题目大意：求出最大的密度子图
解题思路：详见胡伯涛:算法合集之《最小割模型在信息学竞赛中的应用》
第一种解法，最大权闭合的解法
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <vector>
#include <queue>
using namespace std;
#define N 1110
const double INF = 0x3fffffff;
const int D_INF = 0x3f3f3f3f;
const double esp = 1e-8;
struct Edge {
int from, to;
double cap, flow;
Edge() {}
Edge(int from, int to, double cap, double flow): from(from), to(to), cap(cap), flow(flow) {}
};
struct ISAP {
int p[N], num[N], cur[N], d[N];
int t, s, n, m;
bool vis[N];
vector<int> G[N];
vector<Edge> edges;
void init(int n) {
this->n = n;
for (int i = 0; i <= n; i++) {
G[i].clear();
d[i] = D_INF;
}
edges.clear();
}
void AddEdge(int from, int to, double cap) {
edges.push_back(Edge(from, to, cap, 0));
edges.push_back(Edge(to, from, 0, 0));
m = edges.size();
G[from].push_back(m - 2);
G[to].push_back(m - 1);
}
bool BFS() {
memset(vis, 0, sizeof(vis));
queue<int> Q;
d[t] = 0;
vis[t] = 1;
Q.push(t);
while (!Q.empty()) {
int u = Q.front();
Q.pop();
for (int i = 0; i < G[u].size(); i++) {
Edge &e = edges[G[u][i] ^ 1];
if (!vis[e.from] && e.cap - e.flow > esp) {
vis[e.from] = true;
d[e.from] = d[u] + 1;
Q.push(e.from);
}
}
}
return vis[s];
}
double Augment() {
int u = t;
double flow = INF;
while (u != s) {
Edge &e = edges[p[u]];
flow = min(flow, e.cap - e.flow);
u = edges[p[u]].from;
}
u = t;
while (u != s) {
edges[p[u]].flow += flow;
edges[p[u] ^ 1].flow -= flow;
u = edges[p[u]].from;
}
return flow;
}
double Maxflow(int s, int t) {
this->s = s; this->t = t;
double flow = 0;
BFS();
if (d[s] > n)
return 0;
memset(num, 0, sizeof(num));
memset(cur, 0, sizeof(cur));
for (int i = 0; i < n; i++)
if (d[i] < D_INF)
num[d[i]]++;
int u = s;
while (d[s] <= n) {
if (u == t) {
flow += Augment();
u = s;
}
bool ok = false;
for (int i = cur[u]; i < G[u].size(); i++) {
Edge &e = edges[G[u][i]];
if (e.cap - e.flow > esp && d[u] == d[e.to] + 1) {
ok = true;
p[e.to] = G[u][i];
cur[u] = i;
u = e.to;
break;
}
}
if (!ok) {
int Min = n;
for (int i = 0; i < G[u].size(); i++) {
Edge &e = edges[G[u][i]];
if (e.cap - e.flow > esp)
Min = min(Min, d[e.to]);
}
if (--num[d[u]] == 0)
break;
num[d[u] = Min + 1]++;
cur[u] = 0;
if (u != s)
u = edges[p[u]].from;
}
}
return flow;
}
};
ISAP isap;
#define maxn 1010
int n, m, source, sink, Sum;
int x[maxn], y[maxn];
bool vis[maxn];
void init() {
for (int i = 1; i <= m; i++)
scanf("%d%d", &x[i], &y[i]);
}
void build(double mid) {
isap.init(sink);
for (int i = 1; i <= m; i++) {
isap.AddEdge(source, n + i, 1.0);
isap.AddEdge(n + i, x[i], INF);
isap.AddEdge(n + i, y[i], INF);
}
for (int i = 1; i <= n; i++) isap.AddEdge(i, sink, mid);
}
void dfs(int u) {
vis[u] = true;
if (u >= 1 && u <= n) Sum++;
for (int i = 0; i < isap.G[u].size(); i++) {
Edge &e = isap.edges[isap.G[u][i]];
if (!vis[e.to] && e.cap - e.flow > esp) dfs(e.to);
}
}
void solve() {
double l = 0, r = m, mid, ans;
source = 0, sink = n + m + 1;
while (r - l >= 1.0 / n / n) {
mid = (l + r) / 2;
build(mid);
ans = isap.Maxflow(source, sink);
if (1.0 * m - ans < esp) r = mid;
else l = mid;
}
build(l);
isap.Maxflow(source, sink);
memset(vis, 0, sizeof(vis));
Sum = 0;
dfs(source);
printf("%d\n", Sum);
for (int i = 1; i <= n; i++)
if (vis[i])
printf("%d\n", i);
}
int main() {
while (scanf("%d%d", &n, &m) != EOF) {
if (!m) {
printf("1\n1\n");
continue;
}
init();
solve();
}
return 0;
}第二种解法：
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <vector>
#include <queue>
using namespace std;
#define N 1110
#define INF 0x3f3f3f3f
const double Max = 0x3fffffff;
const double esp = 1e-8;
struct Edge {
int from, to;
double cap, flow;
Edge() {}
Edge(int from, int to, double cap, double flow): from(from), to(to), cap(cap), flow(flow) {}
};
struct ISAP {
int p[N], num[N], cur[N], d[N];
int t, s, n, m;
bool vis[N];
vector<int> G[N];
vector<Edge> edges;
void init(int n) {
this->n = n;
for (int i = 0; i <= n; i++) {
G[i].clear();
d[i] = INF;
}
edges.clear();
}
void AddEdge(int from, int to, double cap) {
edges.push_back(Edge(from, to, cap, 0));
edges.push_back(Edge(to, from, 0, 0));
m = edges.size();
G[from].push_back(m - 2);
G[to].push_back(m - 1);
}
bool BFS() {
memset(vis, 0, sizeof(vis));
queue<int> Q;
d[t] = 0;
vis[t] = 1;
Q.push(t);
while (!Q.empty()) {
int u = Q.front();
Q.pop();
for (int i = 0; i < G[u].size(); i++) {
Edge &e = edges[G[u][i] ^ 1];
if (!vis[e.from] && e.cap - e.flow > esp) {
vis[e.from] = true;
d[e.from] = d[u] + 1;
Q.push(e.from);
}
}
}
return vis[s];
}
double Augment() {
int u = t;
double flow = Max;
while (u != s) {
Edge &e = edges[p[u]];
flow = min(flow, e.cap - e.flow);
u = edges[p[u]].from;
}
u = t;
while (u != s) {
edges[p[u]].flow += flow;
edges[p[u] ^ 1].flow -= flow;
u = edges[p[u]].from;
}
return flow;
}
double Maxflow(int s, int t) {
this->s = s; this->t = t;
double flow = 0;
BFS();
if (d[s] > n)
return 0;
memset(num, 0, sizeof(num));
memset(cur, 0, sizeof(cur));
for (int i = 0; i < n; i++)
if (d[i] < INF)
num[d[i]]++;
int u = s;
while (d[s] <= n) {
if (u == t) {
flow += Augment();
u = s;
}
bool ok = false;
for (int i = cur[u]; i < G[u].size(); i++) {
Edge &e = edges[G[u][i]];
if (e.cap - e.flow > esp && d[u] == d[e.to] + 1) {
ok = true;
p[e.to] = G[u][i];
cur[u] = i;
u = e.to;
break;
}
}
if (!ok) {
int Min = n;
for (int i = 0; i < G[u].size(); i++) {
Edge &e = edges[G[u][i]];
if (e.cap - e.flow > esp)
Min = min(Min, d[e.to]);
}
if (--num[d[u]] == 0)
break;
num[d[u] = Min + 1]++;
cur[u] = 0;
if (u != s)
u = edges[p[u]].from;
}
}
return flow;
}
};
ISAP isap;
#define maxn 1010
#define maxm 110
int n, m, source, sink, Sum;
int x[maxn], y[maxn], in[maxm];
bool vis[maxm];
void init() {
memset(in, 0, sizeof(in));
for (int i = 1; i <= m; i++) {
scanf("%d%d", &x[i], &y[i]);
in[x[i]]++; in[y[i]]++;
}
}
void build(double mid) {
isap.init(sink);
for (int i = 1; i <= n; i++) {
isap.AddEdge(source, i, m * 1.0);
isap.AddEdge(i, sink, m + 2 * mid - in[i]);
}
for (int i = 1; i <= m; i++) {
isap.AddEdge(x[i], y[i], 1.0);
isap.AddEdge(y[i], x[i], 1.0);
}
}
void dfs(int u) {
vis[u] = true;
if (u != source || u != sink) Sum++;
for (int i = 0; i < isap.G[u].size(); i++) {
Edge &e = isap.edges[isap.G[u][i]];
if (!vis[e.to] && e.cap - e.flow > esp)
dfs(e.to);
}
}
void solve() {
source = 0, sink = n + 1;
double l = 0, r = m, mid, ans;
while (r - l >= 1.0/n/n) {
mid = (r + l) / 2;
build(mid);
ans = isap.Maxflow(source, sink);
if ((1.0 * n * m - ans) / 2 > esp) l = mid;
else r = mid;
}
build(l);
isap.Maxflow(source, sink);
memset(vis, 0, sizeof(vis));
Sum = 0;
dfs(source);
printf("%d\n", Sum - 1);
for (int i = 1; i <= n; i++)
if (vis[i])
printf("%d\n", i);
}
int main() {
while (scanf("%d%d", &n, &m) != EOF) {
if (!m) {
printf("1\n1\n");
continue;
}
init();
solve();
}
return 0;
}
