J. Ernstberger, A.D. Perkins

16 March 2013

Southeast MAA

A Metaheuristic Search

Technique for Graceful

Labels of Graphs

jernstberger@lagrange.edu

perkins@cse.msstate.edu

● Rosa [7] denes the notion of a graceful label of a

graph.

● Restated by Vassilevska[8]

"A graceful labeling of a graph G with q edges is an

injection from the vertices of G to the set S of

integers {0,1,...q} such that when an edge with

vertices x and y are assigned the label

the resulting edge labelings are distinct."

Graceful Labels and Graphs

Graceful Labels and Graphs, cont.

● Applications of graph labelings (including graceful

labelings) are given in Bloom and Golomb [1].

● A graph that can be characterized via a graceful

label is said to be a graceful graph

Graceful Label, Examples

Figure 1: (Left) A tree diagram with five edges. (Right) A

wheel diagram with eight edges.

Another Tree Example, 35 Edges

Past Work

Ringel-Kotzig Conjecture - "all trees are graceful"

● Eshghi and Azimi [2] - Constrained programming

problem.

● Fang [3] - simulated annealing (a statistical

mechanics lens to minimization) for graceful

labelings of trees.

● Eshghi and Mahmoudzadeh [5] - Metaheuristics

(ant colony) approach for graceful labelings.

● Redl [6] - Integer and constrained programming

problem with specific implementation for speed.

Metaheuristic Approach

Holland[4] defines this concept of a genetic algorithm.

● A population P of m trial solutions (each with n

characteristics) is randomly created.

● A fitness function is defined so that the goodness-of-fit

of each member (possible solution) is measured.

● Those solutions deemed most fit remain until a new

generation. This process is known as elitism.

Metaheuristic Approach, cont.

Image courtesy of National Geographic.

● Offspring are created via the processes mutation

and crossover.

o Mutation is the result of random noise being added to a

population (or individual attributes, the genes).

o Crossover occurs with a probability p and is a direct

swap between genes.

Comparison Example: Zebras

Population Evolution

Courtesy of https://www.indexdata.com/sites/indexdata.com/files/images/zebra.png

Population Evolution, cont.

12 5

94

17

83

11102 6

Population Evolution, (3) - SotF

12 5

94

17

83

11102 6

Population Evolution, (4) - SotF

Fast

Disease

Resistant

Better

Stripes

Fast

Endurance

Disease

Resistant

Population Evolution, (5) - Next Gen!

Fast

Disease

Resistant

Better

Stripes

Fast

Endurance

Disease

Resistant

F,E DR,F

FF DR,F

S,F S,DR

● We use random permutations of the integers in the

set {0,1,...,q} (q is the number of edges of the

graph) to create each member of the population.

The population has m members, on (0,q).

● Corresponding to the population was ,

where E has m members, each on (1,q).

● Each row of E is the computed labeling for the

edges in accordance to the related edge list.

● Practiced "elitism" with varying numbers or

percentages of the elite.

● In our formulation, mutation over the integers and

crossover were equivalent to a swap.

Metaheuristic Approach

Metaheuristic Approach, cont.

● The ith member of the population was

evaluated according to a fitness functional

● Objective is maximize J on (0,1).

Metaheuristic Approach, (3)

● There is no formal theory for the convergence (or

lack thereof) of the genetic algorithm.

● The algorithm cannot state definitively that there is

no graceful label for a graph.

● Trials (for each graph): 100 trials on each of 100

different graphs.

Results

Table 1: Comparison of GA data to the Eshghi, et. al.

ACO[5] and mathematical programming[2] routines.

Results, Tree

Table 2: Metaheuristic search for graceful labelings of

trees of size greater than 25.

● Ratio of increase on time and mean generations

more than doubles (on 2.6x and 2.3x,resp.).

● Due, in part, to the sort used currently.

Future Work

● Explore labelings for large trees

● Generalized Petersen graphs and product graphs

● Computing-efficient fitness functional

● Make software available

● Port to computing-efficient language

References

● Gary S Bloom and Solomon W Golomb.

Applications of numbered undirected graphs.

Proceedings of the IEEE, 65(4):562-570,

1977.

● Kourosh Eshghi and Parham Azimi.

Applications of mathematical programming

in graceful labeling of graphs. Journal of

Applied Mathematics, 2004(1):1-8, 2004.

● Wenjie Fang. A computational approach to

the graceful tree conjecture. arXiv preprint

arXiv:1003.3045, 2010.

References, cont.

● J.H. Holland. Genetic algorithms and the optimal

allocation of trials. SIAM Journal of Computing,

2(2), 1973.

● Houra Mahmoudzadeh and Kourosh Eshghi. A

metaheuristic approach to the graceful labeling

problem. International Journal of Applied

Metaheuristic Computing (IJAMC), 1(4):42-56,

2010.

● Timothy A Redl. Graceful graphs and graceful

labelings: two mathematical programming

formulations and some other new results.

Congressus Numerantium, pages 17-32, 2003.

Refences, (3)

● Alexander Rosa. On certain valuations of the

vertices of a graph. In Theory of Graphs (Internat.

Symposium, Rome, pages 349-355, 1966.

● Virginia Vassilevska. Coding and graceful labeling

of trees.