VNƯ Joumal of Science, Mathcmatics - Physics 23 (2007) 178-182 Algorithm for solution of a routing problem Tran Vu Thieu*, Pham Xuan Hinh Institute o f Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam Received 15 N ovem ber 2006; received in revised fo rm 12 Septem ber 2007 Abstract In this paper we consider a co m binatoria l o p tim iza tio n p ro b le m that is s im ila r to the bottleneck trave ling salesman problem W e show that an o p tim al to u r fo r this problem is pyram idal to u r (1, 3, , , n , , , 4, 2) or consists o f some p yra m id a l subtours The above 7methods can be cxtendeđ to com plete b ip artite graphs P ro b lem statem en t It is well-known that the traveling salesman problem (TSP) is strongly N P-hard (cf [1], p 353) But for some special cases o f the TSP can be solvable in polynom ial time This is the case w here the distance matrix in the TSP fulfills certain additional conditions (e.g the M onge property, Kalmanson matrices, the Demidenko conditions or the Supnick conditions), cf [2-4] In the sequel we will introduce another special case o f the TSP w hich can easily be solvable and show that the optimal tour for this problem is pyram idal tour or consists o f some (at most three) pyramidal subtours Consider a complete graph G = (A, E) w ith vertex set A = {ai, a 2, , an} and eđge set E = AxA Each vertex aj e A has a real num ber t, (i — 1, , n), called the altitude o f vertex a, W e speciíy vertices ab e A (the source) and a* e A (the sink) such that b, e e {1, 2, , n} and tb £ te Consider the following problem, called Problem A for short: P ro b lem A Find a H am iltonian path in the graph from ab to ac (w hich visits every vertex exactly once) so that to m inim ize the highest difference betw een altitudes o f any two successive vertices in the path In other word, am ong perm utations 71 = {i|, i2, , i„} o f the num bers ,2 , , n w ith ii = b, in = e fĩnd a permutation so that to m inim ize the function f(7t)= m a x tị*k - t *ẩầ+\ It is easy to see that such a perm utation corresponds to a H am iltonian path in the graph from ab to a* and that the total num ber o f perm utations is equal to (n-2)! Let us denote by £p the set o f all these permutations Each penmutation 7Ue £P is called a tour from ab to ae and perm utation 71* = argm in{f( 7t) : n s CPị optimaỉ tour from ab to ae Also consider a sim ilar problem: P ro b lem B Find a Ham iltonian cycle in the given graph so that to m ini-m ize the biggest diíĩerence betw een altitudes o f any two successive vertices in the cycle Each o f such cycles is also ■Corresponding author Tel.: 84-4-8351235 E-mail: trvuthieu@yahoo.com 178 T V Thieu, p.x H ìn h / VNU Journal o f Science, Mathematics - Physics 23 (2007) 178-182 179 reíerred to as an optimaỉ tour It is obvious that the num ber o f H am iltonian cycles in the graph is equal to n! As before, this problem is also íòrm ulated as follows: Find a cyclic perm utation T = {i|, in} o f the numbers , , , n so that vvith convention that in+) = i| The solution o f the above stated problem s does not change if we replace t, by ti' = t, + t with t being an arbitrary real number So by taking t > suíĩiciently large we can assume that t, > for all i =1, , , n Furtherm ore, vvithout loss o f generality we can assume (by enum erating the vertices o f the graph if needed) < t , < t ắ t b < ^ t t < t n ( 1) The vertices that have the same value t, can be arranged in an arbitrary order Let in the sequence (1) there be q difĩerent values (q ^ n) We write these values as hi < h < < hq We divide the vertices o f the graph into q classes, denoted by T |, T2, , Tq such that vertex aị belongs to class T|( if and only iftj = hic(i = ,2 , , n; k = 1, , , q) The vertices o f class Ti are first numbered then vertices o f T and íinally vertices o f Tq The vertices o f the same class are numbered in an arbitrary order Basic properties of optimal tours W e now consiđer the problem o f finding an optimal tour from ab to in G We tem porarily assume that all tj's are diíĩerent, that is we enum erated the vertices o f the graph so that ( ^ b < e £ n) < t, < tj < t3< < Vi < to (2) In the sequel we shall rem ove this assum ption Some useful properties o f optimal tours are given in the below stated propositions T heir proofs make use o f the tour im provem ent technique as used in [ ] for proving the existence o f pyramidai tours P ro p o sitio n Let 7t = (i| = b, iĩ, ij, , i„.|, in = e) be an arbitrary optimal tour from b to e If b > and = ik with < k < n then we have a) From b to tour n passes through vertices with decreasing indexes: b = i| > i j > > Ì|c-] > ik= b) Tour n cannot pass through three successive vertices i - 1, i, i + 1or in inverse direction i + 1, i, i - with < i - and i + ắ b Proof a) Conversely, suppose that from b to tour 71 passes through a vertex wiứi index greater than b Then on the path from b to there is two successive vertices r and s such that s < b < r Also on the paứi from to n ứiere exists two successive vertices u and V such that u < b < V, i.e tour 71 is o f the form: 71 = ( b , , r, s , , , , u, V , , n , , e) vvith s < b < r and u < b < V We consứuct a new tour 7i' as indicated in Fig 1: 7i' = ( b , , s , , , , u, V , , r , , n , ,e ) i f v < r o r 7i’ , s^9, •• • ,y *1>, V ,* e) i f r < v ’ • •• = ( b , • •• » • • •• •, •u >, r* » • • • ) **>, • ■• , >nv ,/ ** r Ó>Ị— Ố— Y - © —0 ©— "^ ( * đ