HDU 1068 Girls and Boys（匈牙利算法求最大独立集）

Problem Description

the second year of the university somebody started a study on the romantic relations between the students. The relation “romantically involved” is defined between one girl and one boy. For the study reasons it is necessary to find out the maximum set satisfying the condition: there are no two students in the set who have been “romantically involved”. The result of the program is the number of students in such a set.

The input contains several data sets in text format. Each data set represents one set of subjects of the study, with the following description:

the number of students
the description of each student, in the following format
student_identifier:(number_of_romantic_relations) student_identifier1 student_identifier2 student_identifier3 …
or
student_identifier:(0)

The student_identifier is an integer number between 0 and n-1, for n subjects.
For each given data set, the program should write to standard output a line containing the result.

Sample Input

7
0: (3) 4 5 6
1: (2) 4 6
2: (0)
3: (0)
4: (2) 0 1
5: (1) 0
6: (2) 0 1
3
0: (2) 1 2
1: (1) 0
2: (1) 0

Sample Output

5
2

题意

题目给定一些男女生之间相互的 romantic 关系，要求找出一个最大的集合，使得该集合中的所有男女生之间都不存在 romantic 关系。

分析

一个二分图的最大独立集点数与最大二分匹配个数有直接的关系：

$$\mathrm{最}\mathrm{大}\mathrm{独}\mathrm{立}\mathrm{集}\mathrm{点}\mathrm{数}=\mathrm{顶}\mathrm{点}\mathrm{数}-\mathrm{最}\mathrm{大}\mathrm{二}\mathrm{分}\mathrm{匹}\mathrm{配}\mathrm{对}\mathrm{数}$$

故本题直接转化为求最大二分匹配即可，需要注意的是，题中给出的条件是 1 指向 2，2 也会指向 1，所以最终算出来的匹配数其实是实际对数的两倍，最终被顶点数减去之前首先需要折半。

基础二分匹配练手题。

#include<bits/stdc++.h>
#define sd(n) scanf("%d",&n)
using namespace std;
const int N = 1e3 + 10;
int e[N][N], match[N];
bool book[N];
int n, f[N];

bool dfs(int x) {
	for (int i = 0; i < n; ++i) {
		if (e[x][i] && !book[i]) {
			book[i] = true;
			if (match[i] == -1 || dfs(match[i])) {
				match[i] = x;
				return true;
			}
		}
	}
	return false;
}

int main() {
	while (~scanf("%d", &n)) {
		memset(e, 0, sizeof e);//多组输入，别忘记初始化图啊，TLE几发了。。。
		for (int i = 0; i < n; i++)
			match[i] = -1;
		for (int i = 0; i < n; ++i) {
			int t1, t2, u;
			scanf("%d: (%d)", &t1, &t2);
			for (int j = 0; j < t2; j++){
				scanf("%d", &u);
				e[t1][u] = 1;
			}
		}
		int cnt = 0;
		for (int i = 0; i < n; ++i) {
			memset(book, false, sizeof book);
			if (dfs(i))
				cnt++;
		}
		cout << n - cnt / 2 << endl;
	}
}