The Constrained Windy Postman Problem On Eulerian Graphs

In this thesis,we consider the constrained windy postman problem on Eulerian graphs,which is described as follows.Given an Eulerian graph G=(V,E;w;l,u),where l:E?Z+,u:E?Z+and w:E?R+,w(·)is a cost function where wij is the cost of traversing edge vivj from vi to vj and wji is the cost of traversing edge vjvi from vj to vi for each edge vivj ? E,respectively.We are asked to determine a closed tour C in this Eulerian graph G to traverse each edge e ? E at least l(e)and at most u(e)times,where l(e)?u(e).The objective is to minimize the total cost w(C)=?(vi,vj)?Cwij of the closed tour C.Whenever xij and xji are the numbers of traversing times which(vi,vj)and(vj,vi)occur in C,respectively,where l(e)?xij+xji?u(e),the costs of arcs(vi,vj)and(vj,vi)in the total cost of the closed tour C will be computed xij and xji times,respectively.For the constrained windy postman problem on Eulerian graphs,when l(e)=1 and u(e)is odd for each edge e ? E,we call this version as the odd constrained windy postman problem on Eulerian graphs;when l(e)and u(e)are both even for each edge e ? E,we call this version as the even constrained windy postman problem on Eulerian graphs.We obtain the following two results.(1)For the odd constrained windy postman problem on Eulerian graphs,we design an exact algorithm in time O(m2n).(2)For the even constrained windy postman problem,constructing a new Eulerian graph through copying edges from the given Eulerian graph,we also design an exact algorithm in time O(m2n),where n is the number of vertices in graph G,m is the number of edges in graph G.