「NOI2006」最大获利 - 最大权闭合图

在前期市场调查和站址勘测之后，公司得到了一共 个可以作为通讯信号中转站的地址，建立第 个通讯中转站需要的成本为 （）。另外公司调查得出了所有期望中的用户群，一共 个。关于第 个用户群的信息概括为 , 和 ：这些用户会使用中转站 和中转站 进行通讯，公司可以获益 。（，，）公司可以有选择的建立一些中转站（投入成本），为一些用户提供服务并获得收益（获益之和）。那么如何选择最终建立的中转站才能让公司的净获利最大呢？

链接

CodeVS 1789
BZOJ 1497

题解

裸的最大权闭合图，用最小割。

代码

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

const int MAXN = 5000;
const int MAXM = 50000;

struct Node;
struct Edge;

struct Node {
    Edge *firstEdge, *currentEdge;
    int level, id;
} nodes[MAXN + MAXM + 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) {}
};

struct Dinic {
    bool makeLevelGraph(Node *s, Node *t, int n) {
        for (int i = 0; i < n; 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->to->level == 0 && e->flow < e->capacity) {
                    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->currentEdge; e; e = e->next) {
            if (e->to->level == s->level + 1 && e->flow < e->capacity) {
                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 n) {
        int ans = 0;
        while (makeLevelGraph(&nodes[s], &nodes[t], n)) {
            for (int i = 0; i < n; i++) nodes[i].currentEdge = nodes[i].firstEdge;
            int flow;
            while ((flow = findPath(&nodes[s], &nodes[t])) > 0) ans += flow;
        }

        return ans;
    }
} dinic;

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;
}

int n, m, a[MAXN];

int main() {
    scanf("%d %d", &n, &m);

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

    for (int i = 1; i <= n; i++) {
        int x;
        scanf("%d", &x);

        addEdge(i, t, x);
    }

    int sum = 0;
    for (int i = n + 1; i <= n + m; i++) {
        int a, b, c;
        scanf("%d %d %d", &a, &b, &c);

        sum += c;
        addEdge(s, i, c);

        addEdge(i, a, INT_MAX);
        addEdge(i, b, INT_MAX);
    }

    int maxFlow = dinic(s, t, n + m + 2);
    printf("%d\n", sum - maxFlow);

    return 0;
}