Topic: Find the maximum matching of bipartite graph
Idea: The maximum matching of bipartite graphs has Hungarian algorithm, but it can also be solved by network flow. We set the point set on the left side of the bipartite graph to A, and the point set on the right side of the bipartite graph to B. We connect a directed edge from A to B and the edge weight is 1. We set a super source and a super sink. The super source connects the directed edge to the A set, and the B set connects the directed edge to the super sink (the edge weight is 1). Use dinic algorithm to find the maximum flow. The maximum flow is the maximum matching, and the matching edge is the edge whose flow is not 0 among the forward edges.
code：
#include <bits/stdc++.h>
#define ll long long
#define pi pair<int,int>
#define mk make_pair
#define pb push_back
using namespace std;
const int maxn = 105 inf = 2e9+10000;
struct Edge{
int from,to,cap,flow;
};
struct Dinic{
int n,m,s,t;
vector<Edge>edges;
vector<int>g[maxn];
bool vis[maxn];
int d[maxn];
int cur[maxn];
void init(int n)
{
this->n=n;
for(int i=0;i<=n;i++)g[i].clear();
edges.clear();
}
void AddEdge(int from,int to,int cap)
{
edges.push_back((Edge){
from,to,cap,0});
edges.push_back((Edge){
to,from,0,0});
m=edges.size();
g[from].push_back(m-2);
g[to].push_back(m-1);
}
bool BFS()
{
memset(vis,0,sizeof(vis));
queue<int>q;
q.push(s);
d[s]=0;
vis[s]=1;
while(!q.empty())
{
int x=q.front();
q.pop();
for(int i=0;i<g[x].size();i++)
{
Edge& e=edges[g[x][i]];
if(!vis[e.to]&&e.cap>e.flow)
{
vis[e.to]=1;
d[e.to]=d[x]+1;
q.push(e.to);
}
}
}
return vis[t];
}
int DFS(int x,int a)
{
if(x==t||a==0)return a;
int flow=0,f;
for(int& i=cur[x];i<g[x].size();i++)
{
Edge& e=edges[g[x][i]];
if(d[x]+1==d[e.to]&&(f=DFS(e.to,min(a,e.cap-e.flow)))>0)
{
e.flow+=f;
edges[g[x][i]^1].flow-=f;
flow+=f;
a-=f;
if(a==0)break;
}
}
return flow;
}
int Maxflow(int s,int t,int need)
{
this->s=s;
this->t=t;
int flow=0;
while(BFS())
{
memset(cur,0,sizeof(cur));
flow+=DFS(s,inf);
if(flow>=need)return flow;
}
return flow;
}
vector<int> Mincut()
{
vector<int>ans;
for(int i=0;i<edges.size();i++)
{
Edge& e=edges[i];
if(vis[e.from]&&!vis[e.to]&&e.cap>0)
ans.push_back(i);
}
return ans;
}
void Reduce()
{
for(int i=0;i<edges.size();i++)
edges[i].cap-=edges[i].flow;
}
void ClearFlow()
{
for(int i=0;i<edges.size();i++)
edges[i].flow=0;
}
}sol;
int main()
{
int m,n;
scanf("%d%d",&m,&n);
int u,v;
while(~scanf("%d%d",&u,&v))
{
if(u == -1)break;
sol.AddEdge(u,v,1);
}
int s = n+1;
int t = s+1;
for(int i=1;i<=m;i++) {
sol.AddEdge(s,i,1);
}
for(int i=m+1;i<=n;i++) {
sol.AddEdge(i,t,1);
}
int ans = sol.Maxflow(s,t,inf);
printf("%d\n",ans);
for(int i=0;i<sol.edges.size();i++)
{
if(sol.edges[i].from <= m && sol.edges[i].flow > 0)printf("%d %d\n",sol.edges[i].from,sol.edges[i].to);
}
return 0;
}