003612

In Proceedings of the 3rd International Symposium on Automation, Information and Computing (ISAIC 2022), pages 318-323

ISBN: 978-989-758-622-4; ISSN: 2975-9463

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

As an usual alternative to process exponential-

time algorithms, metaheuristics (Gendreau and

Potvin, 2010) have also been conjectured in the

literature to solve the MIDS problem. Some efforts

exploit GRASP algorithms combined with a path cost

function (GRASP+PC) (Wang et al., 2017), others

use memetic algorithms (Wang et al., 2018) and

local-search-based approaches (Haraguchi, 2018),

including the state-of-the-art metaheuristic approach,

drMIDS (Wang et al., 2020).

Although there are some metaheursitic

approaches for the MIDS problem, we will show that

their results have room for improvements in terms of

solution quality. Here, we develop a new

metaheursitic approach for the MIDS problem, called

GRASP+VD, that uses vertex degree as a greedy

function instead of the path cost used by GRASP+PC

by Wang et al. (2017).

2 PROBLEM FORMULATION

In the subsequent, we shall ponder a graph 𝐺=

(𝑉, 𝐸) as finite, undirected, with no multiple edges,

and unlooped structure. We also consider 𝑁(𝑣) as the

neighborhood of the vertex 𝑣, and the closed

neighborhood as 𝑁



𝑣



=𝑁

(

𝑣

)

∪{𝑣}. Then, we can

define (Allan and Laskar, 1978):

Definition 2.1. A set 𝐷⊆𝑉is a dominating set of 𝐺

if ∀𝑣 ∈ 𝑉 – 𝐷, 𝑁

(

𝑣

)

∩𝐷≠ ∅.

Definition 2.2. A set 𝐼⊆𝑉 is an independent set of

𝐺 if ∀𝑢,𝑣 ∈𝐼,𝑁

(

𝑢

)

∩{𝑣}=∅.

Definition 2.3. A set 𝐼𝐷𝑆 ⊆ 𝑉 is an independent

dominating set of 𝐺 if 𝐼𝐷𝑆 is both an independent and

a dominating set, that is, if it follows both Definitions

2.1 and 2.2.

We are now in position to introduce the MIDS

problem, as follows.

Problem 2.1. Specified a graph 𝐺=

(

𝑉, 𝐸

)

, the

MIDS problem aims to identify the smallest

independent dominating set in 𝐺 , following

Definition 2.3.

With the problem formally introduced, we can

explain how we tackle it with our GRASP+VD

approach.

3 THE GRASP+VD APPROACH

Algorithm 1 demonstrates the main blocks of a

GRASP procedure. On lines 1 to 5, the algorithm runs

𝑛_𝑖𝑡𝑒𝑟 times, where 𝑛_𝑖𝑡𝑒𝑟 is the maximum number

of iterations. On line 2, a solution is fabricated by the

Greedy Randomized Construction (GRC) algorithm,

receiving as input the graph G and α, a threshold

parameter. When α= 0 we have a totally greedy

algorithm and when α= 1 we have a totally random

algorithm. Later, on line 3, the solution passes

through a Local Search phase. Then, on line 4, the

best-known solution 𝑆_𝑏𝑒𝑠𝑡 is updated if 𝑆 is a

reasonable solution (i.e., attends to Definition 2.3)

and better than 𝑆_𝑏𝑒𝑠𝑡. The best-known solution

𝑆_𝑏𝑒𝑠𝑡 is returned on line 6.

Algorithm 1: GRASP

The first step of the GRASP algorithm is the

Greedy Randomized Construction, which is

illustrated in Algorithm 2. Initially, on line 1, the

solution is assumed to be empty. For each iteration of

this stage, the collection of candidates is formed by

all elements of the ground set that can be included to

the partial solution that has being built, while not

preventing the construction of a workable solution. In

this case, the candidate set is the vertex set 𝑉, as

illustrated by line 2. In this case, the greedy function

is the vertex degree: the higher the vertex degree the

lower the incremental cost. The assessment of the

elements by the greedy evaluation function guides to

the making of a restricted candidate list (RCL)

formed by the best elements (this is the greedy aspect

of the heuristic). On line 6, the best elements are

defined by those with greedy function greater or equal

then a parameter α∈



0,1



multiplied by the

difference between the maximum and minimum

value of the greedy function, summed with the

minimum value. The element to be included into the

incomplete solution is randomly selected from those

in the RCL on line 7. Once the selected element is

incorporated on line 8, the candidate list is updated.

The above steps are repeated until there exists at least

one candidate element on lines 3 to 10 (Resende and

Ribeiro, 2010).

A Vertex Degree-Based GRASP Approach for the Minimum Independent Dominating Set Problem

319

Algorithm 2: Greedy Randomized Construction (GRC).

The solutions created by a greedy randomized

construction are not surely optimal. The local search

phase can improve the constructed solution. A local

search algorithm works in an interactive mode by

successively replacing the current solution by a

superior solution in its neighborhood. In this case,

neighborhood refers to the solutions close to the

current solution in the search space, not to be

confused with vertex neighborhood. In this work, we

chose the strategy of removing k elements at random

from the current solution and reconstruct it by using

Algorithm 2 with the remaining solution set. This

simple procedure is illustrated in Algorithm 3.

Algorithm 3: Local Search.

With the GRASP+VD algorithm fully explained,

we can go on to the computational experiments and

results.

4 COMPUTATIONAL

EXPERIMENTS AND RESULTS

The following experiments were done using a

personal computer with processor AMD Ryzen 5

2600 and 16 GB of RAM, running on Windows 10.

The code was implemented in Python. We performed

computational experiments to apply Algorithm 1 into

the DIMACS (Johnson and Trick, 1996) and

BHOSLIB (Xu et al., 2007) benchmark datasets,

obtained in the Network Repository (Rossi and

Ahmed, 2015).

The experiment consists in running the

implemented GRASP+VD algorithm 100 times on

each instance of the datasets, then calculating its

average and minimum results, and comparing to the

results of drMIDS. The outcomes are condensed in

Tables 1, 2, and 3. The first column of the tables

brings the instance name. The second and third

columns bring the minimum and average length of the

independent dominating set extracted by the

GRASP+VD approach, respectively. The fourth

column brings the standard deviation of the length of

the IDS extracted. Finally, the last two columns bring

the minimum and average length of the independent

dominating set extracted by the drMIDS approach,

respectively. The numbers in bold indicate that

GRASP+VD showed equal or better performance in

comparison with drMIDS for that instance. It can be

noticed that GRASP+VD outperforms drMIDS in the

entirety of the BHOSLIB dataset and in 82.7% of the

DIMACS dataset.

These results show that GRASP+VD could be a

better option than drMIDS in terms of finding IDS

with minimum cardinality. By its applications, this

would mean extracting better similarity sets, finding

better energy efficient wireless sensor networks, and

so on. It should be noted that GRASP+VD uses a

simpler information about the vertices, the vertex

degree, than the personalized path cost utilized by

drMIDS as its greedy function.

Table 1: Experimental results of GRASP+VD and drMIDS

on the BHOSLIB dataset.

GRASP+VD drMIDS

Instance min mean std min mean

frb100-40 34.29 0.537 43 44.35

frb30-15-1 33 0 11 11

frb30-15-2 33 0 11 11

frb30-15-3 33.01 0.100 11 11

frb30-15-4 33 0 11 11

frb30-15-5 33.11 0.314 11 11

frb35-17-1 33.19 0.394 13 13

frb35-17-2 33.08 0.273 13 13.03

frb35-17-3 33.07 0.256 13 13

frb35-17-4 33.13 0.338 13 13.29

frb35-17-5 33.06 0.239 13 13.65

frb40-19-1 33.05 0.219 15 15.39

frb40-19-2 33.06 0.239 15 15.03

frb40-19-3 33.19 0.394 15 15.03

ISAIC 2022 - International Symposium on Automation, Information and Computing

320

frb40-19-4 3 3 0 15 15

frb40-19-5 3 3.09 0.288 15 15.19

frb45-21-1 3 3.09 0.288 17 17.77

frb45-21-2 3 3.13 0.338 17 17.87

frb45-21-3 3 3.21 0.409 17 17.39

frb45-21-4 3 3.18 0.386 17 17.55

frb45-21-5 3 3.14 0.349 17 17.45

frb50-23-1 3 3.38 0.488 19 19.94

frb50-23-2 3 3.25 0.435 19 19.9

frb50-23-4 3 3.31 0.465 19 19.9

frb50-23-5 3 3.09 0.288 20 20.03

frb53-24-4 3 3.29 0.456 20 20.9

frb53-24-5 3 3.58 0.496 20 21.06

frb59-26-1 3 3.33 0.473 23 23.61

frb59-26-2 3 3.59 0.494 23 23.9

frb59-26-3 3 3.62 0.488 23 23.84

frb59-26-4 3 3.42 0.554 23 23.94

frb59-26-5 3 3.84 0.368 24 24.19

Table 2: Experimental results of GRASP+VD and drMIDS

on the DIMACS dataset I.

GRASP+VD drMIDS

Instance min mean std min mean

brock200-2 5 5.0 0.1 4 4

brock200-4 33.3 0.5 6 6

brock400-2 33.5 0.5 9 9

brock400-4 33.0 0 9 9

brock800-2 45.0 0.1 8 8

brock800-4 44.9 0.3 8 8

C1000-9 22.4 0.5 25 25.48

C125-9 22 0 14 14

C2000-5 7 7.7 0.4 7 7

C2000-9 33 0 32 32.03

C250-9 22 0 17 17

C4000-5 8 8.8 0.4 8 8

C500-9 22 0 21 21

DSJC1000-5 6 6.9 0.4 6 6

DSJC500-5 5 5.9 0.3 5 5

c-fat200-1 13 13.5 0.5 13 13

c-fat200-2 6 6.4 0.5 6 6

c-fat200-5 3 3 0.0 3 3

c-fat500-1 28 30 0.7 27 27

c-fat500-2 14 14.8 0.5 14 14

c-fat500-5 66 0 6 6

en200-p0-9-44 22 0 16 16

en200-p0-9-55 22 0 16 16

en400-p0-9-55 22 0 20 20

en400-p0-9-65 22 0 20 20

en400-p0-9-75 22 0 20 20

hammin

10-4 88 0 12 12

hammin

6-2 22 0 12 12

hammin

6-4 8 8 0 2 2

hammin

8-2 22 0 32 32

hammin

8-4 8 8 0 4 4

Table 3: Experimental results of GRASP+VD and drMIDS

on the DIMACS dataset II.

GRASP+VD drMIDS

Instance min mean std min mean

ohnson16-2-4 33 0 8 8

ohnson32-2-4 33 0 16 16

ohnson8-2-4 33 0 4 4

ohnson8-4-4 55 0 7 7

keller4 44 0 5 5

keller5 44 0 9 9

keller6 44 0 15 17.16

A Vertex Degree-Based GRASP Approach for the Minimum Independent Dominating Set Problem

321

MANN-a27 2 2 0 27 27

MANN-a45 2 2 0 45 45

MANN-a81 2 2 0 81 81

MANN-a9 2 2 0 9 9

p-hat1500-1 14 16.7 1.1 12 12.71

p-hat1500-2 7 10.0 0.9 7 7.68

p-hat1500-3 4 4.1 0.3 3 3

p-hat300-3 3 3 0.1 3 3

p-hat700-1 12 14.2 0.9 11 11

p-hat700-2 7 8.4 0.9 6 6

p-hat700-3 3 3.5 0.5 3 3

san1000 19 22.1 1.1 4 4

san200-0-7-1 3 3.2 0.4 6 6

san200-0-7-2 4 4.5 0.5 6 6

san200-0-9-2 2 2 0 16 16

san200-0-9-3 2 2 0 15 15

san400-0-5-1 13 15.2 0.9 4 4

san400-0-7-1 4 4.8 0.5 7 7

san400-0-7-2 4 5.0 0.7 7 7

san400-0-7-3 5 5.1 0.3 7 7

5 CONCLUSIONS

The Minimum Independent Dominating Set (MIDS)

problem is a classical graph theory problem. The

solution for this problem has applications in some

areas, like sensors networks and similarity set

extraction. There has been some work regarding

approximated approaches for this problem, but there

was room to improvement.

In this work, we suggested a novel GRASP+VD

approach that uses vertex degree instead of path cost

as greedy function, explained its functioning and

made computational experiments to measure its

performance against the competitor drMIDS

approach. We demonstrated that GRASP+VD

outperforms drMIDS in the entirety of the BHOSLIB

dataset and in 84.2% of the DIMACS dataset.

For future research, we would like to experiment

different types of metaheuristics to compare their

performances with GRASP+VD. We suggest Ant

Colony Optimization and Simulated Annealing as

starting points.

ACKNOWLEDGEMENTS

This work was financed in part by the Coordenação

de Aperfeiçoamento de Pessoal de Nível Superior –

Brasil (CAPES) – Finance Code 001, of Conselho

Nacional de Desenvolvimento Científico e

Tecnológico (CNPq), from Fundação Araucária (FA)

and from Financiadora de Estudos e Projetos

(FINEP).

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