Rainbow Connectivity Using a Rank Genetic Algorithm: Moore …downloads.hindawi.com/journals/jam/2019/4073905.pdf · 2019-07-30 · ResearchArticle Rainbow Connectivity Using a Rank

Research ArticleRainbow Connectivity Using a Rank Genetic AlgorithmMoore Cages with Girth Six

J Cervantes-Ojeda M Goacutemez-Fuentes D Gonzaacutelez-Moreno and M Olsen

Universidad Autonoma Metropolitana Cuajimalpa 05348 Mexico

Correspondence should be addressed to M Gomez-Fuentes mcgomezfuentesaimcom

Received 12 September 2018 Accepted 29 January 2019 Published 3 March 2019

Academic Editor Ali R Ashrafi

Copyright copy 2019 J Cervantes-Ojeda et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A rainbow 119905-coloring of a 119905-connected graph119866 is an edge coloring such that for any two distinct vertices119906 and V of119866 there are at least119905 internally vertex-disjoint rainbow (119906 V)-paths In this work we apply a Rank Genetic Algorithm to search for rainbow 119905-coloringsof the family of Moore cages with girth six (119905 6)-cagesWe found that an upper bound in the number of colors needed to produce arainbow 4-coloring of a (4 6)-cage is 7 improving the one currently known which is 13The computation of the minimumnumberof colors of a rainbow coloring is known to be NP-Hard and the Rank Genetic Algorithm showed good behavior finding rainbow119905-colorings with a small number of colors

1 Introduction and Definitions

Evolutionary algorithms have been applied to a wide varietyof engineering problems [1] and they have also been appliedto mathematics problems For instance Jong and Spears [2]showed that Genetic Algorithms (GA) can be used to solveNP-Complete problems [3] applied a GA to a geometryproblem and [4] solved nonlinear algebraic equations byusing aGA Herewe have successfully applied a Rank GeneticAlgorithm (Rank GA) [5] to the graph theory problem offinding the rainbow connection number of a graph (119903119888(119866))Chakraborty et al [6] proved that for a given graph 119866deciding whether 119903119888(119866) = 2 is NP-Complete and that it isalso NP-Complete to decide whether a given edge-colored(with an unbounded number of colors) graph is rainbowconnectedTherefore our motivation is to try using the RankGA heuristic on this problem To our best knowledge thisproblem has not been approached in this way nor using anyother heuristic algorithms

In a genetic algorithm we have an initial population ofindividuals in our case each individual represents a partic-ular edge-colored graph Each individual of the populationis evaluated according to a fitness function this functionmeasures the ability of the individual for reaching a prede-termined objective Once individuals are evaluated the next

generation is obtained by applying genetic operators to thepopulation inspired by the evolution in nature such as cross-over between individuals selection of the fittest individualsand mutations This procedure is repeated until the geneticalgorithm finds an individual that achieves the requiredobjective or until a maximum number of generations arereached A genetic algorithm can solve mathematical prob-lems that are intractable by exhaustive search as is the case ofthe problem described here

A graph 119866 is 119905-connected if and only if there are at least 119905internally disjoint (119906 V)-paths connecting every two distinctvertices 119906 and V of 119866 [7] A rainbow path is a path 119875 suchthat all the edges of 119875 have different colors (a path 119875 of 119866 is asequence of distinct vertices (V1 V2 V119899) such that V119894 V119894+1is an edge for 119894 isin 1 2 119899 minus 1) An edge coloring of agraph is called a rainbow 119905-coloring if for every pair of distinctvertices 119906 and V there are at least 119905 internally disjoint rainbow(119906 V)-paths The rainbow 119905-connectivity 119903119888119905(119866) (defined byChartrand et al [8]) of a graph 119866 is the minimum integer 119895such that there exists an edge coloring using 119895 colors which isa rainbow 119905-coloring The rainbow 1-connectivity is knownas the rainbow connection number and was introduced byChartrand et al [9]

The rainbow 119905-connectivity of a graph has applicationsin the Cybernetic Security (see Ericksen [10]) For any fixed

HindawiJournal of Applied MathematicsVolume 2019 Article ID 4073905 7 pageshttpsdoiorg10115520194073905

2 Journal of Applied Mathematics

119896 ge 2 deciding if 1199031198881(119866) = 119896 is NP-Complete [6 11]For more references on rainbow connectivity and rainbow 119905-connectivity we refer the reader to the survey by Li [12] et aland the book of Li and Sun [13]

A graph 119866 is 119896-regular if every vertex of 119866 has degree 119896The girth of 119866 is the length of a shortest cycle of 119866 Given twointegers 119896 ge 2 and 119892 ge 3 a (119896 119892)-cage is a 119896-regular graphwith girth 119892 and the minimum possible number of verticesLet 119899(119896 119892) denote the order of a (119896 119892)-cage For referenceson cages see the dynamic survey of Exoo and Jajcay [14]By counting the vertices emerging from a vertex or from anedge the lower bound 1198990(119896 119892) known as theMoore bound isobtained That is 119899(119896 119892) ge 1198990(119896 119892) where1198990 (119896 119892)

=

2(119892minus2)2sum119894=0

(119896 minus 1)119894 = 2 (119896 minus 1)1198922 minus 2119896 minus 2 if 119892 is even1 + (119892minus1)2sum119894=1

119896 (119896 minus 1)119894minus1 = 119896 (119896 minus 1)(119892minus1)2 minus 2119896 minus 2 if 119892 is odd(1)

When 119899(119896 119892) = 1198990(119896 119892) the (119896 119892)-cage is called a Moore(119896 119892)-cage for references onMoore cages see [15]TheMoorecages have been characterized [16ndash18]The existence of (119896 6)-Moore cages is related to the existence of finite geometriesFor instance the (3 6)-cage is the incidence graph of Fanorsquosplane Concerning connectivity of (119896 6)-cages it has beenproved that they are 119896-connected [19]

Chartrand et al [20] showed that the rainbow 3-connectivity of the (3 6)-cage (the Moore cage known asHeawood graph) is between 5 and 7 inclusive RecentlyBalbuena et al [21] proved that it is not 5 and they boundedthe rainbow 119896-connectivity of (119896 6)-cages as follows if 119866 is aMoore (119896 6)-cage then

119896 le 119903119888119896 (119866) le 1198962 minus 119896 + 1 (2)

If 119866 is a (119896 6)-cage then for 119896 = 3 119896 = 4 and 119896 = 5respectively we have 1199031198883(119866) le 7 1199031198884(119866) le 13 and 1199031198885(119866) le21 respectively In this paper we use a genetic algorithm tosearch for rainbow 119896-colorings of (119896 6)-cages with 119896 = 3 and119896 = 4 [21] in order to see if the upper bound for 119903119888119896(119866) can beimproved

The structure of this paper is as follows In Section 2 weexplain how theRankGAwas applied to the rainbow coloringproblem Results are presented in Section 3 and finally theconclusions are drawn in Section 4

2 Finding Rainbow Colorings in a Moore Cagewith a Genetic Algorithm

Finding a particular rainbow 119905-coloring of a graph 119866 using119903 colors (that is a coloring with 119905 internally disjoint rainbowpaths between any pair of vertices) implies that 119903119888119905 (119866) le 119903Weuse an adapted version of the Rank GA [5] to find rainbowcolorings in a Moore cage To do so each individual in thepopulation contains a list of the assigned colors of each edgein the graph and is initialized randomly In the Rank GAthe individuals of the population are ranked from best to

Table 1 Parameters of the Rank GA

Description ValuePopulation size = |119864 (119866)|Number of Generations No limitPopulationrsquos selective pressure 119878 3

worst in terms of their fitness before the genetic operators(selection recombination and mutation) are applied Theapplication of these genetic operators depends on the rank ofeach individual in the populationThe top ranked individualstend to vary less than the bottom ranked ones This is tomake the latter try to escape from local optima of the fitnessfunction Also top ranked individuals tend to be clonedmorethan others who tend to disappear

The adapted RankGA pseudocode that was used is givenin Algorithm 1 and its parameters are shown in Table 1

21 The Fitness Function We detail below the fitness func-tion As this is an interdisciplinary work we express thefitness function mathematically (below) and also in a com-putational way in Algorithm 2

Let 119866 be a graph let Ψ(119866) be the set of all possiblecolorings of the edges 119864(119866) and let 119860 be the set of all pairsof vertices of 119866 that is 119860 = 1198861 1198862 | 1198861 1198862 isin 119881(119866) Thefitness function to be maximized using the RankGA is givenas

119888119900119897119900119903119894119899119892119865119894119905119899119890119904119904 Ψ (119866) 997888rarr R [0 1] (3)

119888119900119897119900119903119894119899119892119865119894119905119899119890119904119904 (120595) = [prodaisinA

119901119886119894119903119865119894119905119899119890119904119904 (a 120595)]1|119860| (4)

where 120595 isin Ψ(119866) and119901119886119894119903119865119894119905119899119890119904119904 119860 times Ψ (119866) 997888rarr R [0 1] (5)

119901119886119894119903119865119894119905119899119890119904119904 (a 120595) = maxP119896aisin119863119875119878

119896(a)119889119901119904119865119894119905119899119890119904119904 (P119896a 120595) (6)

where119863119875119878119896(a) is the set of all setsP119896a of 119896 internally disjointpaths between the vertices in a and

119889119901119904119865119894119905119899119890119904119904 119863119875119878119896 (a) times Ψ (119866) 997888rarr R [0 1] (7)

119889119901119904119865119894119905119899119890119904119904 (P119896a 120595)

= [[

prod119901isinP119896a

119901119886119905ℎ119865119894119905119899119890119904119904 (119901 120595)]]1|P119896a | (8)

where

119901119886119905ℎ119865119894119905119899119890119904119904 P119896a times Ψ (119866) 997888rarr R [0 1] (9)

119901119886119905ℎ119865119894119905119899119890119904119904 (119901 120595) = 119901119886119905ℎ119873119906119898119862119900119897119900119903119904 (119901 120595)1003816100381610038161003816119864 (119901)1003816100381610038161003816 (10)

Journal of Applied Mathematics 3

procedure RankGA119899119906119898119868119899119889119894V119894119889119906119886119897119904 larr997888 |119864(119866)|119875119900119901119906119897119886119905119894119900119899119868119899119894119905119894119886119897119894119911119886119905119894119900119899119864V119886119897119906119886119905119894119900119899 and 119878119900119903119905while not end Criteria met do119877119886119899119896119878119890119897119890119888119905119894119900119899119877119886119899119896119877119890119888119900119898119887119894119899119886119905119894119900119899119864V119886119897119906119886119905119894119900119899 and 119878119900119903119905119877119886119899119896119872119906119905119886119905119894119900119899119864V119886119897119906119886119905119894119900119899 and 119878119900119903119905

procedure Rank Selection119888119897119900119899119890119904 larr997888 119899119906119897119897for 119894 in [0119899119906119898119868119899119889119894V119894119889119906119886119897119904 minus 1] do119903 larr997888 119894(119899119906119898119868119899119889119894V119894119889119906119886119897119904 minus 1)119899 larr997888 lfloor119878(1 minus 119903)(119878minus1)rfloor

for 119895 in [0119899 minus 1] do119888119897119900119899119890119904119886119889119889(119894119899119889119894) ⊳ 119894119899119889119894 is cloned 119899 times119894 larr997888 0while 119888119897119900119899119890119904119904119894119911119890() lt 119899119906119898119868119899119889119894V119894119889119906119886119897119904 do119903 larr997888 119894(119899119906119898119868119899119889119894V119894119889119906119886119897119904 minus 1)119899 larr997888 119878(1 minus 119903)(119878minus1)119891 larr997888 119899 minus lfloor119899rfloor ⊳ 119891 is the fractional part of 119899

if 119903119886119899119889119900119898(0 1) lt 119891 then119888119897119900119899119890119904119886119889119889(119894119899119889119894) ⊳ one extra clone of 119894119899119889119894119894 larr997888 (119894 + 1)mod 119899119906119898119868119899119889119894V119894119889119906119886119897119904119894119899119889 larr997888 119888119897119900119899119890119904 ⊳ replace population119878119900119903119905procedure Rank Recombination

for 119894 in [0119899119906119898119868119899119889119894V119894119889119906119886119897119904 minus 2] step 2 dofor 119895 in [0119899119906119898119866119890119899119890119904 minus 1] do

if 119903119886119899119889119900119898(0 1) lt 05 thenSwitch(119892119894119895 119892119894+1119895) ⊳mating is with 119894119899119889119894+1

procedure Rank Mutationfor 119894 in [0119899119906119898119868119899119889119894V119894119889119906119886119897119904 minus 1] do119903 larr997888 119894(119899119906119898119868119899119889119894V119894119889119906119886119897119904 minus 1)

for 119895 in [0119899119906119898119866119890119899119890119904 minus 1] doif 119903119886119899119889119900119898(0 1) lt 119903 then119892119894119895 larr997888 random integer in the range [0119899119906119898119862119900119897119900119903119904 minus 1]

Algorithm 1 Rank GA

where 119864(119901) is the set of all edges in path 119901 and

119901119886119905ℎ119873119906119898119862119900119897119900119903119904 P119896a times Ψ (119866) 997888rarr R [0 1] (11)

119901119886119905ℎ119873119906119898119862119900119897119900119903119904 (119901 120595) =10038161003816100381610038161003816100381610038161003816100381610038161003816 ⋃119890isin119864(119901)

120595 (119890)10038161003816100381610038161003816100381610038161003816100381610038161003816 (12)

where 120595(119890) is the color that the coloring 120595 assigns to edge 119890The function 119901119886119905ℎ119873119906119898119862119900119897119900119903119904 in (12) calculates the

cardinality of a set made out of the colors120595(119890) assigned to theedges 119890 isin 119864(119901) in path 119901 The function 119901119886119905ℎ119865119894119905119899119890119904119904 in (10)calculates the ratio of the 119901119886119905ℎ119873119906119898119862119900119897119900119903119904 over the length ofpath 119901 being this a value in the range [1|119864(119901)| 1] yielding 1only in case the path has a rainbow coloring and lower valuesas the number of colors used in the path is reduced Thefunction 119889119901119904119865119894119905119899119890119904119904 in (8) calculates the geometric meanof all the 119901119886119905ℎ119865119894119905119899119890119904119904 values of all the 119896 + 1 paths 119901 inthe disjoint path set P119896a yielding a value in the range [0 1]

with 1 representing a rainbow disjoint path set The function119901119886119894119903119865119894119905119899119890119904119904 in (6) returns the maximum 119889119901119904119865119894119905119899119890119904119904 thatcan be obtained from the set of disjoint path sets 119863119875119878119896(a)between the vertices in the vertex pair a yielding 1 if andonly if there exists at least one rainbow disjoint path setP119896a in 119863119875119878119896(a) Finally the function 119888119900119897119900119903119894119899119892119865119894119905119899119890119904119904 in (4)computes the geometric mean of all the 119901119886119894119903119865119894119905119899119890119904119904 valuesyielding a result that is 1 if and only if all pairs of vertices ahave a fitness value of 1 ie the given coloring 120595 is a rainbowcoloring

The pseudocode of the fitness function is given in Algo-rithm 2

22 Finding the Set of Internally Disjoint Paths Sets betweenPairs of Vertices in a Graph The total number of possiblesets 119875119896a of 119896 paths (internally disjoint or not) between thevertices in pair a is very high as 119896 grows so in order to savecomputing time when calculating the fitness of an individual


function 119888119900119897119900119903119894119899119892119865119894119905119899119890119904119904(119888119900119897119900119903119894119899119892)119903119890119904119906119897119905 larr997888 1for Each vertex pair a in the Graph do119903 larr997888 119901119886119894119903119865119894119905119899119890119904119904(a 119888119900119897119900119903119894119899119892)119903119890119904119906119897119905 larr997888 119903119890119904119906119897119905 lowast 119903 ⊳ Accumulate the productreturn 119903119890119904119906119897119905

function 119901119886119894119903119865119894119905119899119890119904119904(a 119888119900119897119900119903119894119899119892)119903119890119904119906119897119905 larr997888 0for Each InternallyDisjointPathsSet119863119875119878119896 of pair a do119903 larr997888 119889119901119904119865119894119905119899119890119904119904(119863119875119878119896 119888119900119897119900119903119894119899119892)

if 119903 gt 119903119890119904119906119897119905 then119903119890119904119906119897119905 larr997888 119903 ⊳ Select best fitnessreturn 119903119890119904119906119897119905

function 119889119901119904119865119894119905119899119890119904119904(119863119875119878119896 119888119900119897119900119903119894119899119892)119903119890119904119906119897119905 larr997888 1for Each Path 119901 in119863119875119878119896 do119903 larr997888 119901119886119905ℎ119865119894119905119899119890119904119904(119901 119888119900119897119900119903119894119899119892)119903119890119904119906119897119905 larr997888 119903119890119904119906119897119905 lowast 119903 ⊳ Accumulate the productreturn 119903119890119904119906119897119905

function 119901119886119905ℎ119865119894119905119899119890119904119904(119901 119888119900119897119900119903119894119899119892)119903119890119904119906119897119905 larr997888 119901119886119905ℎ119873119906119898119862119900119897119900119903119904(119901 119888119900119897119900119903119894119899119892)119901119886119905ℎ119904119894119911119890return 119903119890119904119906119897119905

function 119901119886119905ℎ119873119906119898119862119900119897119900119903119904(119901 119888119900119897119900119903119894119899119892)119888119900119906119899119905 larr997888 0for Each Step 119904 in 119901 do119888 larr997888 119888119900119897119900119903119894119899119892[119904119904119900119906119903119888119890][119904119889119890119904119905]

if not 119906119904119890119889[119888] then ⊳ count colors only once119888119900119906119899119905 larr997888 119888119900119906119899119905 + 1119906119904119890119889[119888] larr997888 119905119903119906119890 ⊳mark color as already usedreturn 119888119900119906119899119905

Algorithm 2 Coloring fitness function

the function 119901119886119894119903119865119894119905119899119890119904119904 in (6) iterates only on the elementsof the set 119863119875119878119896(a) which is the set of path sets P119896a thatcontain 119896 internally disjoint paths between the vertices in aAlgorithm 3 describes a way to calculate and store the set ofall 119863119875119878119896(a)

In this algorithm the procedure 119904119905119900119903119890119863119894119904119895119900119894119899119905119875119886119905ℎ119904119878119890119905119904takes each vertex pair a in 119866 and calls function 119892119890119905119875119886119905ℎ119904in order to get the full set 119875a of paths 119901 between thevertices in a with length 119897 le 119899119906119898119862119900119897119900119903119904 (since longer pathscannot be rainbow colored) Then it calls another function119892119890119905119863119894119904119895119900119894119899119905119875119886119905ℎ119904119878119890119905119904 which tests each set 119875119896a containing 119896paths 119901 isin 119875a between the vertices in a to see if 119875119896a is a disjointpaths set and returns the set 119863119875119878119896(a) of disjoint paths setsP119896a

The function 119892119890119905119875119886119905ℎ119904 is recursive and takes 3 parame-ters a vertex pair a = 1198861 1198862 a graph 119866 and a maximumlength 119898119886119909119871119890119899119892119905ℎ of the paths to be found Figure 1 illus-trates how this function works Paths that go from vertex 1198861to vertex 1198862 are made by adding the step from 1198861 to 119899119890119894119892ℎ119887119900119903119894of 1198861 to all paths that go from 119899119890119894gℎ119887119900119903119894 of 1198861 to 1198862 for each119894 But before doing this vertex 1198861 is marked as visited inorder to avoid paths that visit a vertex more than once Aftercomputing the set of paths vertex 1198861 is unmarked

In Table 2 we show how several measures grow withparameter 119896 First there is the number of vertices |119881(119866)|

in the (119896 6) cage followed by the total number of vertexpairs Then there is the number of vertex pairs with distances119889 = 1 119889 = 2 and 119889 = 3 between them respectivelyWe can see how the total number of paths with length119897 le 7 = 119899119906119898119862119900119897119900119903119904 between vertex pairs changes withdistance 119889 and with 119896 The next measure is the importantone because it shows the drastic increase with 119896 of the totalnumber of paths sets containing 119896 paths (internally disjointand noninternally disjoint) that can be formed with the pathsbetween the pairs of vertices with distances 119889 = 1 119889 =2 and 119889 = 3 between them respectively All these setsneed to be tested to see which of them are disjoint pats setsFinally we show the total number of paths sets that needto be tested and the approximate time to compute this withour existing algorithms and computing power As one cansee the numbers are much bigger as 119896 increases making itpractically impossible at the moment to do the work for119896 = 53 Results

The genetic algorithm was able to find several rainbowcolorings of the (119896 6)-Moore cage for 119896 = 3 and 119896 = 4 using7 colors Using 6 colors the algorithm could not find anyrainbow colorations for neither 119896 = 3 nor 119896 = 4 A samplerainbow coloring that was found can be seen in Figure 2


procedure 119904119905119900119903119890119863119894119904119895119900119894119899119905119875119886119905ℎ119904119878119890119905119904(119866)119894119899119894119905119881119890119903119905119890119909119873119890119894119892ℎ119887119900119903119904( ) ⊳ set the neighbors of each vertex for current 119896for Each a isin 119860 = 1198861 1198862|1198861 1198862 isin 119881(119866) do119901119886119905ℎ119904[a] larr997888 119892119890119905119875119886119905ℎ119904(a 119866 119899119906119898119862119900119897119900119903119904)119889119894119904119895119900119894119899119905119875119886119905ℎ119904119878119890119905119904[a] larr997888 119892119890119905119863119894119904119895119900119894119899119905119875119886119905ℎ119904119878119890119905119904(119901119886119905ℎ119904[a] 119896)

function recursive119892119890119905119875119886119905ℎ119904(a 119866119898119886119909119871119890119899119892119905ℎ)if 119898119886119909119871119890119899119892119905ℎ le 0 then ⊳ wrong length

return 119899119906119897119897if V119894119904119894119905119890119889[1198861] then ⊳ 1198861 has already been visited

return 119899119906119897119897 ⊳ No paths to 1198862if 1198861 = 1198862 then ⊳ destination reached119901 larr997888 1198862 ⊳ single vertex in path

return 119901 ⊳ single path in the setV119894119904119894119905119890119889[1198861] larr997888 119905119903119906119890for 119894 = 1 to 119896 + 1 do119899119890119909119905119875119886119894119903 larr997888 119899119890119894119892ℎ119887119900119903[119894](1198861) 1198862 ⊳ 119894119905ℎ neighbor of 1198861119904119906119887119877119890119904119906119897119905 larr997888 119892119890119905119875119886119905ℎ119904(119899119890119909119905119875119886119894119903119866 119898119886119909119871119890119899119892119905ℎ minus 1)

for Each path 119901 in 119904119906119887119877119890119904119906119897119905 do119899119890119908119875119886119905ℎ larr997888 1198861_119901 ⊳ concatenate paths119903119890119904119906119897119905 larr997888 119903119890119904119906119897119905 cup 119899119890119908119875119886119905ℎ ⊳ aggregate pathV119894119904119894119905119890119889[1198861] larr997888 119891119886119897119904119890return 119903119890119904119906119897119905

function 119892119890119905119863119894119904119895119900119894119899119905119875119886119905ℎ119904119878119890119905119904(119875a 119896)for Each path set 119875119896a of 119896 paths 119901 isin 119875a do ⊳may have huge of iterations

V119894119904119894119905119890119889 larr997888 119891119886119897119904119890for Each internal vertex V of 119901 do

if V119894119904119894119905119890119889[V] thennext 119875119896a

V119894119904119894119905119890119889[V] larr997888 119905119903119906119890119903119890119904119906119897119905 larr997888 119903119890119904119906119897119905 cup 119875119896areturn 119903119890119904119906119897119905

Algorithm 3 Disjoint paths sets between vertices

Length le lengthMax

Length le lengthMax -1

a1

neighbor

neighbor

neighbork+

Paths from neighbor to


Paths from neighbork+ to

a2middot middot middotmiddot middot middot

Figure 1 Recursive finding of the set of paths between vertices 1198861 and 1198862

4 Conclusions

The problem of finding the rainbow 119896-connectivity 119903119888119905(119866)of (119896 6)-cages is intractable by exhaustive search So it hadnot been possible to verify if it is possible to improve theupper bound given by (2) With the Rank GA we could seethat it is unlikely that the upper bound for (3 6)-cages isless than 7 since with 6 colors no solution was found after

having let the algorithm run for a long time We also foundthat for the (4 6)-cage there is an upper bound of 7 colorssince the algorithm found rainbow colorings in this caseThis upper bound improves quite a lot the 13 colors givenby (2) which binds the rainbow 119896-connectivity of (119896 6)-cageswith a quadratic function By finding out that the upperbound for the (4 6)-cage is 7 we know that this upper boundfunction could be one that grows slower than a quadratic


Table 2 Growth with 119896 of the effort to find the sets of disjoint paths sets119896 3 4 5|119881(119866)| 14 26 42vertex pairs 91 325 861vertex pairs with 119889 = 1 21 52 105vertex pairs with 119889 = 2 42 156 420vertex pairs with 119889 = 3 28 117 336paths for pairs with 119889 = 1 17 136 642paths for pairs with 119889 = 2 17 82 257paths for pairs with 119889 = 3 33 244 1025paths sets for pairs with 119889 = 1 680 13633830 887805286368paths sets for pairs with 119889 = 2 680 1749060 8984340696paths sets for pairs with 119889 = 3 5456 144084501 9336731022080total effort 195608 17839699137 3234134601579840approx time 32ms 231min 8 years

0 12

3

4

5

6

7

8

9

1011

12131415

16

17

18

19

20

21

22

23

2425

Figure 2 Sample rainbow coloring with 119896 = 4 and 119899119906119898119862119900119897119900119903119904 = 7

function and may be much slower To find the form of thisfunction the problem should be scaled to Moore cages with119896 ge 5 however for 119896 = 5 the problem is computationallyvery expensive with the algorithms and computing powerthat we currently use Thus as for future work we willconsider different approaches such as taking into account thesymmetries of the graph parallelization and improvement ofalgorithms

We believe that graph theory can benefit from the useof artificial intelligence techniques in some known NP-Complete and NP-Hard problems such as the one presentedin this paper and there may be many of these cases

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they do not have any conflicts ofinterest

Acknowledgments

This work was funded by Universidad Autonoma Metro-politana-Cuajimalpa The authors would like to thank DrRodolfo Rene Suarez Molnar UAM-Cuajimalpa Unit Rectorfor institutional support

References

[1] D Dasgupta and Z Michalewicz Evolutionary Algorithms inEngineering Applications Springer Science and Business Media2013

[2] K A De Jong and W M Spears ldquoUsing genetic algorithms tosolve NP-complete problemsrdquo International Computer GamesAssociation pp 124ndash132 1989

[3] S Jakobs ldquoOn genetic algorithms for the packing of polygonsrdquoEuropean Journal of Operational Research vol 88 no 1 pp 165ndash181 1996

[4] A Pourrajabian R Ebrahimi M Mirzaei and M ShamsldquoApplying genetic algorithms for solving nonlinear algebraicequationsrdquo Applied Mathematics and Computation vol 219 no24 pp 11483ndash11494 2013

[5] J Cervantes and C R Stephens ldquoLimitations of existingmutation rate heuristics and how a rank GA overcomes themrdquoIEEE Transactions on Evolutionary Computation vol 13 no 2pp 369ndash397 2009

[6] SChakraborty E FischerAMatsliah andRYuster ldquoHardnessand algorithms for rainbow connectivityrdquo in Proceedings ofthe 26th International Symposium on Theoretical Aspects ofComputer Science STACS pp 243ndash254 2009 Also see Journalof Combinatorial Optimization vol 21 pp 330ndash347 2011

[7] K Menger ldquoZur allgemeinen Kurventheorierdquo FundamentaMathematicae vol 10 pp 96ndash115 1927

[8] G Chartrand G L Johns K A McKeon and P Zhang ldquoTherainbow connectivity of a graphrdquo Networks An InternationalJournal vol 54 no 2 pp 75ndash81 2009


[9] G Chartrand G L Johns K A McKeon and P ZhangldquoRainbow connection in graphsrdquo Mathematica Bohemica vol133 no 1 pp 85ndash98 2008

[10] A Ericksen ldquoA matter of securityrdquo Graduating Engineer andComputer Careers pp 24ndash28 2007

[11] V B Le and Z Tuza ldquoFinding optimal rainbow connection ishardrdquo 2009

[12] X Li Y Shi and Y Sun ldquoRainbow connections of graphs asurveyrdquoGraphs and Combinatorics vol 29 no 1 pp 1ndash38 2013

[13] X Li and Y Sun Rainbow Connections of Graphs SpringerLondon UK 2013

[14] G Exoo and R Jajcay ldquoDynamic cage surveyrdquo The ElectronicJournal of Combinatorics vol DS16 2013

[15] M Miller and J Siran ldquoMoore graphs and beyond a surveyof the degreediameter problemrdquo The Electronic Journal ofCombinatorics - Dynamic Surveys vol 14 2005

[16] E Bannai and T Ito ldquoOn finite Moore graphsrdquo Journal of theFaculty of Science University of Tokyo vol 20 pp 191ndash208 1973

[17] R M Damerell ldquoOn Moore graphsrdquoMathematical Proceedingsof the Cambridge Philosophical Society vol 74 no 2 pp 227ndash236 1973

[18] A J Hoffman and R R Singleton ldquoOn Moore graphs withdiameters 2 and 3rdquo International Business Machines Journal ofResearch and Development vol 4 pp 497ndash504 1960

[19] X Marcote C Balbuena and I Pelayo ldquoOn the connectivity ofcages with girth five six and eightrdquo Discrete Mathematics vol307 no 11-12 pp 1441ndash1446 2007

[20] G Chartrand G L Johns K A McKeon and P Zhang ldquoOnthe rainbow connectivity of cagesrdquo in Proceedings of the Thirty-Eighth Southeastern International Conference on CombinatoricsGraph Theory and Computing vol 184 pp 209ndash222 2007

[21] C Balbuena J Fresan D Gonzalez-Moreno and M OlsenldquoRainbow connectivity of Moore cages of girth 6rdquo DiscreteApplied Mathematics vol 250 pp 104ndash109 2018

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119896 ge 2 deciding if 1199031198881(119866) = 119896 is NP-Complete [6 11]For more references on rainbow connectivity and rainbow 119905-connectivity we refer the reader to the survey by Li [12] et aland the book of Li and Sun [13]

A graph 119866 is 119896-regular if every vertex of 119866 has degree 119896The girth of 119866 is the length of a shortest cycle of 119866 Given twointegers 119896 ge 2 and 119892 ge 3 a (119896 119892)-cage is a 119896-regular graphwith girth 119892 and the minimum possible number of verticesLet 119899(119896 119892) denote the order of a (119896 119892)-cage For referenceson cages see the dynamic survey of Exoo and Jajcay [14]By counting the vertices emerging from a vertex or from anedge the lower bound 1198990(119896 119892) known as theMoore bound isobtained That is 119899(119896 119892) ge 1198990(119896 119892) where1198990 (119896 119892)

=

2(119892minus2)2sum119894=0

(119896 minus 1)119894 = 2 (119896 minus 1)1198922 minus 2119896 minus 2 if 119892 is even1 + (119892minus1)2sum119894=1

119896 (119896 minus 1)119894minus1 = 119896 (119896 minus 1)(119892minus1)2 minus 2119896 minus 2 if 119892 is odd(1)

When 119899(119896 119892) = 1198990(119896 119892) the (119896 119892)-cage is called a Moore(119896 119892)-cage for references onMoore cages see [15]TheMoorecages have been characterized [16ndash18]The existence of (119896 6)-Moore cages is related to the existence of finite geometriesFor instance the (3 6)-cage is the incidence graph of Fanorsquosplane Concerning connectivity of (119896 6)-cages it has beenproved that they are 119896-connected [19]

Chartrand et al [20] showed that the rainbow 3-connectivity of the (3 6)-cage (the Moore cage known asHeawood graph) is between 5 and 7 inclusive RecentlyBalbuena et al [21] proved that it is not 5 and they boundedthe rainbow 119896-connectivity of (119896 6)-cages as follows if 119866 is aMoore (119896 6)-cage then

119896 le 119903119888119896 (119866) le 1198962 minus 119896 + 1 (2)

If 119866 is a (119896 6)-cage then for 119896 = 3 119896 = 4 and 119896 = 5respectively we have 1199031198883(119866) le 7 1199031198884(119866) le 13 and 1199031198885(119866) le21 respectively In this paper we use a genetic algorithm tosearch for rainbow 119896-colorings of (119896 6)-cages with 119896 = 3 and119896 = 4 [21] in order to see if the upper bound for 119903119888119896(119866) can beimproved

The structure of this paper is as follows In Section 2 weexplain how theRankGAwas applied to the rainbow coloringproblem Results are presented in Section 3 and finally theconclusions are drawn in Section 4

2 Finding Rainbow Colorings in a Moore Cagewith a Genetic Algorithm

Finding a particular rainbow 119905-coloring of a graph 119866 using119903 colors (that is a coloring with 119905 internally disjoint rainbowpaths between any pair of vertices) implies that 119903119888119905 (119866) le 119903Weuse an adapted version of the Rank GA [5] to find rainbowcolorings in a Moore cage To do so each individual in thepopulation contains a list of the assigned colors of each edgein the graph and is initialized randomly In the Rank GAthe individuals of the population are ranked from best to

Table 1 Parameters of the Rank GA

Description ValuePopulation size = |119864 (119866)|Number of Generations No limitPopulationrsquos selective pressure 119878 3

worst in terms of their fitness before the genetic operators(selection recombination and mutation) are applied Theapplication of these genetic operators depends on the rank ofeach individual in the populationThe top ranked individualstend to vary less than the bottom ranked ones This is tomake the latter try to escape from local optima of the fitnessfunction Also top ranked individuals tend to be clonedmorethan others who tend to disappear

The adapted RankGA pseudocode that was used is givenin Algorithm 1 and its parameters are shown in Table 1

21 The Fitness Function We detail below the fitness func-tion As this is an interdisciplinary work we express thefitness function mathematically (below) and also in a com-putational way in Algorithm 2

Let 119866 be a graph let Ψ(119866) be the set of all possiblecolorings of the edges 119864(119866) and let 119860 be the set of all pairsof vertices of 119866 that is 119860 = 1198861 1198862 | 1198861 1198862 isin 119881(119866) Thefitness function to be maximized using the RankGA is givenas

119888119900119897119900119903119894119899119892119865119894119905119899119890119904119904 Ψ (119866) 997888rarr R [0 1] (3)

119888119900119897119900119903119894119899119892119865119894119905119899119890119904119904 (120595) = [prodaisinA

119901119886119894119903119865119894119905119899119890119904119904 (a 120595)]1|119860| (4)

where 120595 isin Ψ(119866) and119901119886119894119903119865119894119905119899119890119904119904 119860 times Ψ (119866) 997888rarr R [0 1] (5)

119901119886119894119903119865119894119905119899119890119904119904 (a 120595) = maxP119896aisin119863119875119878

119896(a)119889119901119904119865119894119905119899119890119904119904 (P119896a 120595) (6)

where119863119875119878119896(a) is the set of all setsP119896a of 119896 internally disjointpaths between the vertices in a and

119889119901119904119865119894119905119899119890119904119904 119863119875119878119896 (a) times Ψ (119866) 997888rarr R [0 1] (7)

119889119901119904119865119894119905119899119890119904119904 (P119896a 120595)

= [[

prod119901isinP119896a

119901119886119905ℎ119865119894119905119899119890119904119904 (119901 120595)]]1|P119896a | (8)

where

119901119886119905ℎ119865119894119905119899119890119904119904 P119896a times Ψ (119866) 997888rarr R [0 1] (9)

119901119886119905ℎ119865119894119905119899119890119904119904 (119901 120595) = 119901119886119905ℎ119873119906119898119862119900119897119900119903119904 (119901 120595)1003816100381610038161003816119864 (119901)1003816100381610038161003816 (10)










Algorithm 1 Rank GA


119901119886119905ℎ119873119906119898119862119900119897119900119903119904 P119896a times Ψ (119866) 997888rarr R [0 1] (11)

119901119886119905ℎ119873119906119898119862119900119897119900119903119904 (119901 120595) =10038161003816100381610038161003816100381610038161003816100381610038161003816 ⋃119890isin119864(119901)

120595 (119890)10038161003816100381610038161003816100381610038161003816100381610038161003816 (12)






























if V119894119904119894119905119890119889[V] thennext 119875119896a



Length le lengthMax


a1

neighbor

neighbor

neighbork+






4 Conclusions





0 12

3

4

5

6

7

8

9

1011

12131415

16

17

18

19

20

21

22

23

2425




Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018

















Algorithm 1 Rank GA


119901119886119905ℎ119873119906119898119862119900119897119900119903119904 P119896a times Ψ (119866) 997888rarr R [0 1] (11)

119901119886119905ℎ119873119906119898119862119900119897119900119903119904 (119901 120595) =10038161003816100381610038161003816100381610038161003816100381610038161003816 ⋃119890isin119864(119901)

120595 (119890)10038161003816100381610038161003816100381610038161003816100381610038161003816 (12)






























if V119894119904119894119905119890119889[V] thennext 119875119896a



Length le lengthMax


a1

neighbor

neighbor

neighbork+






4 Conclusions





0 12

3

4

5

6

7

8

9

1011

12131415

16

17

18

19

20

21

22

23

2425




Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018
































if V119894119904119894119905119890119889[V] thennext 119875119896a



Length le lengthMax


a1

neighbor

neighbor

neighbork+






4 Conclusions





0 12

3

4

5

6

7

8

9

1011

12131415

16

17

18

19

20

21

22

23

2425




Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018

















if V119894119904119894119905119890119889[V] thennext 119875119896a



Length le lengthMax


a1

neighbor

neighbor

neighbork+






4 Conclusions





0 12

3

4

5

6

7

8

9

1011

12131415

16

17

18

19

20

21

22

23

2425




Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018










0 12

3

4

5

6

7

8

9

1011

12131415

16

17

18

19

20

21

22

23

2425




Data Availability




Acknowledgments


References






























Journal of












Journal of







Volume 2018






Volume 2018





























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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Rainbow Connectivity Using a Rank Genetic Algorithm: Moore …downloads.hindawi.com/journals/jam/2019/4073905.pdf · 2019-07-30 · ResearchArticle Rainbow Connectivity Using a Rank