SCU 4439: Vertex Cover(最小顶点覆盖)

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本文介绍了一种利用最大匹配算法解决最小顶点覆盖问题的方法。通过构建图并使用深度优先搜索来寻找最大匹配，从而确定图中至少需要标记多少个顶点才能使得每条边至少有一个顶点被标记。

Vertex Cover

frog has a graph with n vertices v(1),v(2),…,v(n) and m edges (v(a1),v(b1)),(v(a2),v(b2)),…,(v(am),v(bm)).

She would like to color some vertices so that each edge has at least one colored vertex.

Find the minimum number of colored vertices.

Input

The input consists of multiple tests. For each test:

The first line contains 2 integers n,m (2≤n≤500,1≤m≤n(n−1)2). Each of the following m lines contains 2 integers ai,bi (1≤ai,bi≤n,ai≠bi,min{ai,bi}≤30)

Output

For each test, write 1 integer which denotes the minimum number of colored vertices.

Sample Input

Sample Output

    1
    2

题解：
最小顶点覆盖==最大匹配

#include<bits/stdc++.h>
using namespace std;
const int maxn=505;
vector<int>G[maxn];
int match[maxn];
bool vis[maxn];
bool dfs(int v)
{
    vis[v]=1;
    for(int i=0;i<G[v].size();i++)
    {
        int u=G[v][i],w=match[u];
        if(w<0||(!vis[w]&&dfs(w)))
        {
            match[v]=u;
            match[u]=v;
            return 1;
        }
    }
    return 0;
}
int main()
{
    int n,m;
    while(~scanf("%d%d",&n,&m))
    {
        for(int i=0;i<maxn;i++)G[i].clear();
        for(int i=1;i<=m;i++)
        {
            int x,y;scanf("%d%d",&x,&y);
            G[x].push_back(y);
            G[y].push_back(x);
        }
        int res=0;
        memset(match,-1,sizeof(match));
        for(int i=1;i<=n;i++)
        {
            if(match[i]<0)
            {
                memset(vis,0,sizeof(vis));
                if(dfs(i))res++;
            }
        }
        printf("%d\n",res);
    }
    return 0;
}