最小费用最大流

云南大学数学与统计学实验教学中心 实 验 报 告一、实验目的给定网络D=（V,A,C ）,每一条弧(),i j v v A ∈上，除了已给容量ij c ，还给了一个单位流量的费用ij b ，编制程序，找到从发点Vs 到收点Vt 的最小费用最大流。

二、实验环境 VS2010（C++） 三、实验内容 P275例15：以图10-28为例，求最小费用最大流。

弧旁数字为(),ij ij b c四、实验过程A 、最小费用最大流的实现步骤：在最小费用最大流的实现中，其实大的来说就两个步骤：在图中寻找最短路径；路径调整。

当然在这其中涉及到了流量的如何调整，以及图中边的方向变化。

在寻找最短路径的增广链中，需要将从始点到终点的每一条增广链找到！知道没有增广链了程序才能停止。

具体做法如下（课本P275）：由前面的实验可知，寻求最大流的方法是从某个可行流f (我采用的是零流)，找到关于这个流的一条增广链，据此调整f ，对新的可行流试图寻求关于它的增广链，如此反复直到最大流，现在还要求最小费用的最大流，设已知f 是流量v(f)的最小费用流，余下的问题就是去寻求关于f 的最小费用增广链。

构造一个赋权有向图W(f),顶点是原网络图D 上的顶点，而把D 中的每一条弧 变成两个方向上的弧，定义弧上的权 为在网络D中寻求关于f的最小费用增广链等价于在赋权有向图W(f)中，寻求从到的最短路。

因此有如下算法：开始取f( 0 )= 0 , 一般情况下若在第k - 1 步得到最小费用流f( k-1),则构造赋权有向图W ( f(k-1)) , 在W ( f( k - 1 )) 中, 寻求从vs 到vt的最短路。

若不存在最短路(即最短路径是∞) , 则f(k-1)就是最小费用最大流; 若存在最短路, 则在原网络 D 中得到相应的增广链,在增广链μ上对f( k - 1 )进行调整B、将上面算法编写实现运行结果如下图：a、输入各边：输入的结果为红色字体b、初始化后的到结果：上图中得到的图的关系与课本例题中的一样！所以创建图形正确！c、寻找最短增广链：d、最后的最小费用最大流的图如下：图中得到的结果与课本276页的结果一致！说明程序运算得到的结果正确！五、实验总结虽说看课本P276的有点复杂，同时来自己手动做的时候更是会出现思路混乱！但是运用程序来解决后，可以让自己将思路理清，当然前面在做的时候忽略了最小费用最大流中有负值存在，Dijkstra算法失效，得时候P266中的算法来实现！六、源代码源代码如下：//图的顶点结构如下：//-----------------------------------//|vex |formerNode | MinPathWeight |//-----------------------------------//图的边的结构如下：//----------------------//|weight |flows |cost |//----------------------#include<iostream>#include<string>#include<Windows.h>using namespace std;#define Max_vex 100#define M 1000//SetTextC1是原来的颜色//SetTextC2 是设置的输出输入颜色#define SetTextC1 SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE),FOREGROUND_INTENSITY | FOREGROUND_RED | FOREGROUND_GREEN |FOREGROUND_BLUE)#define SetTextC2 SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE),FOREGROUND_INTENSITY | FOREGROUND_GREEN)#define SetTextC3 SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE),FOREGROUND_INTENSITY | FOREGROUND_GREEN | FOREGROUND_BLUE)//靛色#define SetTextC4 SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE),FOREGROUND_INTENSITY | FOREGROUND_RED)//红色#define SetTextC5 SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE),FOREGROUND_INTENSITY | FOREGROUND_RED | FOREGROUND_BLUE)#define SetTextC6 SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE),FOREGROUND_RED |FOREGROUND_GREEN |FOREGROUND_BLUE)//无亮度的白色#define SetTextC7 SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE),FOREGROUND_GREEN)class Arcll{ //边上值public:double weight; //容量double flows; //流量double cost;};class Vex{ //顶点值public:string vexs;//顶点名称int formerVex; //前驱节点的下标double MinPathWeight;//初始点到每个节点的最短路径};class Graphics{ //创建一个图的类private:int vexnum; //顶点数int arcnum; //边数Vex vexs[Max_vex]; //存储定点名称Arcll arcs[Max_vex][Max_vex]; //边数public:void CreateGraphics(Graphics &G);//创建一个图int LocateIndex(Graphics G, string v); // 寻找下标void start(Graphics &G);bool ShorttestPath(Graphics &G,int sindex, int findex);//求最小费用增广链void Display(Graphics G); //打印各边的流量和容量信息void MCMSInfo(Graphics G,Graphics G1); //打印最小费用最大流相关信息};//方法实现//找到相应顶点的下标int Graphics::LocateIndex(Graphics G, string v){int i=0;for(i=0; i<G.vexnum; i++){if(G.vexs[i]pare(v) == 0){return i;}}return -1;}//创建一个有向图void Graphics::CreateGraphics(Graphics &G){int i,j,k;string Vi,Vj;double c,w; //费用、容量SetTextC1;cout<<"-------------------------创建有向图的存储-------------------------------"<<endl;cout<<"请输入顶点数:";SetTextC4; //设置颜色cin>>G.vexnum;SetTextC1;//设置颜色cout<<"请输入图的边数:";SetTextC4;cin>>G.arcnum;SetTextC1;cout<<"请输入顶点名:";SetTextC4;for(i=0; i<G.vexnum; i++){ //初始化顶点cin>>G.vexs[i].vexs;G.vexs[i].formerVex = -1;G.vexs[i].MinPathWeight = 0;}SetTextC1;for(i=0; i<G.vexnum; i++){ //初始化边for(j=0; j<G.vexnum; j++){if(i == j){G.arcs[i][j].cost = 0;} else {G.arcs[i][j].cost = M;}G.arcs[i][j].weight = 0;G.arcs[i][j].flows = 0;}}for(k=0; k<G.arcnum; k++){cout<<"请输入第"<<k+1<<"条边相连的顶点Vi，Vj以及费用、容量:";SetTextC4;cin>>Vi>>Vj>>c>>w;SetTextC1;i = LocateIndex(G,Vi);j = LocateIndex(G,Vj);if(i == -1 || j == -1){cout<<"输入错误！请检查..."<<endl;return;}G.arcs[i][j].weight = w;G.arcs[i][j].cost = c;}cout<<endl;cout<<"各顶点的关系："<<endl;SetTextC2;//绿色for(i=0; i<G.vexnum; i++){if(i==0){cout<<"\t\t"<<G.vexs[i].vexs<<" ";}else {cout<<"\t"<<G.vexs[i].vexs<<" ";}}cout<<endl;SetTextC3;for(i=0; i<G.vexnum; i++){ //初始化边SetTextC2;cout<<"\t"<<G.vexs[i].vexs<<"";SetTextC3;for(j=0; j<G.vexnum; j++){if(G.arcs[i][j].weight == 0){SetTextC6;cout<<"\t"<<"<M,"<<"-1"<<">";SetTextC3;} else {if(G.arcs[i][j].weight > 9){cout<<"\t<"<<G.arcs[i][j].cost<<","<<G.arcs[i][j].weight<<">";} else{cout<<"\t<"<<G.arcs[i][j].cost<<", "<<G.arcs[i][j].weight<<">";}}}cout<<endl;}SetTextC1;cout<<endl<<"有向图创建完毕！"<<endl;}//===================最小费用最大流======================void Graphics::start(Graphics &G){int sindex,findex;//初始点和结束点的下标string svex,fvex;//创建有向图G.CreateGraphics(G);Graphics G1 = G;cout<<"请输入始点和结束点:";SetTextC4;cin>>svex>>fvex;SetTextC1;sindex = LocateIndex(G,svex);//初始点的下标findex = LocateIndex(G,fvex);//结束点的下标while(ShorttestPath(G,sindex,findex));//求最小费用增广链MCMSInfo(G, G1);}bool Graphics::ShorttestPath(Graphics &G,int sindex, int findex){//寻找最短路的增广链int i,j,k = G.vexnum;bool b = true;double min = M;int *r;r = new int[G.vexnum];for(i=0; i<G.vexnum; i++){//寻找赋权图中的最短路G.vexs[i].MinPathWeight = G.arcs[sindex][i].cost;if( i!= sindex && G.vexs[i].MinPathWeight != M){G.vexs[i].formerVex = sindex;}}while(b){b = false;for( i=0; i<G.vexnum; i++){if( i!=sindex){for( j=0; j<G.vexnum; j++){if((G.vexs[i].MinPathWeight + G.arcs[i][j].cost)<G.vexs[j].MinPathWeight&& G.arcs[i][j].cost != M){G.vexs[j].MinPathWeight = G.vexs[i].MinPathWeight +G.arcs[i][j].cost;G.vexs[j].formerVex = i;b = true;}}}}}if(G.vexs[findex].MinPathWeight == M){ //已找到最小费用最大流cout<<endl<<"找不到从"<<G.vexs[sindex].vexs<<"到"<<G.vexs[findex].vexs<<"最短路"<<endl;cout<<"已得到最小费用最大流!!!"<<endl;return false;}else{for( i = findex; i != -1; ){ //打印最短路，并求的增广链的调整值r[--k] = i;j = G.vexs[i].formerVex;if( j == -1 ) break;if(G.arcs[j][i].cost != 0){if((G.arcs[j][i].weight - G.arcs[j][i].flows) < min){min = G.arcs[j][i].weight - G.arcs[j][i].flows;}}else{if(G.arcs[i][j].flows < min ){min = G.arcs[i][j].flows;}}i=j;}cout<<endl<<"最短路: ";SetTextC5;//紫色for( i=k; i<G.vexnum-1; i++){//调整最短路径上的流量值，课本266页的算法cout<<G.vexs[r[i]].vexs<<"\t";if(i== G.vexnum-2){cout<<G.vexs[G.vexnum-1].vexs<<endl;}if(G.arcs[r[i]][r[i+1]].cost!=0){G.arcs[r[i]][r[i+1]].flows = G.arcs[r[i]][r[i+1]].flows + min;G.arcs[r[i+1]][r[i]].cost = -G.arcs[r[i]][r[i+1]].cost;G.arcs[r[i+1]][r[i]].weight = G.arcs[r[i]][r[i+1]].weight;G.arcs[r[i+1]][r[i]].flows = -G.arcs[r[i]][r[i+1]].flows;if(G.arcs[r[i]][r[i+1]].flows == G.arcs[r[i]][r[i+1]].weight){G.arcs[r[i]][r[i+1]].cost = M;}}else{G.arcs[r[i+1]][r[i]].flows = G.arcs[r[i+1]][r[i]].flows - min;G.arcs[r[i]][r[i+1]].cost = -G.arcs[r[i+1]][r[i]].cost;G.arcs[r[i+1]][r[i]].weight = G.arcs[r[i]][r[i+1]].weight;G.arcs[r[i+1]][r[i]].flows = -G.arcs[r[i]][r[i+1]].flows;if(G.arcs[r[i+1]][r[i]].flows == 0){G.arcs[r[i+1]][r[i]].cost = M;}}}SetTextC1;//白色//打印Ccout<<endl<<"调整后的关于费用的赋权图："<<endl;Display(G);return true;}}void Graphics::Display(Graphics G){ //打印各边的流量和容量信息int i,j;SetTextC2;//绿色for(i=0; i<G.vexnum; i++){if(i==0){cout<<"\t\t"<<G.vexs[i].vexs<<" ";}else {cout<<"\t"<<G.vexs[i].vexs<<" ";}}cout<<endl;SetTextC3;for( i = 0; i < G.vexnum; i++){SetTextC2;cout<<"\t"<<G.vexs[i].vexs<<"\t";SetTextC3;for( j = 0; j < G.vexnum; j++){if(G.arcs[i][j].cost == M || G.arcs[i][j].cost == 0){SetTextC6;cout<<"M"<<"\t";SetTextC3;}else{cout<<G.arcs[i][j].cost<<"\t";}}cout<<endl;}SetTextC1;}void Graphics::MCMSInfo(Graphics G, Graphics G1){ //打印最小费用最大流相关信息int i,j;double maxstream = 0;double mincost = 0;cout<<"可行流图如下:"<<endl;SetTextC2;//绿色for(i=0; i<G.vexnum; i++){if(i==0){cout<<"\t\t"<<G.vexs[i].vexs<<" ";}else {cout<<"\t"<<G.vexs[i].vexs<<" ";}}cout<<endl;SetTextC3;for(i = 0; i < G.vexnum; i++){SetTextC2;cout<<"\t"<<G.vexs[i].vexs<<"\t";SetTextC3;for(j = 0; j < G.vexnum; j++){if(G.arcs[i][j].cost<0){G.arcs[i][j].flows = 0;}mincost += G.arcs[i][j].flows * G1.arcs[i][j].cost;if(G.arcs[i][j].flows == 0){SetTextC6;cout<<"0"<<"\t";SetTextC3;} else {cout<<G.arcs[i][j].flows<<"\t";}}cout<<endl;}cout<<endl;SetTextC1;for(j = 0; j < G.vexnum; j++){maxstream = maxstream + G.arcs[0][j].flows;}cout<<"最大流：";SetTextC5;//紫色cout<<maxstream<<endl;SetTextC1;cout<<"最小费用：";SetTextC5;cout<<mincost<<endl;SetTextC1;}int main(){Graphics G;SetTextC7;cout<<"//最小费用最大流算法"<<endl;cout<<"//顶点从vs开始"<<endl;cout<<"//可以求出vs到vt的最小费用最大流"<<endl;cout<<"//其中M表示的无穷大"<<endl;云南大学数学与统计学实验教学中心实验报告SetTextC1;G.start(G);return 0;}第11 页共11 页。