HybridHAM: A Novel Hybrid Heuristic for Finding Hamiltonian Cycle · 2019. 7. 30. · graph....

Research ArticleHybridHAM A Novel Hybrid Heuristic forFinding Hamiltonian Cycle

K R Seeja

Department of Computer Science amp Engineering Indira Gandhi Delhi Technical University for Women Kashmere GateDelhi 110006 India

Correspondence should be addressed to K R Seeja krseejagmailcom

Received 9 May 2018 Revised 7 September 2018 Accepted 25 September 2018 Published 16 October 2018

Academic Editor Jun He

Copyright copy 2018 K R Seeja This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Hamiltonian Cycle Problem is one of the most explored combinatorial problems Being an NP-complete problem heuristicapproaches are found to bemore powerful than exponential time exact algorithms This paper presents an efficient hybrid heuristicthat sits in between the complex reliable approaches and simple faster approaches The proposed algorithm is a combination ofgreedy rotational transformation and unreachable vertex heuristics that works in three phases In the first phase an initial pathis created by using greedy depth first search This initial path is then extended to a Hamiltonian path in second phase by usingrotational transformation and greedy depth first search Third phase converts the Hamiltonian path into a Hamiltonian cycle byusing rotational transformation The proposed approach could find Hamiltonian cycles from a set of hard graphs collected fromthe literature all the Hamiltonian instances (1000 to 5000 vertices) given in TSPLIB and some instances of FHCP Challenge SetMoreover the algorithm has O(n3) worst case time complexity The performance of the algorithm has been compared with thestate-of-the-art algorithms and it was found that HybridHAM outperforms others in terms of running time

1 Introduction

The Hamiltonian Cycle Problem (HCP) is to identify a cyclein an undirected graph connecting all the vertices in thegraph It is considered as a subproblem of the most popularNP-complete problem the Travelling Salesman Problem(TSP) where the problem is to find the minimum weightedHamiltonian cycle Hamiltonian cycles have many applica-tions like reconstructing genome sequences solving gameslike Icosian game finding a knights tour on a chessboard andfinding circular embeddings for regular graphs There is nosingle efficient algorithm for this problem till date The state-of-the-art algorithms are mainly classified into two expo-nential time exhaustive search algorithms and polynomialtime heuristic algorithms While the first category guaranteesgiving solution the latter does not The latter gives solutionin sufficiently less time in most of the cases compared tothe first The algorithms in the first category generally findsome efficient pruning rules for reducing the search spaceand thus improving the running time while those in thesecond category find some general thump rules for finding

the solution in as many problem instances as possible in lesstime The objective of this research is to design a heuristicwhich is quicker than the established sophisticated heuristicsbut which is more reliable than the very fast techniques

Many theorems can be found in literature giving thenecessary and sufficient conditions [1ndash4] for Hamiltoniancycle These conditions are used for checking whether thegraph is Hamiltonian or not A good study [5] on thesetheorems and algorithms for solving HCP is given by Van-degriend amp Basil Rubin amp Frank [6] proposed an exhaustivesearch method for finding all Hamiltonian paths or cyclesin a directed or undirected graph Christophides [7] pro-posed a multipath algorithm that was again an intelligentsearch algorithm with exponential complexity Christophidesalgorithm has been improved by Kocay amp William [8] byproposing two operations for pruning the search spaceMartello [9] proposed a backtrack search algorithm that useslow degree first heuristic for selecting the next vertex Ejovet al [10] solved HCP by solving equations drawn from theadjacent matrix of the graph Even though they could findlong cycles in case of non-Hamiltonian graphs they could

HindawiJournal of OptimizationVolume 2018 Article ID 9328103 10 pageshttpsdoiorg10115520189328103

2 Journal of Optimization

not reduce the exponential time complexity Posarsquos algorithm[11] is considered as the base algorithm for the heuristicalgorithms for HCP Posarsquos idea of rotational transformationand its variants formed the basis of almost all heuristicalgorithms proposed later Angluin and Valiant [12] proposeda much more complicated transformation for directed graphas the rotational transformation is not suitable for directedgraph Bollobas et al [13] proposed a Hamiltonian cyclealgorithm called HAM that uses rotational transformationand cycle extension Various versions of HAM algorithmlike SparseHam [14] and HideHam [15] are also proposedfor different kinds of graphs Brunacci [16] proposed twoalgorithms DB2 and DB2A which consider the HCP as aversion of TSP and the nonedges as highly weighted edgesDB2A algorithm is a modification of DBA algorithm fordirected graphs where directed graph is transformed intoundirected graph Many TSP heuristics like the famous Lin-Kernighan heuristic [17 18] use a technique called ldquok-optrdquotransformation [19 20] which is an exchange of k edgesBaniasadi et al [21] proposed a ldquosnakes and laddersrdquo heuristicfor solving HCP inspired from k-opt transformations Thereare very few published HCP heuristics that sit in the ldquomid-dlerdquo area between sophisticated reliable heuristics and verysimplistic (usually linear or quadratic time) approaches

The proposed heuristic is a combination of greedy depthfirst search unreachable vertex and rotational transforma-tion heuristics The greedy depth first search reduces therunning time considerably The search is greedy since italways selects the unvisited low degree vertex for extendingthe path While the unreachable vertex heuristic reduces thechances of reaching dead end the rotational transformationhelps to come out of the dead end condition Thus theproposedmethod is faster than the complex exact algorithmsand more reliable than the faster heuristics

2 Materials and Methods

21 Hamiltonian Cycle Problem Hamiltonian cycle is a cycleconnecting all the vertices in a given graph only once A graphcontaining at least one Hamiltonian cycle is called Hamil-tonian graph This optimization problem can be formallydefined as follows

Given a graph G=(EV) where E is the set of edges of thegraph V is the set of vertices of the graph and |V| = n

The problem is to find a cycle HC=(v1v2 vn)where all (vi vi+1) and (v1 vn) are elements of E

It is a hard problem that attracted both mathematiciansand computer scientists It is considered as one of the strongrepresentatives of the NP-complete problem

22 Greedy Depth First Search The large search space of theHCP can be explored either breadth wise or depth wiseIn depth first search search is performed in depth wisemanner starting from a given vertex till the point where thesearch cannot proceed further or a dead end is reached Theproposed heuristic uses greedy depth first search to createa Hamiltonian path The path construction starts from thehighest degree vertex because it increases the probabilityof returning to the starting vertex Degree of a vertex is

A

G

F

EDC

B

Figure 1 Unreachable vertex

the number of edges connected to that vertex The othervertices of the path are selected greedily by selecting thesmallest degree neighbour among the unvisited neighboursThe lowest degree vertices are added first into the pathsince they may become unreachable with the selection ofits neighbours For example if the degree of a vertex is twothen both the edges of that vertex must be present in theHamiltonian pathcycle and these edges are added to the pathfirst The vertices that are part of the path constructed sofar are considered as ldquovisitedrdquo Since in Hamiltonian cyclea vertex appears only once only the unvisited neighboursof the current vertex need to be considered In order toguide the greedy depth first search further to longer paths anunreachable vertex heuristic is proposed According to thisheuristic before adding a vertex to the path an unreachablecondition is checked for each of its neighboursThis heuristicis explained in Section 23

23 Unreachable Vertex Heuristic This research proposes anewheuristic namely unreachable vertex heuristic to reducethe chance of reaching a dead end while constructing theHamiltonian path

Definition 1 Let P be the partial path and x be the end vertexon partial path Select the next vertex yisinAdj(x) in the pathsuch that the number of unvisited adjacent vertices of allAdj(y) vertices is greater than one

A vertex is said to be unreachable if all its neighbours arealready a part of the path constructed so far In this case thereis no way to reach the vertex and it becomes unreachable Inthis proposed heuristic if the selection of a vertex is makingany of its unvisited neighbours unreachable then that vertexwill not be added to path

For example consider the graph shown in Figure 1Let the partial path identified so far be A997888rarrC997888rarrD In the

given graphAdj(D) = BE Let the next vertex selected be EAdj(E) = BG F Number of unvisited neighbours of B Gand F are 1 2 and 2 respectively According to the unreachablevertex heuristic E will not be selected as next vertex in thepath since the number of unvisited neighbours of B is one

24 Rotational Transformation Rotational transformation[11] and its variations are found to be powerful heuristics forfinding Hamiltonian cycle It is used to change a path to get anew end vertex as shown in Figure 2 In the figure e is the end

Journal of Optimization 3

a b c d e

a b c d e

Rotational Transformation

Figure 2 Rotational transformation

on which rotational transformation is applied to get a newend vertex

The procedure of rotational transformation is summa-rized below

(1) Find a vertex adjacent to e in the input graph in pathP Let it be vertex b

(2) Create a new path Prsquo

(a) by connecting b to e(b) by reversing the path from c to e

In the proposed algorithm rotational transformation is usedfor two purposes

(1) To come out of the dead end during the constructionof Hamiltonian path

(2) To convert the Hamiltonian path into Hamiltoniancycle

In the first case highest degree end of the path is selected forrotational transformation since it increases the probability ofgetting a new end Similarly for converting path into cycle thelowest degree end vertex of the path is selected for rotationaltransformation since it increases the probability of returningto the higher degree end to make the cycle

3 Proposed Hybrid Heuristic

31 HybridHAM Algorithm The proposed HybridHAMalgorithm works in three steps

(1) Create an initial path(2) Convert the initial path to Hamiltonian path(3) Convert the Hamiltonian path to Hamiltonian cycle

311 Initial Path Creation The path creation starts fromthe highest degree vertex and then adds the smallest degreevertices greedily in order to construct an initial path as longas possible The selection of highest degree initial vertex andthen the smallest degree vertices is meant for constructing

a longer initial path and is given in Table 1 The algorithmchecks the unreachable condition before adding a vertex inthe path This is to reduce the probability of reaching a deadend during the path construction If there are more than onehighest degree vertices then repeat this procedure of creatinginitial path by selecting each one of themas the starting vertexand select the path with maximum number of vertices as theinitial path By following this procedure wemay get an initialpath which is Hamiltonian or near Hamiltonian

312 Conversion of Initial Path into Hamiltonian Path If theinitial path created is not Hamiltonian (less number of ver-tices than the total number of vertices) then select the highestdegree end of the initial path for rotational transformationThis is to increase the probability for getting a new end vertexfor extending the path further to create a Hamiltonian pathExtend the path from the new end vertex by following thegreedy procedure in Section 311 The rotational transforma-tion and greedy path extension are continued until we aregetting a Hamiltonian path At any time during this processrotational transformation could not be applied means thateither the graph is not having Hamiltonian path or thealgorithm fails to identify the Hamiltonian path

313 Conversion of Hamiltonian Path into HamiltonianCycleIf there is an edge connecting the two ends of theHamiltonianpath in the graph then add that edge to the path to get theHamiltonian cycle Otherwise apply rotational transforma-tion to the smallest degree vertex repeatedly until getting anew end vertex which can be connected to the other endvertex to form the Hamiltonian cycle At any time duringthis process rotational transformation could not be appliedmeans that either the graph is not having Hamiltonian cycleor the algorithm fails to identify the Hamiltonian cycle

The vertex selection criteria used in various steps aresummarized in Table 1

32 Example Consider the undirected graph in Figure 3

321 Initial Path Identified in Step 1

0 1 2 3 4 5 6 7 8 12 13 14 22 21 20 1918 15

Here the length of the initial path is 18 which is less thanthe total number of vertices in the graph Since the identifiedpath is not Hamiltonian go to step 2

322 Hamiltonian Path Identified in Step 2

0 7 8 12 14 13 15 17 16 9 1 2 3 4 5 611 23 22 21 20 18 19 10

Here the length of the path is 24 and hence it is Hamil-tonian path Since there is no edge in Figure 3 connectingthe end vertices of the path (ie edge connecting vertex 1 andvertex 11) go to step 3 and apply rotational transformation tofind the new end vertices that are connected in the graph inFigure 3


Table 1 Vertex selection criteria

Vertex Selection Rule Step Reason

Selection of Highest degree initial vertex Step 1 It increases the probability of getting a path to reachback to this vertex to form the cycle

Greedy selection of smallest degreevertices during path construction Step 1

Giving preference to smaller degree vertices increasesthe probability of longer paths For example if thedegree of a vertex adjacent to the current end vertex ofthe path is two then it should be included first in thepath as it is the only possible position of the vertex inthe Hamiltonian path

Selection of the highest degree end of theinitial path for rotational transformation Step 2

It increases the probability for getting a new end vertexfor extending the path further to create a Hamiltonianpath

Selection of the smallest degree endvertex for rotational transformation Step 3

This is done to keep the highest degree end vertex sincethe probability of getting an edge connecting a higherdegree vertex is more compared to a smaller degreevertex

1

2021

22

23 19

18

17

1615

14

1312

11 10

98

7

6

5 4

3

2

0

Figure 3 Input undirected graph

323 Hamiltonian Cycle Identified in Step 3

0 7 8 12 14 13 15 16 17 18 19 20 21 2223 11 6 5 4 10 3 2 9 1

Here the end vertices 1 and 2 are connected in the initialgraph shown in Figure 3 and hence form the Hamiltoniancycle It is shown in Figure 4

33 Pseudocode The input to the algorithm is the adjacencymatrix representation of the graph and the output is the setof vertices corresponding to the Hamiltonian cycle Let 119899 bethe number of vertices in the graph The proposed algorithmcontains three phases and is outlined as follows

HybridHAM()

Figure 4 Hamiltonian cycle

From the input adjacencymatrix create two arrays119881119886 and119881119889 of vertices sorted in the increasing and decreasing order oftheir degrees respectively

Phase 1Create an initial path

(1) Start from one of the highest degree vertex (firstvertex in the array 119881119889) Let it be V119904

(2) Add it to the initial path 119875119894(3) Repeat

(a) Select the next smallest degree neighbour of V119904(from the array 119881119886) Let it be V119894


(b) If the selection of V119894 is not making any of itsunvisited neighbours unreachable then(i) add it to the initial path 119875119894(ii) make V119894 as V119904

Until V119904 becomes a dead end

End of Phase 1

(4) If |119875119894| = 119899 then go to Phase 3ElseRepeat Phase 1 for each of the highest degree vertex ofthe graph and select thelongest P119894 as initial pathEnd if

Phase 2Convert the initial path into Hamiltonian path

(5) Repeat

(a) Select the highest degree end of the path 119875119894 forrotational transformation

(b) Reverse119875119894 if the first vertex in119875119894 is having degreehigher than the last vertex to make the highestdegree vertex as the end vertex of the path Letit be V119909

(c) Apply rotational transformation to 119875119894 using V119909to get a new path

(d) If rotational transformation could not beapplied then either the graph is not havingHamiltonian path or the algorithm fails toidentify the Hamiltonian path and so exit

(e) Extend this new path by using the greedy depthfirst search as in Phase 1

Until |119875119894| = 119899(6) Now 119875119894 is the Hamiltonian Path 119875ℎ Assign 119875ℎ = 119875119894

End of Phase 2

(7) If there is an edge connecting the first and last verticesof 119875ℎ in the graph then119875ℎ is the Hamiltonian cycle and return 119875ℎelsego to Phase 3End if

Phase 3Convert Hamiltonian path into Hamiltonian cycle

(8) Repeat

(a) Select the smallest degree end of the path 119875ℎ forrotational transformation

(b) Reverse 119875ℎ if the first vertex in 119875ℎ is havingdegree higher than the last vertex to make thesmallest degree vertex as the end vertex of thepath Let it be V119910

(c) Apply rotational transformation to 119875ℎ using V119910to get a new path

(d) If rotational transformation could not beapplied then either the graph is not havingHamiltonian Cycle or the algorithm fails toidentify the Hamiltonian path and so exit

Until there is an edge connecting the first and lastvertices of the path 119875ℎ

(9) Now 119875ℎ is the Hamiltonian cycle and return 119875ℎEnd of Phase 3 End of HybridHAM algorithm

34 Worst Case Complexity

(1) Creation of sorted arraysT(n) = O(n2)

(2) Phase 1Greedy depth first search goes through the adjacencymatrix once in theworst case and hence its complexityis O(n2)This search is repeated for each of the highest degreevertex The worst case is all the 119899 vertices are of thesame degreeTherefore T(n) = O(n3)

(3) Phase 2Complexity of rotational transformation at the mostis O(n)Complexity of greedy depth first search is O(n2)Total of these two = O(n2)These two operations are repeated at most 119899 timesTherefore T(n) = O(n3)

(4) Phase 3Complexity of rotational transformation at the mostis O(n)This operation is repeated at most 119899 timesTherefore T(n) = O(n2)

Total worst case complexity of HybridHam T(n) = O(n2) +O(n3)+ O(n3)+ O(n2) =O(1198993)

4 Experiments

The performance evaluation experiments of the proposedalgorithm are conducted in a systemwith 4GBRAMand IntelCore i5 processorThe algorithm is implemented inMATLAB13RA The experiments were carried out on a set of graphscollected from the literature as well as from the TSPLIB

41 Sample Hamiltonian Cycle Graphs The Hamiltoniancycles returned by the proposed algorithm on the collectedset of sample graphs are shown in Figure 5

The running time of the algorithm to solve these sampleinstances is given in Table 2


Example 1 Example 2 Example 3 Example 4





Example 21 Example 22 Example 23

Figure 5 Sample Hamiltonian graphs


Table 2 Running time on sample instances

SlNo Name No of Vertices Running time in Sec1 Example 1 16 00512 Example 2 20 00563 Example 3 11 00514 Example 4 12 00545 Example 5 11 00516 Example 6 20 00497 Example 7 12 00528 Example 8 24 00839 Example 9 24 006310 Example 10 12 005111 Example 11 6 004912 Example 12 12 003113 Example 13 8 002914 Example 14 12 003415 Example 15 19 003016 Example 16 12 003217 Example 17 13 003018 Example 18 16 003819 Example 19 60 004320 Example 20 20 004221 Example 21 25 003122 Example 22 32 003223 Example 23 64 0078

Table 3 Running time on TSPLIB instances

SlNo Name No of vertices

Running time in sec

Concorde HCP SolverSnake and

LadderHeuristic

Hybrid Ham

1 Alb1000 1000 495 072 01 026562 Alb2000 2000 730 329 08 143753 Alb3000a 3000 956 836 344 276564 Alb3000b 3000 994 802 364 157815 Alb3000c 3000 995 843 431 184386 Alb3000d 3000 1014 848 403 164067 Alb3000e 3000 1044 803 429 167198 Alb4000 4000 1345 1784 1389 306259 Alb5000 5000 1724 3085 1412 89844

42 TSPLIB Instances The TSPLIB library [22] containsseven Hamiltonian cycle instances of 1000 2000 3000 and5000 vertices The proposed algorithm could solve all theseproblem instances in a few seconds The running time of theproposed algorithm is comparedwith that ofHCP Solver [23]Concorde TSP Solver [24] and the latest Snake and LadderHeuristic [21] Table 3 gives the time taken by these algorithmsto solve each of the HCP instances

43 FHCP Challenge Set The FHCP Challenge Set [25] isa collection of 1001 Hamiltonian Cycle Problem instances

This dataset is specifically designed to resist the heuristicapproaches HybridHAM is tested on 250 instances of theFHCP Challenge Set Out of these 250 instances the pro-posed heuristic could find 75Hamiltonian paths and 6Hamil-tonian cycles The complete results are shown in Table 4 Thegraphs are too sparse so that the rotation transformationcould not convert the Hamiltonian path into Hamiltoniancycle in Phase 3 of the algorithm Phase 1 and Phase 2 of thealgorithm perform comparatively well with the highly dif-ficult instances of the Challenge Set and hence the Hamil-tonian path It is also found that the proposed highest degree


Table 4 Result on FHCP Challenge Set

SlNo Name No of Vertices No of Edges Output Running Timein Sec

1 Graph 1 66 99 HP 006252 Graph 2 70 106 HC 004693 Graph 3 78 117 HP 006254 Graph 4 84 127 HP 006255 Graph5 90 135 HP 006256 Graph 6 94 142 HC 006257 Graph 7 102 153 HP 007818 Graph8 108 162 HP 006259 Graph 9 114 171 HP 0093810 Graph 11 126 189 HP 0109411 Graph 12 132 199 HP 0109412 Graph 14 142 214 HP 0093813 Graph 15 150 225 HP 0125014 Graph 16 156 235 HP 0093815 Graph 17 162 243 HP 0187516 Graph 18 166 250 HP 0109417 Graph 20 174 261 HP 0234418 Graph 21 180 271 HP 0140619 Graph 22 186 279 HP 0218820 Graph 23 190 286 HP 0156321 Graph 25 204 307 HP 0171922 Graph 26 210 315 HP 0375023 Graph 27 214 322 HP 0203124 Graph 29 228 343 HP 0250025 Graph32 246 369 HP 0406326 Graph 33 252 379 HP 0328127 Graph34 258 387 HP 0468828 Graph 35 262 394 HP 0578129 Graph 36 270 405 HP 0703130 Graph 37 276 415 HP 0375031 Graph 40 294 441 HP 1046932 Graph 41 300 451 HP 0484433 Graph 43 310 466 HP 0703134 Graph 44 312 477 HP 0984435 Graph 45 324 487 HP 0609436 Graph 50 348 523 HP 0906337 Graph 53 366 549 HP 1687538 Graph 54 372 559 HP 0921939 Graph 58 396 595 HP 1703140 Graph 59 400 40001 HC 0265641 Graph 64 416 625 HP 1093842 Graph 65 419 631 HP 1437543 Graph 68 438 657 HP 2921944 Graph69 444 667 HP 2906345 Graph 72 460 52901 HC 0437546 Graph 79 480 57601 HC 0484447 Graph 82 496 745 HP 17031


Table 4 Continued


48 Graph 84 500 62501 HC 0531349 Graph 90 510 65026 HC 0562550 Graph 91 516 775 HP 3515651 Graph 95 540 811 HP 3046952 Graph 96 540 72901 HC 0703153 Graph 99 550 826 HP 5671954 Graph 104 576 865 HP 2859455 Graph 118 636 955 HP 6859456 Graph 122 656 985 HP 4156357 Graph 124 660 991 HP 8281358 Graph 128 677 114583 HC 1359459 Graph 134 724 131045 HC 1531360 Graph 137 736 1105 HP 12953161 Graph 148 816 1225 HP 7953162 Graph 150 823 169333 HC 2750063 Graph 151 828 1243 HP 11562564 Graph 160 896 1345 HP 1465 Graph 162 909 206571 HC 4062566 Graph 168 972 1459 HP 33093867 Graph 169 976 1465 HP 28640668 Graph 176 1020 1531 HP 29437569 Graph 182 1056 1585 HP 22671970 Graph 188 1123 315283 HC 9640671 Graph 190 1136 1705 HP 14484472 Graph 203 1216 1825 HP 40281373 Graph 211 1296 1945 HP 40140674 Graph 233 1456 2185 HP 85468875 Graph 246 1536 2305 HP 1118438

and smallest degree vertex selection criteria suit more graphshaving vertices of different degrees However the algorithmcould find Hamiltonian cycle from medium sparse graphsin very less time (eg Graphs 72 79 84 90 etc in Table 4)Even the winners [26] of the challenge could not solve thecomplete set and they used different algorithms for differenttypes of graphs as their objective was to find solutions ratherthan developing algorithms

It would be worth pointing out that the difficultinstances of FHCP challenge dataset are very rare anddifficult to construct let alone encounter ldquonaturallyrdquo Alsoeven discovering a Hamiltonian path is a difficult (NP-complete) problem and the proposed heuristic succeededin doing so for many instances of the FHCP ChallengeSet

5 Conclusion

This paper proposes a hybrid heuristic for finding Hamil-tonian cycle from undirected graphs The proposed three-stage algorithm uses a combination of three heuristics

greedy depth first search unreachable vertex and rotationaltransformation From the various experiments conductedon different graphs of variable sizes and complexity it isfound that the highest degree heuristic used for selectingthe initial vertex for the creation of longest initial path aswell as Hamiltonian path and the smallest degree heuristicused for the conversion of Hamiltonian path to Hamiltoniancycle are the reasons for getting the solution in a single runThe experimental evaluation also shows that the proposedheuristic is much faster and succeeds in obtaining goodresults in the majority of cases It should be worth noting thatthe increase in speed comes without sacrificing the heuristicsreliability on ldquoeasyrdquo instances of various sizes (includingthe quite large TSPLIB instances) and that it still performsreasonably well on more difficult instances

Data Availability

The data supporting this research are from previouslyreported studies and datasets which have been cited Theprocessed data are available in [22 25]


Conflicts of Interest

The author declares that there is no conflict of interest

Acknowledgments

The research presented in this manuscript is supported byUniversity Grants Commission Major Project Grant FNo42-1362013(SR)

References

[1] G A Dirac ldquoSome theorems on abstract graphsrdquo Proceedings ofthe London Mathematical Society ird Series vol 2 pp 69ndash811952

[2] O Ore ldquoNote on Hamilton circuitsrdquoe American Mathemati-cal Monthly vol 67 55 pages 1960

[3] G Fan ldquoNew sufficient conditions for cycles in graphsrdquo Journalof Combinatorialeory Series B vol 37 no 3 pp 221ndash227 1984

[4] R J Gould ldquoAdvances on the Hamiltonian problemmdasha surveyrdquoGraphs and Combinatorics vol 19 no 1 pp 7ndash52 2003

[5] Vandergriend Basil (1998) Finding Hamiltonian Cycles Algo-rithms graphs and performance [Msc esis] Faculty of Grad-uate Studies and Research University of Alberta 1998

[6] F Rubin ldquoA search procedure for Hamilton paths and circuitsrdquoJournal of the ACM vol 21 pp 576ndash580 1974

[7] N Christofides Grapheory An Algorithmic Approach Com-puter Science and Applied Mathematics Academic Press NewYork NY USA 1975

[8] W Kocay ldquoAn extension of themulti-path algorithm for findingHamilton cyclesrdquoDiscrete Mathematics vol 101 no 1-3 pp 171ndash188 1992

[9] S Martello ldquoAlgorithm 595 An Enumerative Algorithm forFinding Hamiltonian Circuits in a DirectedGraphrdquoACMTran-sactions onMathematical Soware vol 9 no 1 pp 131ndash138 1983

[10] V Ejov J A Filar S K Lucas and J L Nelson ldquoSolvingthe Hamiltonian Cycle problem using symbolic determinantsrdquoTaiwanese Journal of Mathematics vol 10 no 2 pp 327ndash3382006

[11] L Posa ldquoHamiltonian circuits in random graphsrdquo DiscreteMathematics vol 14 no 4 pp 359ndash364 1976

[12] D Angluin and L G Valiant ldquoFast probabilistic algorithms forHamiltonian circuits andmatchingsrdquo in Proceedings of the ninthannual ACM symposium on eory of computing pp 30ndash41ACM 1977

[13] B Bollobas T I Fenner and A M Frieze ldquoAn algorithmfor finding hamilton paths and cycles in random graphsrdquoCombinatorics Probability and Computing vol 7 no 4 pp 327ndash341 1987

[14] A M Frieze ldquoAn algorithm for finding Hamilton cycles inrandom directed graphsrdquo Journal of Algorithms vol 9 no 2 pp181ndash204 1988

[15] A Z Broder A M Frieze and E Shamir ldquoFinding hiddenhamiltonian cyclesrdquoRandomStructuresampAlgorithms vol 5 no3 pp 395ndash410 1994

[16] F A Brunacci ldquoDB2 andDB2A Two useful tools for construct-ing Hamiltonian circuitsrdquo European Journal of OperationalResearch vol 34 no 2 pp 231ndash236 1988

[17] S Lin and B W Kernighan ldquoAn effective heuristic algorithmfor the traveling-salesman problemrdquo Operations Research vol21 pp 498ndash516 1973

[18] K Helsgaun ldquoAn effective implementation of the Lin-Ker-nighan traveling salesman heuristicrdquo European Journal of Oper-ational Research vol 126 no 1 pp 106ndash130 2000

[19] S Lin ldquoComputer solutions of the traveling salesman problemrdquoBell Labs Technical Journal vol 44 pp 2245ndash2269 1965

[20] M M Flood ldquoThe traveling-salesman problemrdquo OperationsResearch vol 4 pp 61ndash75 1956

[21] P Baniasadi V Ejov J A Filar M Haythorpe and S Rosso-makhine ldquoDeterministic ldquosnakes and Laddersrdquo Heuristic forthe Hamiltonian cycle problemrdquo Mathematical ProgrammingComputation vol 6 no 1 pp 55ndash75 2014

[22] TSPLIB-Hamiltonian cycle problem (HCP) accessed on June2013 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[23] PCGRATE accessed on September 2014 httpwwwpcgratecomaboutnpcomprbhcp

[24] CONCORDE accessed on September 2014 httpwww3csstonybrookedusimalgorithimplementconcordeimplementshtml

[25] M Haythorpe ldquoFHCP challenge set the first set of structurallydifficult instances of the hamiltonian cycle problemrdquo Bulletin ofthe Institute of Combinatorics and Its Applications vol 83 pp98ndash107 2018

[26] A Schneider ldquoWhen researchers play to solve 1001 instances oftheHamiltonian Cycle Problemrdquo 2016 httpswwwinriafrencentresophianewswhen-researchers-play-to-solve-1001-instan-ces-of-the-hamiltonian-cycle-problem

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not reduce the exponential time complexity Posarsquos algorithm[11] is considered as the base algorithm for the heuristicalgorithms for HCP Posarsquos idea of rotational transformationand its variants formed the basis of almost all heuristicalgorithms proposed later Angluin and Valiant [12] proposeda much more complicated transformation for directed graphas the rotational transformation is not suitable for directedgraph Bollobas et al [13] proposed a Hamiltonian cyclealgorithm called HAM that uses rotational transformationand cycle extension Various versions of HAM algorithmlike SparseHam [14] and HideHam [15] are also proposedfor different kinds of graphs Brunacci [16] proposed twoalgorithms DB2 and DB2A which consider the HCP as aversion of TSP and the nonedges as highly weighted edgesDB2A algorithm is a modification of DBA algorithm fordirected graphs where directed graph is transformed intoundirected graph Many TSP heuristics like the famous Lin-Kernighan heuristic [17 18] use a technique called ldquok-optrdquotransformation [19 20] which is an exchange of k edgesBaniasadi et al [21] proposed a ldquosnakes and laddersrdquo heuristicfor solving HCP inspired from k-opt transformations Thereare very few published HCP heuristics that sit in the ldquomid-dlerdquo area between sophisticated reliable heuristics and verysimplistic (usually linear or quadratic time) approaches

The proposed heuristic is a combination of greedy depthfirst search unreachable vertex and rotational transforma-tion heuristics The greedy depth first search reduces therunning time considerably The search is greedy since italways selects the unvisited low degree vertex for extendingthe path While the unreachable vertex heuristic reduces thechances of reaching dead end the rotational transformationhelps to come out of the dead end condition Thus theproposedmethod is faster than the complex exact algorithmsand more reliable than the faster heuristics

2 Materials and Methods

21 Hamiltonian Cycle Problem Hamiltonian cycle is a cycleconnecting all the vertices in a given graph only once A graphcontaining at least one Hamiltonian cycle is called Hamil-tonian graph This optimization problem can be formallydefined as follows

Given a graph G=(EV) where E is the set of edges of thegraph V is the set of vertices of the graph and |V| = n

The problem is to find a cycle HC=(v1v2 vn)where all (vi vi+1) and (v1 vn) are elements of E

It is a hard problem that attracted both mathematiciansand computer scientists It is considered as one of the strongrepresentatives of the NP-complete problem

22 Greedy Depth First Search The large search space of theHCP can be explored either breadth wise or depth wiseIn depth first search search is performed in depth wisemanner starting from a given vertex till the point where thesearch cannot proceed further or a dead end is reached Theproposed heuristic uses greedy depth first search to createa Hamiltonian path The path construction starts from thehighest degree vertex because it increases the probabilityof returning to the starting vertex Degree of a vertex is

A

G

F

EDC

B

Figure 1 Unreachable vertex

the number of edges connected to that vertex The othervertices of the path are selected greedily by selecting thesmallest degree neighbour among the unvisited neighboursThe lowest degree vertices are added first into the pathsince they may become unreachable with the selection ofits neighbours For example if the degree of a vertex is twothen both the edges of that vertex must be present in theHamiltonian pathcycle and these edges are added to the pathfirst The vertices that are part of the path constructed sofar are considered as ldquovisitedrdquo Since in Hamiltonian cyclea vertex appears only once only the unvisited neighboursof the current vertex need to be considered In order toguide the greedy depth first search further to longer paths anunreachable vertex heuristic is proposed According to thisheuristic before adding a vertex to the path an unreachablecondition is checked for each of its neighboursThis heuristicis explained in Section 23

23 Unreachable Vertex Heuristic This research proposes anewheuristic namely unreachable vertex heuristic to reducethe chance of reaching a dead end while constructing theHamiltonian path

Definition 1 Let P be the partial path and x be the end vertexon partial path Select the next vertex yisinAdj(x) in the pathsuch that the number of unvisited adjacent vertices of allAdj(y) vertices is greater than one

A vertex is said to be unreachable if all its neighbours arealready a part of the path constructed so far In this case thereis no way to reach the vertex and it becomes unreachable Inthis proposed heuristic if the selection of a vertex is makingany of its unvisited neighbours unreachable then that vertexwill not be added to path

For example consider the graph shown in Figure 1Let the partial path identified so far be A997888rarrC997888rarrD In the

given graphAdj(D) = BE Let the next vertex selected be EAdj(E) = BG F Number of unvisited neighbours of B Gand F are 1 2 and 2 respectively According to the unreachablevertex heuristic E will not be selected as next vertex in thepath since the number of unvisited neighbours of B is one

24 Rotational Transformation Rotational transformation[11] and its variations are found to be powerful heuristics forfinding Hamiltonian cycle It is used to change a path to get anew end vertex as shown in Figure 2 In the figure e is the end


a b c d e

a b c d e






















0 1 2 3 4 5 6 7 8 12 13 14 22 21 20 1918 15



0 7 8 12 14 13 15 17 16 9 1 2 3 4 5 611 23 22 21 20 18 19 10












1

2021

22

23 19

18

17

1615

14

1312

11 10

98

7

6

5 4

3

2

0



0 7 8 12 14 13 15 16 17 18 19 20 21 2223 11 6 5 4 10 3 2 9 1



HybridHAM()










End of Phase 1



(5) Repeat







End of Phase 2



(8) Repeat













4 Experiments

















Running time in sec


LadderHeuristic

Hybrid Ham










Table 4 Continued





5 Conclusion



Data Availability





Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









a b c d e

a b c d e






















0 1 2 3 4 5 6 7 8 12 13 14 22 21 20 1918 15



0 7 8 12 14 13 15 17 16 9 1 2 3 4 5 611 23 22 21 20 18 19 10












1

2021

22

23 19

18

17

1615

14

1312

11 10

98

7

6

5 4

3

2

0



0 7 8 12 14 13 15 16 17 18 19 20 21 2223 11 6 5 4 10 3 2 9 1



HybridHAM()










End of Phase 1



(5) Repeat







End of Phase 2



(8) Repeat













4 Experiments

















Running time in sec


LadderHeuristic

Hybrid Ham










Table 4 Continued





5 Conclusion



Data Availability





Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018


















1

2021

22

23 19

18

17

1615

14

1312

11 10

98

7

6

5 4

3

2

0



0 7 8 12 14 13 15 16 17 18 19 20 21 2223 11 6 5 4 10 3 2 9 1



HybridHAM()










End of Phase 1



(5) Repeat







End of Phase 2



(8) Repeat













4 Experiments

















Running time in sec


LadderHeuristic

Hybrid Ham










Table 4 Continued





5 Conclusion



Data Availability





Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018











End of Phase 1



(5) Repeat







End of Phase 2



(8) Repeat













4 Experiments

















Running time in sec


LadderHeuristic

Hybrid Ham










Table 4 Continued





5 Conclusion



Data Availability





Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018





















Running time in sec


LadderHeuristic

Hybrid Ham










Table 4 Continued





5 Conclusion



Data Availability





Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018













Table 4 Continued





5 Conclusion



Data Availability





Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018











Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








HybridHAM: A Novel Hybrid Heuristic for Finding Hamiltonian Cycle · 2019. 7. 30. · graph....

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Transcript of HybridHAM: A Novel Hybrid Heuristic for Finding Hamiltonian Cycle · 2019. 7. 30. · graph....