Finding Shortest Hamiltonian Path is in P. Abstract

Transcription

1 Finding Shortest Hamiltonian Path is in P Dhananay P. Mehendale Sir Parashurambhau College, Tilak Road, Pune, India bstract The roblem of finding shortest Hamiltonian ath in a eighted comlete grah belongs to the class of NP-Comlete roblems []. In this aer e ill sho that e can obtain shortest Hamiltonian ath in a given eighted comlete grah in olynomial time! We ill be discussing a very simle but useful idea of alying certain chosen sequence of ermutations (actually transositions) on given eighted adacency matrix corresonding to the comlete grah, on oints say, under consideration. This simle and novel algorithm essentially consists of alying certain transositions that ill transform the eighted adacency matrix in such a ay that its vertices are no relabeled and in this relabeled eighted comlete grah the algorithm terminates decisively in roducing the shortest Hamiltonian ath, and this shortest Hamiltonian ath ill be L ( + ) L ( ) Introduction: Let G be a eighted comlete grah ith the vertex set V (G) and edge set E (G) resectively: V (G) = { v, v, L, v } and E (G) = { e, e, L, eq } Let = [ ] denotes the eighted adacency matrix of G. G i Note: lying transosition (m, n) on G is essentially equivalent to interchanging ros as ell as columns, m and n. That is relace m-th ro in G by n-th ro and vice versa and then in thus transformed matrix relace m-th column by n-th column and vice versa (order of these oerations, i.e.

2 hether you interchange ros first and then interchange columns or you interchange columns first and then interchange ros, is immaterial as it roduce same end result). Note that this transformation essentially roduces a ne eighted adacency matrix that ill result due to interchanging labels of vertices lgorithm: v, v in the original eighted comlete grah. m n () If entry at osition (, ) in the matrix, i.e. eight is already smallest in the first ro then roceed to ste. Else, among the eights,,,..., =, find minimum eight, say. ly transosition (, ) on G, roducing ne eighted adacency matrix, say G. () If entry at osition (, ) in the matrix, i.e. eight is already smallest in the second ro then roceed to ste. Else, among the eights,,,..., =, find minimum eight, say. No aly transosition (, ) on G, roducing ne eighted adacency matrix, say G. () If entry at osition (, ) in the matrix, i.e. eight is already smallest in the third ro then roceed to ste. Else, among the eights,,,..., =, find minimum eight, say. No aly transosition (, ) on G, roducing ne eighted adacency matrix, say G. () Continue this rocedure alying aroriate transositions till e finally reach (-)-th ro and among the eights ( ), = ( ),, find minimum eight, say ( ) ( ). No aly transosition (( ), ( ) ) on G( ), roducing ne eighted adacency matrix, say G( ). () Find the sum of eights of edges in the Hamiltonian ath

3 L ( + ) L ( ) Theorem: fter carrying out algorithm.. on given eighted comlete grah the Hamiltonian ath L ( + ) L ( ) reresents the shortest Hamiltonian ath in the given (and conveniently relabeled) eighted comlete grah. Proof: The algorithm begins ith alication of ermutation (transosition) hich brings smallest eight entry in the first ro at osition (, ) in the eighted adacency matrix. This is achieved by transosition of tye (, ), here >. The algorithm then alies transosition hich brings smallest eight entry in the second ro at osition (, ), in the transformed eighted adacency matrix that results after alying transosition mentioned above. This is achieved by transosition of tye (, ), here >. Note that because of its secial form this second transosition doesn t affect the smallest entry achieved at osition (, ) hile bringing smallest entry (eight) in the second ro at osition (, ) by this second transosition! This story continues, i.e. the later alied transositions doesn t affect the results of earlier transositions because of the secial choice of the transositions and at end achieves smallest ossible eights in the ros at ositions on the diagonal neighboring the rincile diagonal, i.e. at ositions (, ), (, ),., (-, ), of the evolved eighted adacency matrix, evolved through the successive transositions of secially chosen tye. Note that this neighboring diagonal reresents the eights on the Hamiltonian ath L ( + ) L ( ) Examle: We consider folloing eighted adacency matrix reresenting a eighted comlete grah and find the shortest Hamiltonian ath in its relabeled coy ill be in the form

4 + ) ( ) ( L L Consider folloing eighted adacency matrix in hich entries are actually the eights of the corresonding edges: () Since entry at osition (, ) is already smallest in the first ro e roceed to next ste. () Since entry in osition (, ) = is smallest in second ro e aly transosition (, ) on the above matrix that results into matrix () Since entry in osition (, ) = is smallest in third ro e aly transosition (, ) on the above matrix that results into matrix

5 Clearly, in this transformed eighted adacency matrix the Hamiltonian ath ill be the shortest and has total eight i,( i+ ) = i= Conclusion: It is clear to see that this algorithm requires checking at most (-) ros of gradually decreasing lengths, (-), (-),.,etc for finding the minimum entry in these ros. The algorithm further needs at most (-) transosition oerations to be carried out on the eighted adacency matrix under consideration. The algorithm is clearly in P, i.e. of olynomial time comlexity! References. Christos H. Paadimitriou, Comutational Comlexity, Page, ddison-wesley Publishing Comany, Inc.,.