传送门 AtCoder ABC239G Builder Takahashi
将原图中每个节点拆为入点 v v v 与出点 v ′ v' v′,对于原图任一边 ( u , v ) (u,v) (u,v) 则 u ′ → v , v → u u'\rightarrow v, v\rightarrow u u′→v,v→u 连一条容量为 ∞ \infty ∞ 的边,对于原图每一个点, v → v ′ v\rightarrow v' v→v′ 连一条容量为 c v c_v cv 的边。此时答案为新图的最小割。
对于最小割集的求解,求解最大流后,从源点出发在残余网络中 DFS,对所有可达的点打上标记,最终满足 v v v 被标记而 v ′ v' v′ 未被标记的节点则属于最小割集。
#include
using namespace std;
using ll = long long;
constexpr ll INF = 1e18;
struct MaxFlow {
struct Edge {
int to;
ll cap;
int rev;
};
vector<int> iter, level;
vector<vector<Edge>> g;
MaxFlow(int n) : iter(n), level(n), g(n) {}
void add_edge(int from, int to, ll cap) {
g[from].push_back({to, cap, (int)g[to].size()});
g[to].push_back({from, 0, (int)g[from].size() - 1});
}
void bfs(int s) {
fill(level.begin(), level.end(), -1);
queue<int> q;
level[s] = 0;
q.push(s);
while (!q.empty()) {
int v = q.front();
q.pop();
for (auto [to, cap, _] : g[v]) {
if (cap > 0 && level[to] == -1) {
level[to] = level[v] + 1;
q.push(to);
}
}
}
}
ll dfs(int v, int t, ll f) {
if (v == t) {
return f;
}
for (int &i = iter[v]; i < (int)g[v].size(); ++i) {
auto &e = g[v][i];
if (e.cap > 0 && level[v] < level[e.to]) {
int d = dfs(e.to, t, min(f, e.cap));
if (d > 0) {
e.cap -= d;
g[e.to][e.rev].cap += d;
return d;
}
}
}
return 0;
}
ll max_flow(int s, int t) {
ll flow = 0;
for (;;) {
fill(iter.begin(), iter.end(), 0);
bfs(s);
if (level[t] == -1) {
return flow;
}
ll f;
while ((f = dfs(s, t, INF)) > 0) {
flow += f;
}
}
}
};
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
int n, m;
cin >> n >> m;
MaxFlow flow(n * 2);
for (int i = 0; i < m; ++i) {
int u, v;
cin >> u >> v;
u -= 1, v -= 1;
flow.add_edge(v + n, u, INF);
flow.add_edge(u + n, v, INF);
}
for (int v = 0; v < n; ++v) {
int c;
cin >> c;
flow.add_edge(v, v + n, c);
}
cout << flow.max_flow(0 + n, n - 1) << '\n';
vector<int> used(2 * n);
auto dfs = [&](auto dfs, int v) -> void {
used[v] = 1;
for (auto [to, cap, _] : flow.g[v]) {
if (cap > 0 && !used[to]) {
dfs(dfs, to);
}
}
};
dfs(dfs, 0 + n);
vector<int> vs;
for (int v = 0; v < n; ++v) {
if (used[v] && !used[v + n]) {
vs.push_back(v);
}
}
cout << (int)vs.size() << '\n';
for (int v : vs) {
cout << v + 1 << ' ';
}
cout << '\n';
return 0;
}