【SDOI2017】新生舞会

题面

题解

一眼\(0/1\)分数规划

二分答案\(mid\)，我们要\(\sum\limits_i a^{'}_i - mid\sum\limits_i b_i^{'}\)最大

那么我们将\(a_{i,j}-mid\times b_{i,j}\)作为\((i,j)\)的边权

跑一遍二分图最大权匹配即可。

代码

// luogu-judger-enable-o2
#include<cstdio>
#include<cstring>
#include<cctype>
#include<cmath>
#include<algorithm>
#include<queue>
#define RG register
#define file(x) freopen(#x".in", "r", stdin);freopen(#x".out", "w", stdout);
#define clear(x, y) memset(x, y, sizeof(x))

inline int read()
{
    int data = 0, w = 1; char ch = getchar();
    while(ch != '-' && (!isdigit(ch))) ch = getchar();
    if(ch == '-') w = -1, ch = getchar();
    while(isdigit(ch)) data = data * 10 + (ch ^ 48), ch = getchar();
    return data * w;
}

const int N(110), maxn(5e5 + 10);
const double eps(1e-8);

struct edge { int next, to, cap; double dis; } e[maxn];
int head[maxn], e_num = -1, S, T, a[N][N], b[N][N];
int pre[maxn], pre_e[maxn], vis[maxn], n;
double dis[maxn], cost;

inline void add_edge(int from, int to, int cap, double dis)
{
    e[++e_num] = (edge) {head[from], to, cap, dis}; head[from] = e_num;
    e[++e_num] = (edge) {head[to], from,  0, -dis}; head[to]   = e_num;
}

std::queue<int> q;
void MinCostMaxFlow()
{
    cost = 0;
    while(1)
    {
        std::fill(dis + S, dis + T + 1, -1e20); clear(vis, 0);
        vis[S] = 1, dis[S] = 0, q.push(S);
        while(!q.empty())
        {
            int x = q.front(); q.pop();
            for(RG int i = head[x]; ~i; i = e[i].next)
            {
                int to = e[i].to; double ds = e[i].dis + dis[x];
                if(e[i].cap > 0 && dis[to] < ds)
                {
                    dis[to] = ds, pre[to] = x, pre_e[to] = i;
                    if(!vis[to]) vis[to] = 1, q.push(to);
                }
            }
            vis[x] = 0;
        }

        if(dis[T] == -1e20) return;
        int cap = 1e9;
        for(RG int i = T; i ^ S; i = pre[i])
            cap = std::min(cap, e[pre_e[i]].cap);
        cost += 1. * cap * dis[T];
        for(RG int i = T; i ^ S; i = pre[i])
            e[pre_e[i]].cap -= cap, e[pre_e[i] ^ 1].cap += cap;
    }
}

bool check(double mid)
{
    e_num = -1, S = 1, T = (n << 1) + 2;
    for(RG int i = S; i <= T; i++) head[i] = -1;
    for(RG int i = 1; i <= n; i++) add_edge(S, i + 1, 1, 0);
    for(RG int i = n + 2; i < T; i++) add_edge(i, T, 1, 0);
    for(RG int i = 1; i <= n; i++) for(RG int j = 1; j <= n; j++)
        add_edge(i + 1, j + n + 1, 1, 1. * a[i][j] - 1. * b[i][j] * mid);
    MinCostMaxFlow(); return fabs(cost) <= eps || cost > eps;
}

int main()
{
    n = read();
    for(RG int i = 1; i <= n; i++)
        for(RG int j = 1; j <= n; j++)
            a[i][j] = read();
    for(RG int i = 1; i <= n; i++)
        for(RG int j = 1; j <= n; j++)
            b[i][j] = read();
    double l = -1e7, r = 1e7;
    while(fabs(r - l) > eps)
    {
        double mid = (l + r) / 2;
        if(check(mid)) l = mid;
        else r = mid;
    }
    printf("%.6lf\n", r);
    return 0;
}