【图论】二分图匹配(Hungary算法&KM算法&hopcroft-karp算法)

匈牙利算法板子(用邻接表写可以做到O(n*m))：

const int maxn = 510;
int cnt_x,cnt_y;
int G[maxn][maxn];
int link[maxn];
bool vis[maxn];

bool find(int u){
    for(int v=1;v<=cnt_y;v++){
        if(G[u][v]&&!vis[v]){
            vis[v]=true;
            if(link[v]==-1||find(link[v])){
                link[v]=u;
                return true;
            }
        }
    }
    return false;
}

int Hungary(){
    int ans=0;
    memset(link,-1,sizeof(link));
    for(int u=1;u<=cnt_x;u++){
        memset(vis,0,sizeof(vis));
        if(find(u)) ans++;
    }
    return ans;
}

---

KM算法の O(n^3)的板子：

#include
#define ll long long
#define endl '\n'
#define mem(a) memset(a,0,sizeof(a))
#define IO ios::sync_with_stdio(false);cin.tie(0);
using namespace std;
const int INF=0x3f3f3f3f;
const int mod=1e9+7;
const int maxn=1e3+7;
int wx[maxn],wy[maxn];
int cx[maxn],cy[maxn];
int visx[maxn],visy[maxn];
int cntx,cnty;
int Map[maxn][maxn];
int slack[maxn];
bool find(int u){
	visx[u]=1;
	for(int v=1;v<=cnty;v++){
		if(!visy[v]&&Map[u][v]!=INF){
			int t=wx[u]+wy[v]-Map[u][v];
			if(t==0){
				visy[v]=1;
				if(cy[v]==-1||find(cy[v])){
					cx[u]=v;
					cy[v]=u;
					return true;
				}
			}
			else if(t>0){
				slack[v]=min(slack[v],t);
			}
		}
	}
	return false;
}
int KM(){
	memset(cx,-1,sizeof(cx));
	memset(cy,-1,sizeof(cy));
	memset(wx,0,sizeof(wx));
	memset(wy,0,sizeof(wy));
	for(int i=1;i<=cntx;i++){
		for(int j=1;j<=cnty;j++){
			if(Map[i][j]==INF) continue;
			wx[i]=max(wx[i],Map[i][j]);
		}
	}
	for(int i=1;i<=cntx;i++){
		memset(slack,INF,sizeof(slack));
		while(1){
			memset(visx,0,sizeof(visx));
			memset(visy,0,sizeof(visy));
			if(find(i)) break;
			int d=INF;
			for(int j=1;j<=cnty;j++){
				if(!visy[j]&&d>slack[j]) d=slack[j];
			}
			for(int j=1;j<=cntx;j++){
				if(visx[j]) wx[j]-=d;
			}
			for(int j=1;j<=cnty;j++){
				if(visy[j]) wy[j]+=d;
				else slack[j]-=d;
			}
		}
	}
	int ans=0;
	for(int i=1;i<=cntx;i++){
		if(cx[i]!=-1) ans+=Map[i][cx[i]];
	}
	return ans;
}
int main(){
	while(cin>>cntx>>cnty){
		for(int i=1;i<=cntx;i++){
			for(int j=1;j<=cnty;j++){
				cin>>Map[i][j];
			}
		}
		cout<<KM()<<endl;
	}
	return 0;
}

---

O(n^4)的板子，看上去易懂，但是不是很推荐，适合入门:
ps（n左m右）

#include
#define ll long long
#define endl '\n'
#define mem(a) memset(a,0,sizeof(a))
#define IO ios::sync_with_stdio(false);cin.tie(0);
using namespace std;
const int INF=0x3f3f3f3f;
const int mod=1e9+7;
const int maxn=1e3+7;
int w[maxn][maxn];
int line[maxn],usex[maxn],usey[maxn],cx[maxn],cy[maxn];
int ans,n,m;
bool find(int x){
	usex[x]=1;
	for(int i=1;i<=m;i++){
		if((usey[i]==0)&&(cx[x]+cy[i]==w[x][i])){
			usey[i]=1;
			if(line[i]==0||find(line[i])){
				line[i]=x;
				return true;
			}
		}
	}
	return false;
}
int KM(){
	for(int i=1;i<=n;i++){
		while(true){
			int d=INF;
			memset(usex,0,sizeof(usex));
			memset(usey,0,sizeof(usey));
			if(find(i)) break;
			for(int j=1;j<=n;j++){
				if(usex[j]){
					for(int k=1;k<=m;k++){
						if(!usey[k]) d=min(d,cx[j]+cy[k]-w[j][k]);
					}
				}
			}
			if(d==INF) return -1;
			for(int j=1;j<=n;j++){
				if(usex[j]) cx[j]-=d;
			}
			for(int j=1;j<=m;j++){
				if(usey[j]) cy[j]+=d;
			}
		}
	}
	ans=0;
	for(int i=1;i<=m;i++){
		ans+=w[line[i]][i];
	}
	return ans;
}
int main(){
	while(cin>>n>>m){
		memset(cx,0,sizeof(cx));
		memset(cy,0,sizeof(cy));
		memset(w,0,sizeof(w));
		memset(line,0,sizeof(line));
		for(int i=1;i<=n;i++){
			int d=0;
			for(int j=1;j<=n;j++){
				cin>>w[i][j];
				d=max(d,w[i][j]);
			}
			cx[i]=d;
		}
		cout<<KM()<<endl;
	}
	return 0;
}

---

hopcroft-karp算法(O(sqrt(n)m) )：

#include
#define ll long long
#define rep(i,a,n) for(int i=a;i<=n;i++)
#define per(i,n,a) for(int i=n;i>=a;i--)
#define endl '\n'
#define mem(a) memset(a,0,sizeof(a))
#define IO ios::sync_with_stdio(false);cin.tie(0);
using namespace std;
const int INF=0x3f3f3f3f;
const int mod=1e9+7;
const int maxn=505;
int bmap[maxn][maxn];
int cx[maxn],cy[maxn],dx[maxn],dy[maxn];
bool vis[maxn];
int nx,ny,k,dis;
bool searchpath(){
	queue<int> q;
	dis=INF;
	memset(dx,-1,sizeof(dx));
	memset(dy,-1,sizeof(dy));
	for(int i=1;i<=nx;i++){
		if(cx[i]==-1){
			q.push(i);
			dx[i]=0;
		}
	}
	while(!q.empty()){
		int u=q.front();
		q.pop();
		if(dx[u]>dis) break;
		for(int v=1;v<=ny;v++){
			if(bmap[u][v]&&dy[v]==-1){
				dy[v]=dx[u]+1;
				if(cy[v]==-1) dis=dy[v];
				else{
					dx[cy[v]]=dy[v]+1;
					q.push(cy[v]);
				}
			}
		}
	}
	return dis!=INF;
}
int find(int u){
	for(int v=1;v<=ny;v++){
		if(!vis[v]&&bmap[u][v]&&dy[v]==dx[u]+1){
			vis[v]=1;
			if(cy[v]!=-1&&dy[v]==dis) continue;
			if(cy[v]==-1||find(cy[v])){
				cy[v]=u;cx[u]=v;
				return true;
			}
		}
	}
	return false;
}
int MaxMatch(){
	int res=0;
	memset(cx,-1,sizeof(cx));
	memset(cy,-1,sizeof(cy));
	while(searchpath()){
		memset(vis,0,sizeof(vis));
		for(int i=1;i<=nx;i++){
			if(cx[i]==-1){
				res+=find(i);
			}
		}
	}
	return res;
}
int main(){
	while(cin>>k>>nx>>ny&&k){
		memset(bmap,0,sizeof(bmap));
		for(int i=1;i<=k;i++){
			int u,v;cin>>u>>v;
			bmap[u][v]=1;
		}
		cout<<MaxMatch()<<endl;
	}
}