要选出一些点,这些点之间没有相邻边且要求权值之和最大,求这个权值
分析:二分图带权最大独立集.
用最大流最小割定理求解.其建图思路是:将所有格点编号,奇数视作X部,偶数视作Y部,建立源点S和汇点T, S向X部的点建边,Y部向T建边,容量为该点权值.
相邻的一对点(肯定是一奇一偶),由X中的点向Y中的点建边,容量为正无穷.
最后跑出最大流,|带权最大独立集| = |点权之和| - |最小割| = |点权之和| - |最大流|

#include<iostream>
#include<cstring>
#include<stdio.h>
#include<algorithm>
#include<string>
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 intstruct 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 G[55][55];
int dir[4][2] = {1,0,-1,0,0,1,0,-1};int main()
{#ifndef ONLINE_JUDGEfreopen("in.txt","r",stdin);freopen("out.txt","w",stdout);#endifint N,M;int u,v,tmp;while(scanf("%d %d",&N, &M)==2){int S = 0,T = N*M+1;int sum  =0;F.init();for(int i=1;i<=N;++i){for(int j=1;j<=M;++j){scanf("%d",&tmp);G[i][j] = tmp;sum+=tmp;}}for(int i=1;i<=N;++i){for(int j=1;j<=M;++j){int id = (i-1)*M +j;if((i+j)&1){F.AddEdge(S,id,G[i][j]);for(int k=0;k<4;++k){int nx = i+ dir[k][0];int ny = j+ dir[k][1];if(nx<1 || nx>N || ny<1 ||ny>M) continue;int nid = (nx-1)*M + ny;F.AddEdge(id,nid,INF);}}else{F.AddEdge((i-1)*M+j,T,G[i][j]);}}}int flow = F.maxFlow_sap(S,T,T+1);int res = sum - flow;printf("%d\n",res);}return 0;
}