maximum_weight_closure.hpp¶
Maximum-weight vertex set closed under implications (from selected implies to selected), using one s-t min cut.
求有向依赖图的最大权闭合子集；把“选一个就必须选后继”的收益选择转成最小割。
Implementation¶
#ifndef NOYA_MAXIMUM_WEIGHT_CLOSURE_HPP
#define NOYA_MAXIMUM_WEIGHT_CLOSURE_HPP 1
/// @complexity Time: One maximum-flow computation.
/// Space: O(V + E) plus the flow workspace.
#include "atcoder/maxflow.hpp"
#include <cassert>
#include <cstdint>
#include <limits>
#include <utility>
#include <vector>
namespace noya {
struct maximum_weight_closure_result {
std::int64_t weight = 0;
std::vector<int> selected;
};
/// @brief Maximum-weight vertex set closed under implications (from selected
/// implies to selected), using one s-t min cut.
inline maximum_weight_closure_result
maximum_weight_closure(const std::vector<std::int64_t> &weight,
const std::vector<std::pair<int, int>> &implications) {
int n = int(weight.size());
int source = n;
int sink = n + 1;
atcoder::mf_graph<std::int64_t> flow(n + 2);
__int128 positive_sum_wide = 0;
for (std::int64_t value : weight) {
if (value > 0) {
positive_sum_wide += value;
}
}
assert(positive_sum_wide < std::numeric_limits<std::int64_t>::max());
std::int64_t positive_sum = std::int64_t(positive_sum_wide);
std::int64_t infinity = positive_sum + 1;
for (int vertex = 0; vertex < n; vertex++) {
if (weight[vertex] > 0) {
flow.add_edge(source, vertex, weight[vertex]);
} else if (weight[vertex] < 0) {
assert(weight[vertex] != std::numeric_limits<std::int64_t>::min());
flow.add_edge(vertex, sink, -weight[vertex]);
}
}
for (auto [from, to] : implications) {
assert(0 <= from && from < n);
assert(0 <= to && to < n);
if (from != to) {
flow.add_edge(from, to, infinity);
}
}
std::int64_t cut = flow.flow(source, sink);
auto reachable = flow.min_cut(source);
maximum_weight_closure_result result;
result.weight = positive_sum - cut;
for (int vertex = 0; vertex < n; vertex++) {
if (reachable[vertex]) {
result.selected.push_back(vertex);
}
}
return result;
}
} // namespace noya
#endif // NOYA_MAXIMUM_WEIGHT_CLOSURE_HPP
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <limits>
#include <queue>
#include <utility>
#include <vector>
/// @complexity Time: One maximum-flow computation.
/// Space: O(V + E) plus the flow workspace.
namespace atcoder {
namespace internal {
template <class T> struct simple_queue {
std::vector<T> payload;
int pos = 0;
void reserve(int n) { payload.reserve(n); }
int size() const { return int(payload.size()) - pos; }
bool empty() const { return pos == int(payload.size()); }
void push(const T& t) { payload.push_back(t); }
T& front() { return payload[pos]; }
void clear() {
payload.clear();
pos = 0;
}
void pop() { pos++; }
};
} // namespace internal
} // namespace atcoder
namespace atcoder {
template <class Cap> struct mf_graph {
public:
mf_graph() : _n(0) {}
explicit mf_graph(int n) : _n(n), g(n) {}
int add_edge(int from, int to, Cap cap) {
assert(0 <= from && from < _n);
assert(0 <= to && to < _n);
assert(0 <= cap);
int m = int(pos.size());
pos.push_back({from, int(g[from].size())});
int from_id = int(g[from].size());
int to_id = int(g[to].size());
if (from == to) to_id++;
g[from].push_back(_edge{to, to_id, cap});
g[to].push_back(_edge{from, from_id, 0});
return m;
}
struct edge {
int from, to;
Cap cap, flow;
};
edge get_edge(int i) {
int m = int(pos.size());
assert(0 <= i && i < m);
auto _e = g[pos[i].first][pos[i].second];
auto _re = g[_e.to][_e.rev];
return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
}
std::vector<edge> edges() {
int m = int(pos.size());
std::vector<edge> result;
for (int i = 0; i < m; i++) {
result.push_back(get_edge(i));
}
return result;
}
void change_edge(int i, Cap new_cap, Cap new_flow) {
int m = int(pos.size());
assert(0 <= i && i < m);
assert(0 <= new_flow && new_flow <= new_cap);
auto& _e = g[pos[i].first][pos[i].second];
auto& _re = g[_e.to][_e.rev];
_e.cap = new_cap - new_flow;
_re.cap = new_flow;
}
Cap flow(int s, int t) {
return flow(s, t, std::numeric_limits<Cap>::max());
}
Cap flow(int s, int t, Cap flow_limit) {
assert(0 <= s && s < _n);
assert(0 <= t && t < _n);
assert(s != t);
std::vector<int> level(_n), iter(_n);
internal::simple_queue<int> que;
auto bfs = [&]() {
std::fill(level.begin(), level.end(), -1);
level[s] = 0;
que.clear();
que.push(s);
while (!que.empty()) {
int v = que.front();
que.pop();
for (auto e : g[v]) {
if (e.cap == 0 || level[e.to] >= 0) continue;
level[e.to] = level[v] + 1;
if (e.to == t) return;
que.push(e.to);
}
}
};
auto dfs = [&](auto self, int v, Cap up) {
if (v == s) return up;
Cap res = 0;
int level_v = level[v];
for (int& i = iter[v]; i < int(g[v].size()); i++) {
_edge& e = g[v][i];
if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
Cap d =
self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
if (d <= 0) continue;
g[v][i].cap += d;
g[e.to][e.rev].cap -= d;
res += d;
if (res == up) return res;
}
level[v] = _n;
return res;
};
Cap flow = 0;
while (flow < flow_limit) {
bfs();
if (level[t] == -1) break;
std::fill(iter.begin(), iter.end(), 0);
Cap f = dfs(dfs, t, flow_limit - flow);
if (!f) break;
flow += f;
}
return flow;
}
std::vector<bool> min_cut(int s) {
std::vector<bool> visited(_n);
internal::simple_queue<int> que;
que.push(s);
while (!que.empty()) {
int p = que.front();
que.pop();
visited[p] = true;
for (auto e : g[p]) {
if (e.cap && !visited[e.to]) {
visited[e.to] = true;
que.push(e.to);
}
}
}
return visited;
}
private:
int _n;
struct _edge {
int to, rev;
Cap cap;
};
std::vector<std::pair<int, int>> pos;
std::vector<std::vector<_edge>> g;
};
} // namespace atcoder
namespace noya {
struct maximum_weight_closure_result {
std::int64_t weight = 0;
std::vector<int> selected;
};
/// @brief Maximum-weight vertex set closed under implications (from selected
/// implies to selected), using one s-t min cut.
inline maximum_weight_closure_result
maximum_weight_closure(const std::vector<std::int64_t> &weight,
const std::vector<std::pair<int, int>> &implications) {
int n = int(weight.size());
int source = n;
int sink = n + 1;
atcoder::mf_graph<std::int64_t> flow(n + 2);
__int128 positive_sum_wide = 0;
for (std::int64_t value : weight) {
if (value > 0) {
positive_sum_wide += value;
}
}
assert(positive_sum_wide < std::numeric_limits<std::int64_t>::max());
std::int64_t positive_sum = std::int64_t(positive_sum_wide);
std::int64_t infinity = positive_sum + 1;
for (int vertex = 0; vertex < n; vertex++) {
if (weight[vertex] > 0) {
flow.add_edge(source, vertex, weight[vertex]);
} else if (weight[vertex] < 0) {
assert(weight[vertex] != std::numeric_limits<std::int64_t>::min());
flow.add_edge(vertex, sink, -weight[vertex]);
}
}
for (auto [from, to] : implications) {
assert(0 <= from && from < n);
assert(0 <= to && to < n);
if (from != to) {
flow.add_edge(from, to, infinity);
}
}
std::int64_t cut = flow.flow(source, sink);
auto reachable = flow.min_cut(source);
maximum_weight_closure_result result;
result.weight = positive_sum - cut;
for (int vertex = 0; vertex < n; vertex++) {
if (reachable[vertex]) {
result.selected.push_back(vertex);
}
}
return result;
}
} // namespace noya