[TJOI2015] 线性代数 - 最大权闭合子图

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\(D=(AB-C)A^T\\
=\sum_{i=1}^n(\sum_{j=1}^na_jb_{j,i}-c_i)a_i\\
=\sum_{i=1}^n\sum_{j=1}^na_ia_jb_{i,j}-\sum_{i=1}^na_ic_i\)

对每一对 \(i,j\)，同时选获得 \(b_{ij}+b_{ji}\)

某个 \(i\) 不选，额外损失 \(c_i\)

考虑最大权闭合子图

\(S \to (i,j)= b_{ij}+b_{ji}\)

\((i,j) \to i (j) = \infty\)

\(i \to T= c_i\)

跑最大流即可，最后用 \(\sum b_{ij}\) 减去答案

#include <bits/stdc++.h>
using namespace std;

const int inf = 1e+9;

namespace flow {

const int maxn = 300005;
const int inf = 1e+9;

int dis[maxn], ans, cnt = 1, s, t, pre[maxn * 10], nxt[maxn * 10], h[maxn], v[maxn * 10];
std::queue<int> q;
void make(int x, int y, int z) {
    pre[++cnt] = y, nxt[cnt] = h[x], h[x] = cnt, v[cnt] = z;
    pre[++cnt] = x, nxt[cnt] = h[y], h[y] = cnt;
}
bool bfs() {
    memset(dis, 0, sizeof dis);
    q.push(s), dis[s] = 1;
    while (!q.empty()) {
        int x = q.front();
        q.pop();
        for (int i = h[x]; i; i = nxt[i])
            if (!dis[pre[i]] && v[i])
                dis[pre[i]] = dis[x] + 1, q.push(pre[i]);
    }
    return dis[t];
}
int dfs(int x, int flow) {
    if (x == t || !flow)
        return flow;
    int f = flow;
    for (int i = h[x]; i; i = nxt[i])
        if (v[i] && dis[pre[i]] > dis[x]) {
            int y = dfs(pre[i], min(v[i], f));
            f -= y, v[i] -= y, v[i ^ 1] += y;
            if (!f)
                return flow;
        }
    if (f == flow)
        dis[x] = -1;
    return flow - f;
}
int solve(int _s,int _t) {
    s=_s;
    t=_t;
    ans = 0;
    for (; bfs(); ans += dfs(s, inf));
    return ans;
}
}

int n,b[505][505],c[505];

int id_node(int p) {
    return 2+p;
}

int id_pair(int p,int q) {
    return 2+p*n+q;
}

int main() {
    scanf("%d",&n);
    int sum = 0;
    for(int i=1;i<=n;i++) {
        for(int j=1;j<=n;j++) {
            scanf("%d",&b[i][j]);
            sum += b[i][j];
            flow::make(1,id_pair(i,j),b[i][j]);
            flow::make(id_pair(i,j),id_node(i),inf);
            flow::make(id_pair(i,j),id_node(j),inf);
        }
    }
    for(int i=1;i<=n;i++) {
        scanf("%d",&c[i]);
        flow::make(id_node(i),2,c[i]);
    }
    cout<<sum - flow::solve(1,2);
}