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Luogu P4553 80人环游世界

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题目描述

https://www.luogu.com.cn/problem/P4553

Solution

注意到限制每个点被经过恰好 $a_i$ 次，所以我们考虑上下界

建图：

$s$ 连 $s’$，下界为 $0$，上界为 $m$，费用为 $0$

$s’$ 连 $u$，下界为 $0$，上界为 $m$，费用为 $0$

$u$ 连 $t$，下界为 $0$，上界为 $m$，费用为 $0$

$u$ 连 $u’$，下界为 $a_u$，上界为 $a_u$，费用为 $0$

$u’$ 连 $v$，下界为 $0$，上界为 $m$，费用为 $u$ 和 $v$ 的距离

跑有源汇上下界最小费用最大流即可

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#include <iostream>
#include <queue>
#define maxn 210
#define INF 1000000000
using namespace std;

int n, m, a[maxn];

struct Edge {
    int to, next, w, fi;
} e[1000000]; int c1, head[maxn], in[maxn], out[maxn];
inline void add_edge(int u, int v, int w, int fi) {
    e[c1].to = v; e[c1].w = w; e[c1].fi = fi; 
    e[c1].next = head[u]; head[u] = c1++;
}

inline void Add_edge(int u, int v, int l, int r, int fi) {
    add_edge(u, v, r - l, fi); add_edge(v, u, 0, -fi);
    in[v] += l; out[u] += l; 
}

int la[maxn], pre[maxn], fl[maxn], dis[maxn], s, _s, t, ss, tt; bool vis[maxn];
bool spfa() {
    fill(dis, dis + maxn, INF); dis[s] = 0;
    queue<int> Q; Q.push(s); fl[s] = INF; pre[t] = -1;
    while (!Q.empty()) {
        int u = Q.front(); Q.pop(); vis[u] = 0; 
        for (int i = head[u]; ~i; i = e[i].next) {
            int v = e[i].to, w = e[i].w, fi = e[i].fi; 
            if (w > 0 && dis[v] > dis[u] + fi) {
                dis[v] = dis[u] + fi; pre[v] = u;
                fl[v] = min(fl[u], w); la[v] = i;
                if (vis[v]) continue; vis[v] = 1;
                Q.push(v);
            }
        }
    }
    return ~pre[t];
}

int mcmf() {
    int mf = 0, mc = 0;
    while (spfa()) {
        mf += fl[t]; 
        mc += fl[t] * dis[t];
        int now = t;
        while (now != s) {
            e[la[now]].w -= fl[t];
            e[la[now] ^ 1].w += fl[t];
            now = pre[now];
        }
    }
    return mc;
} 

int main() { fill(head, head + maxn, -1); 
    ios::sync_with_stdio(false);
    cin.tie(nullptr); cout.tie(nullptr);

    cin >> n >> m;
    s = 0; _s = 2 * n + 1; t = _s + 1; ss = t + 1; tt = ss + 1;
    Add_edge(s, _s, 0, m, 0); 
    for (int i = 1; i <= n; ++i) cin >> a[i];
    for (int i = 1; i <= n; ++i) {
        Add_edge(_s, i, 0, m, 0);
        Add_edge(i, i + n, a[i], a[i], 0);
        Add_edge(i + n, t, 0, m, 0); 
        for (int j = i + 1; j <= n; ++j) {
            int x; cin >> x;
            if (~x) Add_edge(i + n, j, 0, m, x); 
        }
    }
    Add_edge(t, s, 0, INF, 0); 
    for (int i = s; i <= t; ++i) {
        if (in[i] > out[i]) Add_edge(ss, i, 0, in[i] - out[i], 0);
        else Add_edge(i, tt, 0, out[i] - in[i], 0); 
    }
    swap(s, ss); swap(t, tt); int ans = mcmf(); swap(s, ss); swap(t, tt); 
    for (int i = head[s]; ~i; i = e[i].next)
        if (e[i].to == t) e[i].w = e[i ^ 1].w = 0;
    for (int i = head[ss]; ~i; i = e[i].next) e[i].w = e[i ^ 1].w = 0;
    for (int i = head[tt]; ~i; i = e[i].next) e[i].w = e[i ^ 1].w = 0;
    cout << ans + mcmf() << "\n"; 
    return 0;
}