POJ - 2125 Destroying The Graph (最小点权覆盖)

题意:给一张图,现在要删去所有的边,删去一个点的所有入边和所有出边都有其对应\(W_{i+}\)和\(W_{i-}\).求删去该图的最小花费,并输出解
分析:简而言之就是用最小权值的点集去覆盖所有的边.
模型转化到网络流的建图中,将图中的边视作点,并将其一拆为二,出点作为X部,入点作为Y部,若有边(u,v)则由建边u->v+N,容量正无穷.
将原图中的点视作边,入边的花费即建边i+N->T,容量为其花费;出边的花费建边S->i,容量也是为其花费.
跑出S->T的最大流,|最大流| = |最小割| = |最小点权覆盖|.

还需输出每条边的解决方案.在残余网上,与S相连的点,若满流则表示没有选择该点;与T相连的点,若满流则表示选择该点.从源点dfs一次后,再遍历从S出发和从T出发的所有弧即可.
*其性质和最大权闭合子图很相似.

#include<iostream>
#include<cstring>
#include<stdio.h>
#include<algorithm>
#include<string>
#include<cmath>
#include<vector>
using namespace std;
const int INF = 0x3f3f3f3f;
const int MAXN=3010;//点数的最大值
const int MAXM=400010;//边数的最大值
#define captype int

struct SAP_MaxFlow{
    struct Edge{
        int from,to,next;
        captype cap;
    }edges[MAXM];
    int tot,head[MAXN];
    int gap[MAXN];
    int dis[MAXN];
    int cur[MAXN];
    int pre[MAXN];

    void init(){
        tot=0;
        memset(head,-1,sizeof(head));
    }
    void AddEdge(int u,int v,captype c,captype rc=0){
        edges[tot] = (Edge){u,v,head[u],c};  head[u]=tot++;
        edges[tot] = (Edge){v,u,head[v],rc}; head[v]=tot++;
    }
    captype maxFlow_sap(int sNode,int eNode, int n){//n是包括源点和汇点的总点个数，这个一定要注意
        memset(gap,0,sizeof(gap));
        memset(dis,0,sizeof(dis));
        memcpy(cur,head,sizeof(head));
        pre[sNode] = -1;
        gap[0]=n;
        captype ans=0;
        int u=sNode;
        while(dis[sNode]<n){
            if(u==eNode){
                captype Min=INF ;
                int inser;
                for(int i=pre[u]; i!=-1; i=pre[edges[i^1].to])
                if(Min>edges[i].cap){
                    Min=edges[i].cap;
                    inser=i;
                }
                for(int i=pre[u]; i!=-1; i=pre[edges[i^1].to]){
                    edges[i].cap-=Min;
                    edges[i^1].cap+=Min;
                }
                ans+=Min;
                u=edges[inser^1].to;
                continue;
            }
            bool flag = false;
            int v;
            for(int i=cur[u]; i!=-1; i=edges[i].next){
                v=edges[i].to;
                if(edges[i].cap>0 && dis[u]==dis[v]+1){
                    flag=true;
                    cur[u]=pre[v]=i;
                    break;
                }
            }
            if(flag){
                u=v;
                continue;
            }
            int Mind= n;
            for(int i=head[u]; i!=-1; i=edges[i].next)
            if(edges[i].cap>0 && Mind>dis[edges[i].to]){
                Mind=dis[edges[i].to];
                cur[u]=i;
            }
            gap[dis[u]]--;
            if(gap[dis[u]]==0) return ans;
            dis[u]=Mind+1;
            gap[dis[u]]++;
            if(u!=sNode) u=edges[pre[u]^1].to;  //退一条边
        }
        return ans;
    }
}F;

int vis[MAXN];
void dfs(int u)
{
    vis[u] = 1;
    for(int i=F.head[u];~i;i=F.edges[i].next){
        int v= F.edges[i].to;
        if(!vis[v] && F.edges[i].cap>0){
            dfs(v);
        }
    }
}

int main()
{
	#ifndef ONLINE_JUDGE
        freopen("in.txt","r",stdin);
        freopen("out.txt","w",stdout);
    #endif
    int N,M;
    int u,v,tmp;
    while(scanf("%d %d",&N, &M)==2){
        F.init();
        int s = 0,t = N*2+1;
        for(int i=1;i<=N;++i){
            scanf("%d",&tmp);
            F.AddEdge(i+N,t,tmp);
        }
        for(int i=1;i<=N;++i){
            scanf("%d",&tmp);
            F.AddEdge(s,i,tmp);
        }
        for(int i =1;i<=M;++i){
            scanf("%d %d",&u, &v);
            F.AddEdge(u,v+N,INF);
        }
        int flow = F.maxFlow_sap(s,t,t+1);
        memset(vis,0,sizeof(vis));
        dfs(s);
        vector<int> res;
        for(int i=F.head[s];~i;i=F.edges[i].next){
            int u = s;
            int v = F.edges[i].to;
            if(vis[u] && !vis[v]){
                res.push_back(v);
            }
        }
        for(int i=F.head[t];~i;i=F.edges[i].next){
            int u = t;
            int v = F.edges[i].to;
            if(vis[v] && !vis[u]){
                res.push_back(v);
            }
        }
        int sz = res.size();
        printf("%d\n%d\n",flow,sz);
        for(int i=0;i<sz;++i){
            if(res[i]<=N){
                printf("%d -",res[i]);
            }
            else{
                printf("%d +",res[i]-N);
            }
            printf("\n");
        }
    }
    return 0;
}