「COGS 14」搭配飞行员 - 二分图匹配

从一个二分图中选出尽量多的边，使得任意两条边没有公共点。

链接

COGS 14

题解

二分图匹配，可以用匈牙利~也可以用带花树~，然而我不会写。

所以要把二分图最大匹配转化成最大流 ……

设二分图左右两列分别为 X 和 Y，建立超级源点 S，从 S 向 X 中的每个点连一条边，容量为 1，建立超级汇点 T，从 Y 中的每个点向 T 连一条边，容量为 1。最后对于原图的每一条边 (u, v)（假设 u 在左侧 v 在右侧），连接一条由 u 指向 v 的有向边，容量为 1。然后跑一遍最大流就是最大匹配啦！

代码

#include <cstdio>
#include <climits>
#include <algorithm>
#include <queue>

const int MAXN = 100;

struct Node;
struct Edge;

struct Node {
    Edge *firstEdge;
    int level;
} nodes[MAXN + 2];

struct Edge {
    Node *from, *to;
    int capacity, flow;
    Edge *next, *reversedEdge;

    Edge(Node *from, Node *to, int capacity) : from(from), to(to), capacity(capacity), flow(0), next(from->firstEdge) {}
};

int n, n1;

inline void addEdge(int from, int to, int capacity) {
    nodes[from].firstEdge = new Edge(&nodes[from], &nodes[to], capacity);
    nodes[to].firstEdge = new Edge(&nodes[to], &nodes[from], 0);

    nodes[from].firstEdge->reversedEdge = nodes[to].firstEdge, nodes[to].firstEdge->reversedEdge = nodes[from].firstEdge;
}

struct Dinic {
    bool makeLevelGraph(Node *s, Node *t) {
        for (int i = 0; i < n + 1; i++) nodes[i].level = 0;

        std::queue<Node *> q;
        q.push(s);
        s->level = 1;

        while (!q.empty()) {
            Node *v = q.front();
            q.pop();

            for (Edge *e = v->firstEdge; e; e = e->next) {
                if (e->flow < e->capacity && e->to->level == 0) {
                    e->to->level = v->level + 1;
                    if (e->to == t) return true;
                    else q.push(e->to);
                }
            }
        }

        return false;
    }

    int findPath(Node *s, Node *t, int limit = INT_MAX) {
        if (s == t) return limit;

        for (Edge *e = s->firstEdge; e; e = e->next) {
            if (e->to->level == s->level + 1 && e->capacity > e->flow) {
                int flow = findPath(e->to, t, std::min(limit, e->capacity - e->flow));
                if (flow > 0) {
                    e->flow += flow;
                    e->reversedEdge->flow -= flow;
                    return flow;
                }
            }
        }

        return 0;
    }

    int operator()(int s, int t) {
        int ans = 0;
        while (makeLevelGraph(&nodes[s], &nodes[t])) {
            int flow;
            while ((flow = findPath(&nodes[s], &nodes[t])) > 0) ans += flow;
        }

        return ans;
    }
} dinic;

int main() {
    freopen("flyer.in", "r", stdin);
    freopen("flyer.out", "w", stdout);

    scanf("%d %d", &n, &n1);

    const int s = 0, t = n + 1;

    for (int i = 1; i <= n1; i++) addEdge(s, i, 1);
    for (int i = n1 + 1; i <= n; i++) addEdge(i, t, 1);

    while (!feof(stdin)) {
        int u, v;
        scanf("%d %d", &u, &v);
        if (u > v) std::swap(u, v);

        addEdge(u, v, 1);
    }

    printf("%d\n", dinic(s, t));

    fclose(stdin);
    fclose(stdout);

    return 0;
}