The Problem Of Constructing Constrained Paths In Digraphs

In this thesis,we mainly consider the problem of constructing constrained paths in digraphs.This problem consists of two basic parts,i.e.,the problem of constrained path augmentation and the problem of constructing constrained paths.The problem of constrained path augmentation is formatted as follows:Given a weighted digraph D =(V,A;C)and a s-t.directed path Ps,t,cost function c:A R+,we are asked to find a subset Al(?)A ?A(Ps,t),such that there exists a s-t directed path in the reduced subgraph Ps,t ? A1—{a}(or the reduced subgraph Ps t ? A1—{u}),for each arc a ?A(Ps,t)(or each vertex u? V(Ps,t)—?s,t?),the objective is to minimize the sum of costs of arcs in A1,i.e.,c(A1)= ?e?A1 c(e)is minimized.For these problems,we design two polynomial-time algorithms to solve them and their complexities are both O(mn),where m represents the number of arcs A and n represents the number of vertices V.Based on the problems mentioned-above,we consider the problem of constructing constrained paths in digraphs,which is formatted as follow:Given a weighted digraph D =(V,A;w,c),L is the length of each stock piece of material and co is the sale price of each stock piece,w:A ?R+is a length function and c:A?R+ is a cost function,we are asked to construct a subset A1(?)A A(Ps,t),such that there exists a s-t directed path in the reduced subgraph Ps,tA1-{a}(or the reduced subgraph Ps,t? A1-{u}),for each arc a?A(Ps,t)(or each vertex u? V(Ps,t)-{s,t},the objective is to minimize the total construction cost of A1,i.e.cost{A1)= ?ecA1 c(e)|k(A1).c0 is minimized,where k(A1)is the number of stock pieces required to construct all arcs in A 1.We design two 2-approximation algorithms to solve these two problems and their complexities are both O(mn),where m represents the number of arcs A and n represents the number of vertices V.