Ant Colony Optimization Approaches for the Tree t-Spanner Problem

Manisha Israni and Shyam Sundar

Department of Computer Applications, National Institute of Technology Raipur, Raipur (Chhattisgarh), India

Keywords:

Tree Spanner, Weighted Graph, Swarm Intelligence, Ant Colony Optimization.

Abstract:

A tree t-spanner of a given connected graph is a spanning tree T in which the ratio of distance between every

pair of vertices is at most t times their distance in the graph, where t is a parameter known as stretch factor

of T . The tree t-spanner problem deals with ﬁnding a spanning tree in a connected graph whose stretch

factor is minimum amongst all spanning trees of the graph. For unweighted graph, this problem is N P-Hard

for any ﬁxed t ≥ 4, whereas for weighted graph, this problem is N P -Hard for any ﬁxed t > 1. This paper

concerns this problem for connected, undirected, and weighted graph and proposes three variants of ant colony

optimization (ACO) approach for this problem. ACO approach is a swarm intelligence technique inspired by

the foraging behavior of real ants. All three variants of ACO approach have been tested on a set of randomly

generated graph instances. Computational results show the effectiveness of all three variants of ACO approach.

1 INTRODUCTION

Given a connected graph G(V, E), a tree t-spanner is

a spanning tree (say T ) in which the distance between

every two vertices (say u and v) in T is at most t times

their distance in G. t is a parameter called stretch fac-

tor of T and it is determined by the maximum stretch

taken over all pairs of vertices in G, where the stretch

of a pair of vertices (say u and v) in T is the ratio of

the distance between u and v in T to their distance in

G. The tree t-spanner problem aims to ﬁnd a span-

ning tree of G whose stretch factor (t) is minimum

amongst all spanning trees of G.

The term t-spanner of a graph G, which was ﬁrst

coined by Peleg and Ullman (Emek and Peleg, 2008),

is a spanning subgraph (H) of G in which the dis-

tance between every pair of vertices is at most t times

their distance in G. This notion (spanner) describes an

property about the approximation of pairwise vertex-

to-vertex distances in the original graph by spanning

subgraph (H). This approximation distance quality is

measured by the parameter t ≥ 1, which is referred to

as the stretch factor of the t-spanner. Spanners with

such distance approximation property make them of

practical relevances in various areas such as commu-

nication networks, distributed systems, motion plan-

ning, network design, and parallel machine architec-

tures (Alth

ofer et al., 1993; Awerbuch et al., 1992;

Bhatt et al., 1986; Liestman and Shermer, 1993; Pe-

leg and Ullman, 1987; Peleg and Upfal, 1988). For

example, spanners can be used to build synchronizers

for transforming synchronous algorithms into asyn-

chronous ones (Emek and Peleg, 2008).

In a connected graph, if a spanning subgraph (say

T ) is both t-spanner and tree, then T is referred to

as a tree t-spanner. Note that a spanning tree of the

graph is always a tree spanner, where a tree spanner

is a tree t-spanner for some t ≥ 1. Tree spanners also

ﬁnd practical relevance. For example, tree spanners

of small stretch factors can be applied in performing

multi-source broadcast in a network (Awerbuch et al.,

1992), which can greatly simplify the message rout-

ing at the cost of only small delay in message delivery.

Later, Cai and Corneil (Cai and Corneil, 1995) stud-

ied graph theoretic, algorithmic, and complexity is-

sues about tree spanners of weighted and unweighted

connected graphs. They proved on edge-weighted

graphs that a tree 1-spanner, if it exists, is a mini-

mum spanning tree and can be found in polynomial

time. They also proved on edge-weighted graphs that

the problem of ﬁnding a tree t-spanner is N P -Hard

for any ﬁxed t > 1. In case of unweighted graphs,

the problem of ﬁnding a tree t-spanner is N P -Hard

for any ﬁxed integer t ≥ 4. For connected, undi-

rected and unweighted graph, this problem is referred

to as minimum max-stretch spanning tree problem

(Emek and Peleg, 2008; Dragan and K

ohler, 2014).

In literature, approximation algorithms on some spe-

ciﬁc types of unweighted graph (Cai, 1992; Peleg and

Tendler, 2001; Venkatesan et al., 1997) have been de-

veloped for the tree t-spanner problem; however, such

speciﬁc types of graph may not be encountered in ev-

Israni M. and Sundar S.

Ant Colony Optimization Approaches for the Tree t-Spanner Problem.

DOI: 10.5220/0006490002000206

In Proceedings of the 9th International Joint Conference on Computational Intelligence (IJCCI 2017), pages 200-206

ISBN: 978-989-758-274-5

ery situation. One can ﬁnd various related tree span-

ner problems in details in (Liebchen and W

unsch,

2008).

This paper concerns the tree t-spanner problem

for connected, undirected and edge-weighted graph

G(V, E, w), where V is a set of vertices; E is a set

of edges; and for each edge, there exists an edge

weight which is a positive rational number. Here-

after, the tree t-spanner problem for connected, undi-

rected and edge-weighted graph G(V, E, w) will be re-

ferred to as [Tree] t-SP. Since [Tree] t-SP is N P -

Hard for t > 1, metaheuristic techniques are the ap-

propriate approaches that can ﬁnd high quality solu-

tion in a reasonable time. In literature, only genetic al-

gorithm (GA) (Moharam and Morsy, 2017) has been

proposed so far for the [Tree] t-SP. GA (Moharam

and Morsy, 2017) is based on generational population

model which uses multi-point crossover and mutation

genetic operators for generating offsprings.

Ant colony optimization (ACO) approach (Dorigo

et al., 1991; Dorigo and St

utzle, 2004) is swarm-

based metaheuristic technique that has been ap-

plied successfully to many combinatorial optimiza-

tion problems. It is inspired by the foraging behavior

of ants. This paper proposes three variants of ACO

approach for the [Tree] t-SP. The proposed three vari-

ants of ACO approach have been tested on a set of

randomly generated graph instances (see description

Section 4.1). On a set of randomly generated graph

instances, computational results show the effective-

ness of the proposed three variants of ACO approach

for this problem.

The rest of this paper is organized as follows: Sec-

tion 2 and 3, respectively, describes a brief introduc-

tion of ACO approach and presents three variants of

ACO approach for the [Tree] t-SP. Computational re-

sults are reported in Section 4. Finally, Section 5 con-

tains some concluding remarks.

2 ACO APPROACH

Ant colony optimization (ACO) approach is a swarm-

based metaheuristic technique derived from the ob-

servation of real ants’ foraging behavior. In nature,

ants wander randomly in search of food and deposit

a chemical substance called pheromones on the paths

while wandering. Other ants can smell pheromones

and tend to choose their way probabilistically based

on paths marked by strong pheromone concentrations,

resulting a pheromone-trail. Pheromones on the paths

also evaporate with time. While returning to the nest

from the food source, ants tend to choose the short-

est path marked by strong pheromone concentrations

in comparison to longer paths, resulting in further in-

crement on pheromone concentrations on such short-

est path. Such strong pheromone trail on this short-

est path stimulates more and more ants to follow

this shortest path again, which in turn, further rein-

forces the pheromone trail on this shortest path. Af-

ter some time, the whole ants in the colony follow

that pheromone trail which leads to the shortest path

from nest to food source and vice versa. Hence such

behavior for searching the shortest path between nest

and food source emerges from the cooperation among

individuals of the whole colony.

This feature of real ant colonies has been emulated

in ant colony optimization (ACO) approach for the so-

lution of hard combinatorial optimization problems.

In ACO approach, each artiﬁcial ant in the colony

builds a solution through incremental procedure that

takes the help of heuristic information of the prob-

lem under consideration and pheromone trails that are

accumulated through search experience. Dorigo et

al. (Dorigo et al., 1991) developed the ﬁrst ACO ap-

proach called Ant System. Later, different variants of

ACO have been proposed in the literature. One can

ﬁnd a good survey of different variants of ACO algo-

rithms and their applications in (Dorigo and St

utzle,

2004). Readers can ﬁnd some recent work on ACO

approach (Sundar and Singh, 2013; Sundar et al.,

2012; Monteiro et al., 2015)

3 THREE VARIANTS OF ACO

APPROACH FOR THE [Tree] t-SP

This section presents three variants of ACO approach

for the [Tree] t-SP. All components of each variant of

ACO approach are the same except the updating rule

regarding the pheromone trails component.

Initially, all pairs of shortest paths for a given

graph is precomputed. Also, pheromone value at each

edge of G is initialized with a high value (here the

value is set to 10 which is determined empirically) so

that ACO approach performs a wider exploration of

the search space during initial iterations before start-

ing biasing the search. Then, at every iteration, each

artiﬁcial ant in the colony constructs a spanning tree

and computes the ﬁtness of its constructed solution;

and the update of pheromone trails is performed.

3.1 Construction of Solution

Each artiﬁcial ant, say k, constructs a spanning tree

(solution), say T , on G(V, E, w) with the help of

heuristic information and pheromone value. Initially,

T is an empty set (Φ). S is another set which is ini-

tially empty (Φ). A vertex, also called root vertex (say

r) of T to be grown, is selected randomly from V and

is added to S. For each vertex x in V , the attribute

(r, x) that is the distance from the root r of T to x is

maintained. For the root vertex r, d

(r, r) is set to 0.

Hereafter, similar to Prim’s algorithm (Prim, 1957),

iteratively, ant k, at each step, selects an edge con-

necting a vertex, say u, in S to a vertex, say v, in V \S;

however, instead of selecting an edge with minimum

cost, an edge is selected according to a probabilis-

tic action choice rule based on heuristic information

and pheromone values. The probability of selecting

an edge (prob

) by ant k is determined as follows:

prob

[τ

]

[η

]

∑

∈S

∑

∈V \S

[τ

]

[η

]

(1)

where τ

is the pheromone concentration on the

edge e

; η

is the heuristic value that is avail-

able a priori, and that is equal to the weight on edge

; and α and β are two parameters which deter-

mine the relative inﬂuence of the pheromone trail and

the heuristic information respectively. By this prob-

abilistic action choice rule, the probability of select-

ing a particular edge (u, v) increases with the value

of associated pheromone trail τ

and of the heuris-

tic information value η

. The parameters α and β

together help in searching high quality solution in the

search space. Hence, being biased by higher heuristic

and pheromone value, this probabilistic action choice

rule plays an important role in exploring various good

(in terms of ﬁtness) spanning trees in the search space

on a graph instance. The selected edge e

is added

to T and v is added to S. Update d

(r, v) through

“d

(r, v) = d

(r, u) + w(u, v)”. This process is re-

peated until V \S becomes empty. At this juncture,

a spanning tree T is constructed.

3.2 Fitness Computation

Once the spanning tree T that is also a tree spanner

is constructed, its ﬁtness, in terms of its stretch fac-

tor t, is computed. Before computing ﬁtness, lowest

common ancestors for pairs of vertices in T with the

root vertex r, are preprocessed by least common an-

cestor algorithm (Harel and Tarjan, 1984), where low-

est common ancestor for any two vertices, say u and

v, in T is the common ancestor of vertices u and v

that is located farthest from the root, and is referred

to as LCA(u,v). In doing so, one can get the lowest

common ancestor query for any pair of vertices say u

and v in T , i.e., LCA(u, v) in constant-time. Distance

(u, v) between any two vertices u and v in T can be

computed as

(u, v) = d

(r, u) + d

(r, v) − 2 × (d

(r, LCA(u, v)))

(2)

One can easily determine the maximum stretch factor

t of T taken over all pairs of vertices x, y in G, i.e.,

(u, v)

∀(u, v) ∈ G (3)

3.3 Update of Pheromone Trails

Once all ants have constructed their solutions at

the current iteration (say it) of ACO approach, the

pheromone trails are updated at the end of the cur-

rent iteration, i.e. it. We follow three ways

for the update of pheromone trails based on local

pheromone update, global pheromone update, and

mixed pheromone update that also deﬁne three vari-

ants of ACO approach for the [Tree] t-SP.

 Local Pheromone Update: First, decrease the

pheromone value on each edge, say e

, of G by a

constant factor, and then augment the pheromone

on those edges of G that are selected by the itera-

tion’s best solution (S

) or that are selected by the

iteration’s best ant. The pheromone trails which

are updated by the iteration’s best ant on the edges

of G are as follows:

(it + 1) = ρ τ

(it) + ∆τ

(4)

where 0 < ρ ≤ 1 is the persistence rate, and ∆τ

the amount of pheromone that the iteration’s best

ant deposits on the edges of the iteration’s best so-

lution (S

). The parameter ρ is used to avoid un-

limited accumulation of pheromone trails as the

ACO approach progresses, and simultaneously it

also helps the ACO approach in forgetting bad de-

cisions taken in past. In addition, if an edge is not

selected by the ants, the pheromone value of this

edge decreases exponentially in the number of it-

erations. ∆τ

is deﬁned as follows:

∆τ

(

aug1

, if e

∈ S

;

0, otherwise.

(5)

where p

aug1

is a parameter to be determined

empirically. This equation explains that the better

the solution S

in terms of ﬁtness is, the more

pheromone the edges that are part of this solution

receive. Hence edges that are part of solution with

higher ﬁtness are more likely to be selected by

ants in future iterations of the approach, thereby

receiving more reinforcement. Hereafter, this

variant of ACO approach will be referred to as

ACO Local.

 Global Pheromone Update: It is similar to

local pheromone update except augmenting

pheromone. In this variant of ACO approach,

augment pheromone on those edges of G that

are present in the best-so-far solution (say S

)

found by best-so-far ant since the start of ACO ap-

proach. The pheromone trails which are updated

by the best-so-far ant on the edges of G are as fol-

lows:

(it + 1) = ρ τ

(it) + ∆τ

(6)

where ∆τ

is the amount of pheromone best-so-

far ant deposit on the edges of best-so-far solution

). ∆τ

is deﬁned as follows:

∆τ

(

aug2

, if e

∈ S

;

0, otherwise.

(7)

where p

aug2

is a parameter to be determined

empirically. Hereafter, this variant of ACO

approach will be referred to as ACO Global.

 Mixed Pheromone Update: It is similar to lo-

cal pheromone update except augmenting the

pheromone. In this variant of ACO approach,

two ways in mixed way are applied in augment-

ing the pheromone: ﬁrst augment the pheromone

on those edges of G that are present in the iter-

ation’s best solution (S

) or that are selected by

iteration’s best ant; and second, at every P

gen-

erations, augment the pheromone on those edges

of G that are present in the best-so-far solution

(say S

) found by best-so-far ant since the start of

ACO approach, where P

is a parameter to be de-

termined empirically. The update of pheromone

trails in this approach is as follows:

(it + 1) = ρ τ

(it) + ∆τ

mixed

(8)

where ∆τ

mixed

is the sum of amount of pheromone

( ∆τ

, i.e. p

aug1

) augmenting on the edges of S

and amount of pheromone (∆τ

, i.e. p

aug1

) aug-

menting on the edges of S

at every iter, where

∆τ

is similar to equation 5 and ∆τ

, which

is applied at every P

generations, is similar to

equation 7. Note that here ∆τ

at every P

gen-

erations is additional reinforcement. Hereafter,

this variant of ACO approach will be referred to

as ACO Mixed.

Algorithm 1: Pseudo-code of ACO approach for the

[Tree] t-SP.

Compute all pairs of shortest paths of G;

Initialize pheromone value at each edge of G;

while Termination criteria is not satisﬁed do

for k ← 1 to pop

ant

Each ant k constructs a spanning tree of

G with the help of heuristic information

and pheromone value;

Perform update of pheromone trails;

The lower pheromone trail limit, i.e. τ

min

is ex-

plicitly set for the proposed three variants of ACO ap-

proach. The pseudo-code of ACO approach for the

[Tree] t-SP is given by Algorithm 1, where pop

ant

the size of colony (population) of ants.

4 COMPUTATIONAL RESULTS

The proposed three variants of ACO approach

(ACO Local, ACO Global, and ACO Mixed) for the

[Tree] t-SP have been implemented in C and tested

on a set of randomly generated graph instances. All

experiments have been performed on a Linux with the

conﬁguration of 3.2 GHz × 4 Intel Core i5 processor

with 4 GB RAM. Each variant of ACO approach has

been executed 10 independent runs on each instance.

We have allowed each variant of ACO approach to

execute for 1000 generations.

Subsequent subsections discuss about the genera-

tion of graph instances and the parameter tuning. In

addition, a comparison study of all three variants of

ACO approach on graph instances is also discussed.

4.1 Graph Instances

Since instances used in GA (Moharam and Morsy,

2017) for the [Tree] t-SP are not available, and it

should be also noted that authors (Moharam and

Morsy, 2017) tested the effectiveness of their GA on

only two instances (i.e. ﬁrst instance is a graph of

50 vertices and second one is a graph of 100 ver-

tices (one can see Table 5 of (Moharam and Morsy,

2017)). To test our proposed ACO approaches, two

set of graph instances G(V, E, w) with V = 50, 100 are

generated randomly in 500 × 500 plane. A graph in-

stance is generated as follows: each point represent-

ing a vertex in G is selected randomly in 500 × 500

plane. The Euclidean distance between two vertices

, v

) is its edge weight. For each set, three different

complete graph instances are generated, and with the

help of each generated graph G

, E

), further three

sparse graphs with different edge density, i.e., |E

|, |E

| are generated, where |E

| = 0.8 × |E

| = 0.6 × |E

|; and |E

| = 0.4 × |E

|. Note that

0.8 × |E

| (0.6 ×|E

| and 0.4 × |E

|) means 20% (40%

and 60%) random edges of E

are not considered into

and E

). All graph instances are represented

in A1 A2 A3 (see Table 2, where A1 is the total num-

ber of vertices of that graph (see column Vertex of Ta-

ble 2); A2 is X % (see column Edge of Table 2) ran-

dom edges of total number of edges in its correspond-

ing complete graph with A1 vertices are not consid-

ered in current graph instance; and A3 presents dif-

ferent graph instance (in terms of edge density) with

the same number of vertices (A1). In Table 2, con-

sider 50 0.0 1 instance which is a complete graph

with A1 = 50, A2 = 0.0 and A3 = 1. Three different

instances with different edge-density have been gen-

erated from this graph by doing changes in A2, i.e.

A2 = {0.2, 0.4, 0.6}. The generated graph instances

corresponding to 50 0.0 1 are 50 0.2 1, 50 0.4 1 and

50 0.6 1.

Hence, 24 graph instances have been generated

from the two sets of graph instances with |V | ∈ {50,

100}.

Table 1: Possible values of each parameter used in ACO

approaches for the [Tree] t-SP.

Parameter Description Characteristics

ant pop The population of ants {10, 20, 30, 40}

min

Value of τ

min

{0.001, 0.05, 0.01, 0.005}

α A parameter {0, 1, 2}

β A parameter {0, 1, 2}

P A parameter {0.05, 0.1,0.2}

ρ A parameter {0.975, 0.98, 0.985, 0.99}

A parameter {10, 20, 30}

4.2 Parameter Tuning

ACO approaches are stochastic approaches, and pa-

rameters used in our proposed three variants of ACO

approach for the [Tree] t-SP play signiﬁcant roles in

ﬁnding high quality solutions. It is to be noted that de-

termining a perfect tuning of various parameters used

is a difﬁcult task; however, it is always possible to

determine approximate parameter values that provide

good results overall. pop

ant

, α, β, ρ, τ

min

, and P ∈

aug1

, p

aug2

} (based on ACO Local, ACO Global,

and ACO Mixed) are the parameters that have been

used in the three variants of ACO approach for this

problem. Various possible values related to each pa-

rameter have been considered from our initial experi-

ments and available literatures (one can see Table 1).

On investigation, pop

ant

= 30, α = 1, β = 2, ρ = 0.975,

min

= 0.001, P = 0.05, and P

= 20 are the values

of parameters that approximate high quality solutions

overall; however, these values are in no way optimal

200 400

600

800 1,000

Generation

Average Solution Quality

50 0.4 1

50 0.4 2

100 0.2 1

100 0.4 3

(a) ACO Local

200 400

600

800 1,000

Generation

Average Solution Quality

50 0.4 1

50 0.4 2

100 0.2 1

100 0.4 3

(b) ACO Global

200 400

600

800 1,000

Generation

Average Solution Quality

50 0.4 1

50 0.4 2

100 0.2 1

100 0.4 3

Figure 1: Improvement of average solution quality over suc-

cessive generations.

parameter values for all instances. One may study

(Birattari, 2005; Eiben and Smith, 2011) for a de-

tailed coverage of parameter tuning in metaheuristic

techniques in general. It should be noted that all val-

ues of parameters are same for each variant of ACO

approach, as it was observed experimentally that each

variant of ACO approach produces high quality solu-

tion overall.

Table 2: Results of three variants of ACO approach on different graph instances for the [Tree] t-SP.

Instance Characteristics ACO Local ACO Global ACO Mixed

Vertex Edge Best Avg SD ATET Best Avg SD ATET Best Avg SD ATET

50 0.0 1 50 0% 4.92 4.97 0.04 6.53 4.73 5.19 0.29 6.74 4.92 4.97 0.05 6.54

50 0.2 1 50 20% 4.49 4.58 0.08 8.15 4.51 4.63 0.10 8.40 4.44 4.56 0.06 8.16

50 0.4 1 50 40% 4.53 4.57 0.04 6.66 4.53 4.77 0.25 6.88 4.44 4.54 0.06 6.66

50 0.6 1 50 60% 4.64 4.83 0.21 4.65 4.74 5.08 0.21 4.81 4.57 4.71 0.13 4.66

50 0.0 2 50 0% 5.02 5.19 0.17 6.52 5.11 5.32 0.20 6.75 5.02 5.27 0.13 6.53

50 0.2 2 50 20% 4.69 4.79 0.11 7.93 4.70 4.96 0.20 8.24 4.69 4.82 0.16 7.93

50 0.4 2 50 40% 3.98 4.04 0‘06 6.61 4.13 4.47 0.24 6.85 3.98 4.19 0.23 6.62

50 0.6 2 50 60% 4.05 4.07 0.03 4.57 4.04 4.42 0.27 4.73 4.05 4.06 0.02 4.59

50 0.0 3 50 0% 5.45 5.63 0.09 6.61 5.45 5.91 0.39 6.77 5.45 5.69 0.13 6.56

50 0.2 3 50 20% 4.87 4.91 0.02 8.22 4.85 5.01 0.15 8.47 4.87 4.91 0.02 8.22

50 0.4 3 50 40% 4.87 4.91 0.02 8.22 4.85 5.01 0.15 8.49 4.87 4.91 0.02 8.22

50 0.6 3 50 60% 4.38 4.52 0.08 4.64 4.43 4.70 0.23 4.79 4.26 4.50 0.14 4.64

100 0.0 1 100 0% 6.54 6.91 0.22 45.37 6.71 7.04 0.24 45.55 6.67 6.83 0.17 45.28

100 0.2 1 100 20% 6.71 6.83 0.08 62.40 6.66 6.92 0.24 64.66 6.28 6.66 0.18 62.36

100 0.4 1 100 40% 6.49 6.83 0.22 51.20 6.24 6.92 0.46 52.81 6.42 6.68 0.18 51.00

100 0.6 1 100 60% 6.80 7.40 0.33 34.51 6.64 7.09 0.29 35.65 6.77 7.22 0.37 34.55

100 0.0 2 100 0% 6.66 6.76 0.08 45.50 6.66 6.99 0.29 45.47 6.39 6.55 0.13 45.48

100 0.2 2 100 20% 6.23 6.47 0.19 62.83 6.04 6.80 0.40 62.76 6.18 6.39 0.11 62.82

100 0.4 2 100 40% 6.06 6.19 0.16 50.84 6.10 6.61 0.49 50.92 6.09 6.20 0.15 50.95

100 0.6 2 100 60% 5.97 6.12 0.16 34.51 6.21 6.65 0.33 35.74 5.98 6.19 0.18 34.51

100 0.0 3 100 0% 8.07 8.73 0.15 45.69 7.51 8.12 0.41 45.52 7.60 8.08 0.21 45.59

100 0.2 3 100 20% 7.71 7.93 0.20 63.16 7.49 8.83 0.61 63.30 7.58 7.89 0.17 63.14

100 0.4 3 100 40% 7.70 7.89 0.18 51.19 7.27 7.90 0.46 52.98 7.51 7.78 0.17 51.04

100 0.6 3 100 60% 7.53 7.93 0.17 34.38 7.25 7.53 0.28 35.67 7.38 7.73 0.23 34.39

4.3 Performance Evaluation

Table 2 reports the results of each variant of

ACO approach (i.e., ACO Local, ACO Global, and

ACO Mixed) for the [Tree] t-SP on a set of graph

instances. In Table 2, column Instance denotes the

name of each graph instance in A1 A2 A3 format;

columns Vertex and Edge respectively denote the to-

tal number of vertices and edges of the graph; and

columns Best, Avg and SD and ATET, respectively de-

note the best value, the average value, the standard

deviation and the average total execution obtained by

ACO Local, ACO Global, and ACO Mixed over 10

runs.

One can also observe in Table 2 that among all

three variants of ACO approach, ACO Local, in terms

of Best, is better on 3, is equal on 4 and is worse on

17 graph instances in comparison to other variants of

ACO approach; ACO Local, in terms of Avg, is better

on 6, is equal on 3 and is worse on 15 graph instances

in comparison to other variants of ACO approach;

ACO Global, in terms of Best, is better on 11, is equal

on 1 and is worse on 12 graph instances in compari-

son to other variants of ACO approach; ACO Global,

in terms of Avg, is better on 2 and is worse on 22 graph

instances in comparison to other variants of ACO ap-

proach; ACO

Mixed, in terms of Best, is better on 6,

is equal on 4 and is worse on 14 graph instances in

comparison to other variants of ACO approach; and

ACO Mixed, in terms of Avg, is better on 13, is equal

on 3 and is worse on 8 graph instances in comparison

to other variants of ACO approach. Based on experi-

mental observation, ACO Global is better in terms of

Best and in terms of Avg ACO Mixed is better; how-

ever one can also notice that the results obtained by

all three variant of ACO approaches are close to each

other.

4.4 Convergence Analysis

To examine the effectiveness of the proposed three

variants of ACO approach for the [Tree] t-SP, we have

performed experiments based on convergence of each

proposed variant of ACO approach. For this, four

graph instances i.e. 50 0.4 1, 50 0.4 2, 100 0.2 1 and

100 0.4 3 have been selected from the set of consid-

ered graph instances. Figure 1-(a-c) respectively de-

pict the emergence of average solution quality (av-

erage value (Avg) based on 10 runs) over succes-

sive generations. The curves in Figure 1-(a-c) clearly

demonstrate that the proposed three variants of ACO

approach for this problem converge rapidly towards

the high quality solution.

5 CONCLUSIONS

This paper concerns the tree t-spanner problem for

connected, undirected and weighted graph and pro-

poses three variants of ACO approach. All compo-

nents of each variant of ACO approach are the same

except the updating rule regarding the pheromone

trails component. All three variants of ACO ap-

proach have been tested on a set of randomly gen-

erated graph instances. Experimental results demon-

strate the effectiveness of solution quality obtained by

all three variants of ACO approach on each instance.

It was also found empirically that the performance

of all three variants of ACO approach are close to

each other. Convergence analysis on some instances

show that all three variants of ACO approach con-

verge rapidly towards the high quality solutions over

successive generations for such instances.

As a future work, other metaheuristic techniques

will be developed for this problem, as this problem

is under-studied in the domain of metaheuristic tech-

niques.

ACKNOWLEDGEMENTS

This work is supported in part by a grant (grant num-

ber YSS/2015/000276) from the Science and Engi-

neering Research Board – Department of Science &

Technology, Government of India.

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