网络流小结(HNOI2019之前)

\(\text{一:Dinic最大流}\)

最坏复杂度 \({\mathcal O(n^2m)}\) 一般可以处理 \(10^4\) ~ \(10^5\) 的网络。

struct Edge {
    int v, w, nxt;
} E[M * 2];

int head[N], tot = 1;//记得从1存边

inline void add(int u, int v, int w) {
    E[++tot] = (Edge) { v, w, head[u] }; 
    head[u] = tot;
}

int n, m, d[N], S, T;

//bfs找增广路
bool bfs() {
    queue<int> q;
    memset(d, 0, sizeof(int) * (n + 1));
    d[S] = 1;
    q.push(S);
    while (q.size()) {
        int u = q.front();
        q.pop();
        for (register int i = head[u]; i; i = E[i].nxt) {
            if (E[i].w and d[E[i].v] == 0) {
                d[E[i].v] = d[u] + 1;
                if (E[i].v == T) {
                    return true;
                }
                q.push(E[i].v);
            }
        }
    }
    return false;
}

int Dinic(int u, int flow) {
    if (u == T) {
        return flow;
    }
    int k, rest = flow;
    for (register int i = head[u]; i; i = E[i].nxt) {
        int v = E[i].v;
        if (E[i].w and d[v] == d[u] + 1) {
            k = Dinic(v, min(E[i].w, rest));
            if (k == 0) {
                d[v] = 0;
            }
            E[i].w -= k;
            E[i ^ 1].w += k;
            rest -= k;
        }
    }
    return flow - rest;
}

//主函数中

for (register int u, v, w, i = 1; i <= m; ++ i) {
    cin >> u >> v >> w;
    add(u, v, w);//建边
    add(v, u, 0);//反悔的边，记得w赋为0
}

int maxflow = 0, flow = 0;
    while (bfs()) 
        while (flow = Dinic(S, INF))
            maxflow += flow;

二：\(\text{Dinic费用流}\)

复杂度不知道，\(10^3\) ~ \(10^4\) 应该能跑。

struct Edge {
    int v, w, cost, nxt;
} E[M * 2];

int head[N], tot = 1;//同上

inline void add(int u, int v, int w, int cost) {
    E[++tot] = (Edge) { v, w, cost, head[u] }; //多加一维费用
    head[u] = tot;
}

bool vis[N];
int n, m, d[N], S, T, mincost;

inline bool spfa() {//SPFA找增广路，要求增广路费用最小
    queue<int> q;
    memset(d, 0x3f, sizeof(int)*(n+1));
    memset(vis, 0, (n+1));
    d[S] = 0;
    vis[S] = 1;
    q.push(S);
    while (q.size()) {
        int u = q.front(), v;
        q.pop();
        vis[u] = 0;
        for (register int i = head[u]; i; i = E[i].nxt) {
            v = E[i].v;
            if (E[i].w and d[v] > d[u] + E[i].cost) {
                d[v] = d[u] + E[i].cost;
                if (vis[v]) continue;
                q.push(v);
                vis[v] = 1;
            }
        }
    }
    return d[T] != d[0];
}

int Dinic(int u, int flow) {
    if (u == T) {
        return flow;
    }
    vis[u] = 1;//别忘了vis
    int k, rest = flow;
    for (register int i = head[u]; i and rest; i = E[i].nxt) {
        int v = E[i].v;
        if ((!vis[v] or v == T) and E[i].w and d[u] + E[i].cost == d[v]) {//这里多了个限制条件。
            k = Dinic(v, min(E[i].w, rest));
            if (k == 0) {
                d[v] = 0;
            }
            rest -= k;
            E[i].w -= k;
            E[i ^ 1].w += k;
            mincost += E[i].cost * k;//累计贡献（费用）
        }
    }
    return flow - rest;
}

//主函数

for (register int i = 1; i <= m; ++ i) {
    int u, v, c, w;
    cin >> u >> v >> w >> c;
    add(u, v, w, c);
    add(v, u, 0, -c);//反悔的边cost为负
}

int maxflow = 0, flow;
while (spfa()) {//与最大流略有不同。
    vis[T] = 1;
    while (vis[T]) {
        memset(vis, 0, (n+1));
        maxflow += Dinic(S, INF);
    }
}

三：一些题目：

DAG最小路径覆盖

建拆点二分图，跑dinic(或匈牙利)，答案 = 点数 - 最大流(最大匹配数)。

重点在如何递归输出路径，看代码。

/*
拆点的时候要拆成i与i+n

最好不要拆成i<<1 and i<<1|1

否则会出现玄学错误
*/
#include<bits/stdc++.h>
#pragma GCC optimize(2)

using namespace std;

const int N = 5000, M = 6007;

int n, m, s = 0, t = 3500;
int ver[M<<1], d[N<<1], edge[M<<1], nxt[M<<1], head[N<<1];
int pre[N<<1], succ[N<<1], tot = 1;

void addEdge(int u, int v, int w) {
    ver[++tot] = v; edge[tot] = w; nxt[tot] = head[u]; head[u] = tot;
}

int bfs() {
    queue<int> q;
    memset(d, 0, sizeof d);
    q.push(s); d[s] = 1;
    while (q.size()) {
        int v, u = q.front(); q.pop();
        for (int i = head[u]; i; i = nxt[i]) {
            if (edge[i] and !d[v = ver[i]]) {
                d[v] = d[u] + 1;
                if (v == t) return 1;
                q.push(v);
            }
        }
    }
    return 0;
}
int dinic(int u, int flow) {
    if (u == t) return flow;
    int k, rest = flow, v;
    for (int i = head[u]; i and rest; i = nxt[i]) {
        if (edge[i] and d[v = ver[i]] == d[u] + 1) {
            k = dinic(v, min(edge[i], rest));
            if (!k) d[v] = 0;
            rest -= k;
            edge[i] -= k;
            edge[i ^ 1] += k;
            if (v != t and k and u != s) {//记录路径
                succ[u] = v - n;
                pre[v - n] = u;
            }
        }
    }
    return flow - rest;
}

int vis[N<<1];

void put(int x) {//递归输出路径
    if (!x) return;
    if (pre[x] != x) put(pre[x]);
    vis[x] = 1;
    printf("%d ", x);
}

int main() {
    scanf("%d%d", &n, &m);
    for (int i = 1; i <= m; ++ i) {
        int u, v;
        scanf("%d", &u);
        scanf("%d", &v);
        addEdge(u, v + n, 1);
        addEdge(v + n, u, 0);
    }
    for (int i = 1; i <= n; ++ i) {
        addEdge(s, i, 1); addEdge(i, s, 0);
        addEdge(i + n, t, 1); addEdge(t, i + n, 0);
    }
    for (int i = 1; i <= n; ++ i)
        pre[i] = succ[i] = i;
    int maxflow = 0, flow;
    while (bfs())
        while (flow = dinic(s, 0x3f3f3f3f))
            maxflow += flow;
    int ans = n - maxflow;
    for (int i = n; i >= 1; -- i) {
        if (succ[i] == i and !vis[i]) {//输出路径
            put(i);
            puts("");
        }
    }
    printf("%d\n", ans);
}

试题库问题

对于每一个题型 \(i\) ，都可以选择一个题目 \(j\) ，那么就从 \(i\) 到 \(j\) 连一条边

那么就是二分图多重匹配

可以拆点匈牙利做, 但效率低下

考虑网络流，

对于一个左部节点 \(L\) 它要匹配 \(x\) 个右部节点，就从源点连一条容量为 \(x\) 的边。

#include<bits/stdc++.h>
#pragma GCC optimize(2)

using namespace std;

const int N = 50000, oo = 0x3f3f3f3f;

int tot = 1, k, n, s, t;
int ver[N], edge[N], nxt[N], head[N];
vector<int> succ[N];

void add(int u, int v, int w) {
    ver[++tot] = v;
    edge[tot] = w;
    nxt[tot] = head[u];
    head[u] = tot;
    ver[++tot] = u;
    edge[tot] = 0;
    nxt[tot] = head[v];
    head[v] = tot;
}

int d[N];

bool bfs() {
    queue<int> q;
    memset(d, 0, sizeof d);
    d[s] = 1; q.push(s);
    while (q.size()) {
        int u = q.front(), y; q.pop();
        for (int i = head[u]; i; i = nxt[i]) {
            if (edge[i] and !d[y = ver[i]]) {
                d[y] = d[u] + 1;
                if (y == t) return 1;
                q.push(y);
            }
        }
    }
    return 0;
}

int dinic(int u, int flow) {
    if (u == t) return flow;
    int k, rest = flow;
    for (int i = head[u]; i and rest; i = nxt[i]) {
        int v = ver[i];
        if (edge[i] and d[v] == d[u] + 1) {
            k = dinic(v, min(edge[i], rest));
            if (!k) d[v] = 0;
            edge[i] -= k;
            edge[i ^ 1] += k;
            rest -= k;
        }
    }
    return flow - rest;
}

int main() {
    int m = 0;
    scanf("%d%d", &k, &n);
    s = 0, t = 5000; 
    for (int i = 1; i <= k; ++ i) {
        int x; scanf("%d", &x); m += x;
        add(s, i, x);
    }
    for (int i = 1; i <= n; ++ i) {
        int num; scanf("%d", &num);
        for (int j = 1; j <= num; ++ j) {
            int x; scanf("%d", &x);
            add(x, i + k, 1);
        }
        add(i + k, t, 1);
    }
    int flow, maxflow = 0;
    while (bfs())
        while (flow = dinic(s, +oo))
            maxflow += flow;
    if (m != maxflow) 
    {cout << "No Solution!" << endl; return 0;}
    for (int i = 1; i <= k; ++ i) {
        printf("%d: ", i);
        for (int j = head[i]; j; j = nxt[j]) {
            if (!edge[j] and ver[j] != s and ver[j] != t)//根据残量网络输出。
                printf("%d ", ver[j] - k);
        }
        puts("");
    }
}

最长不下降子序列问题

第一问直接做 \(LIS\)

第二问建模：若 \(f[i]\) 能转移到 \(f[j]\) , 才向 \(i\) 到 \(j\) 连边.

第三问直接在残量网络上将 \(s\) 到 \(1\) 与 \(n\) 到 \(t\) 扩容(要讨论)，继续找增广路。

#include<bits/stdc++.h>
#pragma GCC optimize(2)
using namespace std;

const int N = 50000, oo = 0x7f7f7f7f;

int n, s, t;
int tot = 1, edge[N], ver[N], nxt[N], d[N], f[N], head[N];
int a[N];

void add(int u, int v, int w) {
    ver[++tot] = v;
    edge[tot] = w;
    nxt[tot] = head[u];
    head[u] = tot;
    ver[++tot] = u;
    edge[tot] = 0;
    nxt[tot] = head[v];
    head[v] = tot;
}

int bfs() {
    memset(d, 0, sizeof d);
    queue<int> q;
    q.push(s); d[s] = 1;
    while (q.size()) {
        int u = q.front(); q.pop();
        for (int i = head[u]; i; i = nxt[i]) {
            if (edge[i] and !d[ver[i]]) {
                d[ver[i]] = d[u] + 1;
                if (ver[i] == t) return 1;
                q.push(ver[i]);
            }
        }
    }
    return 0;
}

int dinic(int u, int flow) {
    if (u == t) return flow;
        int k, rest = flow, v;
    for (int i = head[u]; i and rest; i = nxt[i]) {
        if (edge[i] and d[v = ver[i]] == d[u] + 1) {
            k = dinic(v, min(edge[i], rest));
            if (!k) d[v] = 0;
            edge[i] -= k;
            edge[i ^ 1] += k;
            rest -= k;
        }
    }
    return flow - rest;
}
int main() {
    scanf("%d", &n);
    for (int i = 1; i <= n; ++ i) 
        scanf("%d", a + i);

    for (int i = 1; i <= n; ++ i) {
        f[i] = 1;
        for (int j = 1; j < i; ++ j)
            if (a[i] >= a[j])
                f[i] = max(f[j] + 1, f[i]);	
    }
    int ans = 0;
    for (int i = 1; i <= n; ++ i)
        ans = max(ans, f[i]);
    cout << ans << endl;

    int maxflow = 0, flow;
    for (int i = 1; i <= n; ++ i) 
        for (int j = i + 1; j <= n; ++ j) 
            if (a[i] <= a[j] and f[j] == f[i] + 1)
                add(i, j, 1);
    s = 0, t = 1000;
    for (int i = 1; i <= n; ++ i) {
        if (f[i] == 1)    
            add(s, i, 1); 
        if (f[i] == ans)  //想一想这里为什么不是"else if(f[i]==ans)"
            add(i, t, 1); //调了我好久
    }
    while (bfs())
        while (flow = dinic(s, +oo))
            maxflow += flow;
    cout << maxflow << endl;

    add(s, 1, +oo); 
    if (f[n] == ans)//注意，只有满足f[n]==ans的才能连一条+oo的边到t
        add(n, t, +oo);
    while (bfs())
        while (flow = dinic(s, +oo)) //新加边后残量网络上跑的最大流
            maxflow += flow;		//加上第二问的答案就是第三问答案
    cout << maxflow << endl;
}

方格取数问题

二分图带权最大独立集

\(Ans = sum - maxflow\)

先对每个格子黑白相间地染色，分成左部节点与右部节点。

对于每个左部节点和右部节点都有一个权值，要求选出的点权值最大且没有连边

建边方法：

\(1. s \xRightarrow{val[L]} L\)

\(2. L \xRightarrow{inf} R\)

\(3. R \xRightarrow{val[R]}t\)

跑出来的最大流就是答案。

#include<bits/stdc++.h>
#pragma GCC optimize(2)

using namespace std;

const int N = 50000, oo = 0x3f3f3f3f;

int n, m, s, t;
int tot = 1, edge[N], ver[N], nxt[N], head[N];
int val[200][200], id[200][200];

void add(int u, int v, int w) {
    ver[++tot] = v;
    edge[tot] = w;
    nxt[tot] = head[u];
    head[u] = tot;
    ver[++tot] = u;
    edge[tot] = 0;
    nxt[tot] = head[v];
    head[v] = tot;
}

const int tx[] = {1,0,-1,0};
const int ty[] = {0,1,0,-1};

void ADD(int x, int y) {
    for (int i = 0; i < 4; ++ i) {
        int nx = x + tx[i], ny = y + ty[i];
        if (nx > 0 and nx <= n and ny > 0 and ny <= m)
            add(id[x][y], id[nx][ny], +oo);
    }
}
int d[N];

int bfs() {
    memset(d, 0, sizeof d);
    queue<int> q; q.push(s);
    d[s] = 1;
    while (q.size()) {
        int u = q.front(); q.pop();
        for (int i = head[u]; i; i = nxt[i]) {
            if (edge[i] and !d[ver[i]]) {
                d[ver[i]] = d[u] + 1;
                if (ver[i] == t) return 1;
                q.push(ver[i]);
            }
        }
    }
    return 0;
}

int dinic(int u, int flow) {
    if (u == t) return flow;
    int k, rest = flow;
    for (int i = head[u]; i and rest; i = nxt[i]) {
        if (edge[i] and d[ver[i]] == d[u] + 1) {
            k = dinic(ver[i], min(edge[i], rest));
            if (!k) d[ver[i]] = 0;
            rest -= k;
            edge[i] -= k;
            edge[i ^ 1] += k;
        }
    }
    return flow - rest;
}
int main() {
    int cnt = 0, sum = 0;
    scanf("%d%d", &n, &m);
    for (int i = 1; i <= n; ++ i)
        for (int j = 1; j <= m; ++ j) {
            int x; scanf("%d", &x);
            sum += x;
            id[i][j] = ++cnt;
            val[i][j] = x;
        }
    s = 0, t = cnt + 11;
    for (int i = 1; i <= n; ++ i) {
        for (int j = 1; j <= m; ++ j) {
            if (i + j & 1) {
                ADD(i,j);
                add(s, id[i][j], val[i][j]);
            } else 
                add(id[i][j], t, val[i][j]);
        }
    }
    int maxflow = 0, flow;
    while (bfs()) 
        while (flow = dinic(s, +oo))
            maxflow += flow;
    printf("%d\n", sum - maxflow);
    return 0;
}

晨跑

由于每个点（除了1，n）只能经过一次，所以考虑拆点，然后连一条容量为1的边，跑最小费用最大流。

一个点只可以选一次，可以尝试拆点，连容量为1的边。

#include<bits/stdc++.h>
#pragma GCC optimize(2)

using namespace std;

const int N = 500000, oo = 0x3f3f3f3f;

int tot = 1, n, m, s, t, mincost;
int head[N], nxt[N], ver[N], edge[N], cost[N], d[5000];
bool vis[5000];

void add(int u, int v, int w, int c) {
    ver[++tot] = v;
    edge[tot] = w;
    cost[tot] = c;
    nxt[tot] = head[u];
    head[u] = tot;
}

bool spfa() {
    memset(d, 0x3f, sizeof d);
    memset(vis, 0, sizeof vis);
    queue<int> q; q.push(s);
    vis[s] = 1; d[s] = 0;
    while (q.size()) {
        int u = q.front(); q.pop(); vis[u] = 0;
        for (int i = head[u]; i; i = nxt[i]) {
            if (edge[i] and d[ver[i]] > d[u] + cost[i]) {
                d[ver[i]] = d[u] + cost[i];
                if (!vis[ver[i]])
                    q.push(ver[i]), vis[ver[i]] = 1;
            }
        }
    }
    return d[t] != d[0];
}

int dinic(int u, int flow) {
    if (u == t) return flow;
    int k, rest = flow;
    vis[u] = 1;
    for (int i = head[u]; i and rest; i = nxt[i]) {
        if ((!vis[ver[i]] or ver[i] == t) and edge[i] and d[ver[i]] == d[u] + cost[i]) {
            k = dinic(ver[i], min(edge[i], rest));
            if (!k) d[ver[i]] = 0;
            edge[i] -= k;
            edge[i ^ 1] += k;
            rest -= k;
            mincost += cost[i] * k;
        }
    }
    return flow - rest;
}

int main() {
    scanf("%d%d", &n, &m);
    s = 1; t = n + n;
    add(1, n + 1, +oo, 0), add(n + 1, 1, 0, 0);
    add(n, n + n, +oo, 0), add(n + n, n, 0, 0);
    for (int i = 2; i < n; ++ i)
        add(i, i + n, 1, 0), add(i + n, i, 0, 0);
    for (int i = 1; i <= m; ++ i) {
        int x, y, z; scanf("%d%d%d", &x, &y, &z);
        add(x + n, y, 1, z); add(y, x + n, 0, -z);
    }
    int maxflow = 0;
    while (spfa()) {
        vis[t] = 1;
        while (vis[t]) {
            memset(vis, 0, sizeof vis);
            maxflow += dinic(s, +oo);
        }
    }
    cout << maxflow << " " << mincost << endl;
}

[SCOI2007]修车

对于一个修车工先后修 \(1\) ～ \(n\) 的车，对答案的贡献就是：

\(W_n × 1 + W_{n-1} × 2 + \dots + W_1 × n\)

所以我们考虑将每个工人拆成 \(n\) 个阶段的点，然后建二分图

对于每台车（左部节点）

它可以被这些工人中任意一位，任意一时刻被修理

则从这台车 \(x\) 连向右边的工人 \(i\) 的第 \(j\) 个阶段

对答案的贡献就是 \(W(i,x) * j\)

跑最小费用最大流即可

在直接建图不好算答案的时候不妨算贡献。

对于一个点有不同阶段，拆点的思想一定要有。

#include<bits/stdc++.h>
#define id(i,j) (i-1) * n + j
#pragma GCC optimize(2)
using namespace std;
const int N = 400000, oo = 0x3f3f3f3f;
int tot = 1, n, m, s, t;
int ver[N], nxt[N], edge[N], cost[N], head[N];
int d[N], vis[N], mincost, maxflow;
void add(int u, int v, int w, int c) {
    ver[++tot] = v;
    edge[tot] = w;
    cost[tot] = c;
    nxt[tot] = head[u];
    head[u] = tot;
    ver[++tot] = u;
    edge[tot] = 0;
    cost[tot] = -c;
    nxt[tot] = head[v];
    head[v] = tot;
}
int spfa() {
    queue<int> q;
    memset(vis, 0, sizeof vis);
    memset(d, 0x3f, sizeof d);
    d[s] = 0; vis[s] = 1; q.push(s);
    while (q.size()) {
        int u = q.front(); q.pop(); vis[u] = 0;
        for (int i = head[u]; i; i = nxt[i]) {
            int v = ver[i];
            if (edge[i] and d[v] > d[u] + cost[i]) {
                d[v] = d[u] + cost[i];
                if (!vis[v]) {
                    vis[v] = 1;
                    q.push(v);
                }
            }
        }
    }
    return d[t] != d[N - 3];
}
int dinic(int u, int flow) {
    if (u == t) return flow;
    int k, rest = flow;
    vis[u] = 1;
    for (int i = head[u]; i and rest; i = nxt[i]) {
        int v = ver[i];
        if ((!vis[v] or v == t) and edge[i] and d[v] == d[u] + cost[i]) {
            k = dinic(v, min(edge[i], rest));
            if (!k) d[v] = 0;
            rest -= k;
            edge[i] -= k;
            edge[i ^ 1] += k;
            mincost += k * cost[i];
        }
    }
    return flow - rest;
}
int main() {
    scanf("%d%d", &m, &n);
    for (int i = 1; i <= n; ++ i) 
        for (int j = 1; j <= m; ++ j) {
            int x; scanf("%d", &x);
            for (int k = 1; k <= n; ++ k) {
                add(i, n + id(j, k) , 1, x * k);
            }
        }
    s = 0; t = n * m + n + 1;
    for (int i = 1; i <= n; ++ i)
        add(s, i, 1, 0);
    for (int i = 1; i <= m; ++ i)
        for (int j = 1; j <= n; ++ j)
            add(n + id(i,j), t, 1, 0);
    while (spfa()) {
        vis[t] = 1;
        while (vis[t]) {
            memset(vis, 0, sizeof vis);
            maxflow += dinic(s, +oo);
        }
    }
    printf("%.2lf\n", 1.0 * mincost / n);
}

[NOI2012]美食节

[SCOI2007]修车 的加强版

考虑到 \(spfa\) 是基于边数的，且每次做一次spfa最多只能找到一条增广路

所以想到优化边数

对于一个厨师第 \(i\) 阶段做第 \(j\) 道菜， 对于同一 \(j\) 发现费用是随 \(i\) 单增的，

所以包含这个厨师做 \(j\) 这个菜的增广路找到的顺序一定是 \((i,j)\) , \((i+1, j)\) , \((i+2, j)\dots\)

所以我们每跑完一次 \(spfa\) 再加 \(i+1\) 的那条边，优化了边数，就可以 \(AC\) 了。

网络流如果复杂度过不去，要考虑适时加边，优化边数。

详见代码

#include<iostream>
#include<cstdio>
#include<cstring>
#include<queue>
using namespace std;
#define N 120000
#define inf 0x3f3f3f3f
int tot = 1, n, m, s, t, sum, ans;
int ver[N], edge[N], cost[N], nxt[N], incf[N], d[N], vis[N], pre[N], head[N];
void add(int u, int v, int w, int c) {
    ver[++tot] = v; edge[tot] = w; cost[tot] = c; nxt[tot] = head[u]; head[u] = tot;
    ver[++tot] = u; edge[tot] = 0; cost[tot] = -c;nxt[tot] = head[v]; head[v] = tot;
}
int p[N], c[3000][3000];
int Ek() {
    memset(d, 0x3f, sizeof d); memset(vis, 0, sizeof d); pre[t] = 0;
    d[s] = 0; vis[s] = 1; queue<int> q; q.push(s);
    incf[s] = inf;
    while (q.size()) {
        int u = q.front(); q.pop(); vis[u] = 0;
        for (int i = head[u]; i; i = nxt[i]) {
            int v = ver[i];
            if (edge[i] and d[v] > d[u] + cost[i]) {
                d[v] = d[u] + cost[i]; pre[v] = i;
                incf[v] = min(incf[u], edge[i]);
                if (!vis[v]) 
                    vis[v] = 1, q.push(v);
            }
        }
    }
    if (!pre[t]) return 0;
    for (int u = t; u != s; u = ver[pre[u] ^ 1]) {
        int e = pre[u];
        edge[e] -= incf[t];
        edge[e ^ 1]+= incf[t];
        ans += incf[t] * cost[e];
    }
    return 1;
}
int dish[N], cook[N];
int main() {
    ios::sync_with_stdio(0);
    cin >> n >> m;//n种菜，m个厨师
    for (int i = 1; i <= n; ++ i) cin >> p[i], sum += p[i];//sum个阶段
    s = 0, t = N - 2;
    for (int i = 1; i <= n; ++ i)
     	add(s, i + sum * m, p[i], 0);//对于每种菜，从源点连一条容量为该种菜数量的边
    for (int i = 1; i <= n; ++ i) 
        for (int j = 1; j <= m; ++ j) {
            cin >> c[i][j];
            add(i + sum * m, (j - 1) * sum + 1, 1, c[i][j]);
            //第一阶段连边
        }
    for (int i = 1; i <= m; ++ i) 
        add((i - 1) * sum + 1, t, 1, 0);//向汇点连边
    for (int i = 1; i <= m; ++ i) 
        for (int j = 1; j <= sum; ++ j) {
            int tmp = (i - 1) * sum + j;
            dish[tmp] = j; cook[tmp] = i;//一个映射: tmp这个点是第j个菜(阶段), 由i这个厨师来做
        }
    while (Ek()) {
        int tmp = ver[pre[t] ^ 1];
        add(tmp + 1, t, 1, 0);//下个阶段连边
        for (int i = 1; i <= n; ++ i) 
            add(i + m * sum, tmp + 1, 1, c[i][cook[tmp]] * (dish[tmp] + 1));
    }
    cout << ans << endl;
}

[ZJOI2010]网络扩容

第一问裸的最大流，

第二问考虑将每条路都扩容，

就在原图的基础上每个点再加一条容量为 \(inf\)，有费用的边；

表示可以无线扩容，每扩容一点流量，就花费一点价值；

为了限制扩容的流量为 \(k\)，从 \(n \xRightarrow{(k,0)} n+1, t \leftarrow n+1\);

从 \(s\) 到 \(t\) 跑最小费用最大流即可

网络流重边：一条边没有费用，一条边有费用，可以表示当一个东西超出某一限制后才会产生费用。

#include<bits/stdc++.h>
using namespace std;
const int N = 60000, inf = 0x3f3f3f3f;
int n, m, incf[N], s, t, a[N], b[N], c[N]; 
int tot = 1, head[N];
int ver[N], edge[N], cost[N], nxt[N];
bool vis[N];
int pre[N];
void add(int u, int v, int w, int c) {
    ver[++tot] = v;
    edge[tot] = w;
    cost[tot] = c;
    nxt[tot] = head[u];
    head[u] = tot;
    ver[++tot] = u;
    edge[tot] = 0;
    cost[tot] = -c;
    nxt[tot] = head[v];
    head[v] = tot;
}
int d[N];
int bfs() {
    memset(d, 0, sizeof d);
    queue<int> q; q.push(s);
    d[s] = 1;
    while (q.size()) {
        int u = q.front(); q.pop();
        for (int i = head[u]; i; i = nxt[i]) {
            int v = ver[i];
            if (edge[i] and !d[v]) {
                d[v] = d[u] + 1;
                if (v == t) return 1;
                q.push(v);
            }
        }
    }
    return 0;
}
int mincost = 0;
int dinic(int u, int flow) {
    if (u == t) return flow;
    int k, rest = flow;
    for (int i = head[u]; i and rest; i = nxt[i]) {
        int v = ver[i];
        if (edge[i] and d[v] == d[u] + 1) {
            k = dinic(v, min(edge[i], rest));
            if (!k) d[v] = 0;
            rest -= k;
            edge[i] -= k;
            edge[i ^ 1] += k;
        }
    }
    return flow - rest;
}
int Dinic(int u, int flow) {
    if (u == t) return flow;
    int k, rest = flow;
    vis[u] = 1;
    for (int i = head[u]; i and rest; i = nxt[i]) {
        int v = ver[i];
        if ((!vis[v] or v == t) and edge[i] and d[v] == d[u] + cost[i]) {
            k = Dinic(v, min(edge[i], rest));
            if (!k) d[v] = 0;
            rest -= k;
            edge[i] -= k;
            edge[i ^ 1] += k;
            mincost += k * cost[i];
        }
    }
    return flow - rest;
}
int spfa() {
    memset(d, 0x3f, sizeof d);
    memset(vis, 0, sizeof vis);
    queue<int> q; q.push(s);
    int M = d[0];
    d[s] = 0; vis[s] = 1;
    while (q.size()) {
        int u = q.front(); q.pop(); vis[u] = 0;
        for (int i = head[u]; i; i = nxt[i]) {
            int v = ver[i];
            if (edge[i] and d[v] > d[u] + cost[i]) {
                d[v] = d[u] + cost[i];
                if (!vis[v])
                    q.push(v), vis[v] = 1;
            }
        }
    }
    return d[t] < M;
}
int zkw() {
    while (spfa()) {
        vis[t] = 1;
        while (vis[t]) {
            memset(vis, 0, sizeof vis);
            Dinic(s, inf);
        }
    }
    return mincost;
}
int z[N];
int main() {
    ios::sync_with_stdio(0);
    int k;
    cin >> n >> m >> k;
    for (int i = 1; i <= m; ++ i) {
        cin >> a[i] >> b[i] >> c[i] >> z[i];
        add(a[i], b[i], c[i], 0);
    }
    s = 1, t = n;
    int ans1 = 0, flow;
    while (bfs()) 
        while (flow = dinic(s, inf))
            ans1 += flow;
    cout << ans1 << " ";
    t ++; tot = 1;
    memset(head, 0, sizeof head);
    for (int i = 1; i <= m; ++ i) {
        add(a[i], b[i], c[i], 0);
        add(a[i], b[i], inf, z[i]);
    }
    add(t - 1, t, k + ans1, 0);
    cout << zkw() << endl;
}