1. 最大流
const int N = 200; const int M = N*N; int head[N], deep[N], cur[N]; int w[M], to[M], nx[M]; int tot; void add(int u, int v, int val){ w[tot] = val; to[tot] = v; nx[tot] = head[u]; head[u] = tot++; w[tot] = 0; to[tot] = u; nx[tot] = head[v]; head[v] = tot++; } int bfs(int s, int t){ queue<int> q; memset(deep, 0, sizeof(deep)); q.push(s); deep[s] = 1; while(!q.empty()){ int u = q.front(); q.pop(); for(int i = head[u]; ~i; i = nx[i]){ if(w[i] > 0 && deep[to[i]] == 0){ deep[to[i]] = deep[u] + 1; q.push(to[i]); } } } return deep[t] > 0; } int Dfs(int u, int t, int flow){ if(u == t) return flow; for(int &i = cur[u]; ~i; i = nx[i]){ if(deep[u]+1 == deep[to[i]] && w[i] > 0){ int di = Dfs(to[i], t, min(w[i], flow)); if(di > 0){ w[i] -= di, w[i^1] += di; return di; } } } return 0; } int Dinic(int s, int t){ int ans = 0, tmp; while(bfs(s, t)){ for(int i = 0; i <= t; i++) cur[i] = head[i]; while(tmp = Dfs(s, t, inf)) ans += tmp; } return ans; } void init(){ memset(head, -1, sizeof(head)); tot = 0; }
2. 二分图最优匹配
1 const int N = 210; 2 int val[N][N]; 3 LL lx[N], ly[N], slack[N]; 4 int linky[N]; 5 LL pre[N]; 6 bool vis[N], visx[N],visy[N]; 7 void bfs(int k){ 8 LL px, py = 0,yy = 0, d; 9 memset(pre, 0, sizeof(LL) * (n+2)); 10 memset(slack, inf, sizeof(LL) * (n+2)); 11 linky[py]=k; 12 do{ 13 px = linky[py],d = INF, vis[py] = 1; 14 for(int i = 1; i <= n; i++) 15 if(!vis[i]){ 16 if(slack[i] > lx[px] + ly[i] - val[px][i]) 17 slack[i] = lx[px] + ly[i] -val[px][i], pre[i]=py; 18 if(slack[i]<d) d=slack[i],yy=i; 19 } 20 for(int i = 0; i <= n; i++) 21 if(vis[i]) lx[linky[i]] -= d, ly[i] += d; 22 else slack[i] -= d; 23 py = yy; 24 }while(linky[py]); 25 while(py) linky[py] = linky[pre[py]] , py=pre[py]; 26 } 27 void KM(){ 28 memset(lx, 0, sizeof(int)*(n+2)); 29 memset(ly, 0, sizeof(int)*(n+2)); 30 memset(linky, 0, sizeof(int)*(n+2)); 31 for(int i = 1; i <= n; i++) 32 memset(vis, 0, sizeof(bool)*(n+2)), bfs(i); 33 } 34 void input(){ 35 for(int i = 1; i <= n; i++){ 36 for(int j = 1; j <= n; j++) 37 scanf("%d", &val[i][j]); 38 } 39 int main(){ 40 input(); 41 KM(); 42 LL ans = 0; 43 for(int i = 1; i <= n; ++i) 44 ans += lx[i] + ly[i]; 45 printf("%lld ", ans); 46 return 0; 47 }
结果是 尽可能多匹配情况下的 最大值， 如果需要求最小值， 在input 的时候将值取反， 最后输出的答案的时候也要取反。
3.最小花费最大流
KM+spfa
const int N = 1000; const int M = N * N; int head[N], to[M], ct[M], w[M], nt[M]; int d[N], vis[N]; int pre[N], id[N]; int tot; void add(int u, int v, int flow, int cost){ to[tot] = v; ct[tot] = cost; w[tot] = flow; nt[tot] = head[u]; head[u] = tot++; to[tot] = u; ct[tot] = -cost; w[tot] = 0; nt[tot] = head[v]; head[v] = tot++; } void init(){ memset(head, -1, sizeof(head)); tot = 0; } int spfa(int s, int t){ queue<int> q; memset(d, inf, sizeof(d)); memset(vis, 0, sizeof(vis)); memset(pre, -1, sizeof(pre)); d[s] = 0; q.push(s); while(!q.empty()){ int u = q.front(); q.pop(); vis[u] = 0; for(int i = head[u]; ~i; i = nt[i]){ if(w[i] > 0 && d[to[i]] > d[u] + ct[i]){ d[to[i]] = d[u] + ct[i]; pre[to[i]] = u; id[to[i]] = i; if(!vis[to[i]]){ vis[to[i]] = 1; q.push(to[i]); } } } } return d[t] < inf; } int MinCostFlow(int s, int t){ int Mi = inf; int sum = 0; int tt = 0; while(spfa(s, t)){ Mi = inf; for(int i = t; i != s; i = pre[i]) Mi = min(Mi, w[id[i]]); for(int i = t; i != s; i = pre[i]){ w[id[i]] -= Mi; w[id[i]^1] += Mi; } tt += Mi; sum += d[t] * Mi; } return sum; }
如果需要最大的花费， 建边的时候将值取反就了。
4.上下界网络流
无源汇有上下界限制的网络流
1 #include<bits/stdc++.h> 2 using namespace std; 3 #define Fopen freopen("_in.txt","r",stdin); freopen("_out.txt","w",stdout); 4 #define LL long long 5 #define ULL unsigned LL 6 #define fi first 7 #define se second 8 #define pb push_back 9 #define lson l,m,rt<<1 10 #define rson m+1,r,rt<<1|1 11 #define max3(a,b,c) max(a,max(b,c)) 12 #define min3(a,b,c) min(a,min(b,c)) 13 #define _S(X) cout << x << ' '; 14 #define __S(x) cout << x << endl; 15 typedef pair<int,int> pll; 16 const int INF = 0x3f3f3f3f; 17 const LL mod = (int)1e9+7; 18 const int N = 20100; 19 const int _M = 500100; 20 int head[N]; 21 int M[N]; 22 int w[_M], to[_M], nx[_M], id[_M]; 23 int B[_M]; 24 int n, m, _u, _v, _w; 25 int tot, s, t, ss, tt; 26 int deep[N], cur[N]; 27 void add(int u, int v, int val){ 28 w[tot] = val; 29 to[tot] = v; 30 nx[tot] = head[u]; 31 head[u] = tot++; 32 } 33 34 void init(){ 35 memset(head, -1, sizeof(int)*(n+3)); 36 memset(M, 0, sizeof(int)*(n+3)); 37 tot = 0; 38 s = 1; 39 t = n; 40 ss = n + 1; 41 tt = n + 2; 42 } 43 44 int bfs(int s, int t){ 45 queue<int> q; 46 memset(deep, 0, sizeof(int)*(n+3)); 47 q.push(s); 48 deep[s] = 1; 49 while(!q.empty()){ 50 int u = q.front(); 51 q.pop(); 52 for(int i = head[u]; ~i; i = nx[i]){ 53 if(w[i] > 0 && deep[to[i]] == 0){ 54 deep[to[i]] = deep[u] + 1; 55 q.push(to[i]); 56 } 57 } 58 } 59 if(deep[t] > 0) return 1; 60 return 0; 61 } 62 int Dfs(int u, int t, int flow){ 63 if(u == t) return flow; 64 for(int &i = cur[u]; ~i; i = nx[i]){ 65 if(deep[u]+1 == deep[to[i]] && w[i] > 0){ 66 int di = Dfs(to[i], t, min(w[i], flow)); 67 if(di > 0){ 68 w[i] -= di, w[i^1] += di; 69 return di; 70 } 71 } 72 } 73 return 0; 74 } 75 int Dinic(int s, int t){ 76 int ans = 0, tmp; 77 while(bfs(s, t)){ 78 for(int i = 1; i <= n+2; i++) cur[i] = head[i]; 79 while(tmp = Dfs(s, t, INF)) ans += tmp; 80 } 81 return ans; 82 } 83 int main(){ 84 while(~scanf("%d%d", &n, &m)){ 85 init(); 86 int b, c; 87 for(int i = 1; i <= m; i++){ 88 scanf("%d%d%d%d", &_u, &_v, &b, &c); 89 id[i] = tot; B[i] = b; 90 add(_u,_v,c-b); add(_v,_u,0); 91 M[_u] -= b; M[_v] += b; 92 } 93 int sum = 0; 94 for(int i = 1; i <= n; i++){ 95 if(M[i] > 0) add(ss, i, M[i]), add(i, ss, 0), sum += M[i]; 96 if(M[i] < 0) add(i, tt, -M[i]), add(tt, i, 0); 97 } 98 int ans = Dinic(ss, tt); 99 if(ans == sum){ 100 puts("YES"); 101 for(int i = 1; i <= m; i++) 102 printf("%d ",w[id[i]^1] + B[i]); 103 } 104 else puts("NO"); 105 } 106 return 0; 107 }
有源汇有上下界最大流
1 #include<bits/stdc++.h> 2 using namespace std; 3 #define Fopen freopen("_in.txt","r",stdin); freopen("_out.txt","w",stdout); 4 #define LL long long 5 #define ULL unsigned LL 6 #define fi first 7 #define se second 8 #define pb push_back 9 #define lson l,m,rt<<1 10 #define rson m+1,r,rt<<1|1 11 #define max3(a,b,c) max(a,max(b,c)) 12 #define min3(a,b,c) min(a,min(b,c)) 13 #define _S(X) cout << x << ' '; 14 #define __S(x) cout << x << endl; 15 typedef pair<int,int> pll; 16 const int INF = 0x3f3f3f3f; 17 const LL mod = (int)1e9+7; 18 const int N = 1000; 19 const int _M = 50010; 20 int head[N]; 21 int M[N]; 22 int w[_M], to[_M], nx[_M]; 23 int n, m, _u, _v, _w; 24 int tot, s, t, ss, tt; 25 int deep[N], cur[N]; 26 void add(int u, int v, int val){ 27 w[tot] = val; 28 to[tot] = v; 29 nx[tot] = head[u]; 30 head[u] = tot++; 31 } 32 33 void init(){ 34 memset(head, -1, sizeof(int)*(n+3)); 35 memset(M, 0, sizeof(int)*(n+3)); 36 tot = 0; 37 ss = n + 1; 38 tt = n + 2; 39 } 40 41 int bfs(int s, int t){ 42 queue<int> q; 43 memset(deep, 0, sizeof(int)*(n+3)); 44 q.push(s); 45 deep[s] = 1; 46 while(!q.empty()){ 47 int u = q.front(); 48 q.pop(); 49 for(int i = head[u]; ~i; i = nx[i]){ 50 if(w[i] > 0 && deep[to[i]] == 0){ 51 deep[to[i]] = deep[u] + 1; 52 q.push(to[i]); 53 } 54 } 55 } 56 if(deep[t] > 0) return 1; 57 return 0; 58 } 59 int Dfs(int u, int t, int flow){ 60 if(u == t) return flow; 61 for(int &i = cur[u]; ~i; i = nx[i]){ 62 if(deep[u]+1 == deep[to[i]] && w[i] > 0){ 63 int di = Dfs(to[i], t, min(w[i], flow)); 64 if(di > 0){ 65 w[i] -= di, w[i^1] += di; 66 return di; 67 } 68 } 69 } 70 return 0; 71 } 72 int Dinic(int s, int t){ 73 int ans = 0, tmp; 74 while(bfs(s,t)){ 75 for(int i = 1; i <= n+2; i++) cur[i] = head[i]; 76 while(tmp = Dfs(s, t, INF)) ans += tmp; 77 } 78 return ans; 79 } 80 int main(){ 81 while(~scanf("%d%d", &n, &m)){ 82 init(); 83 scanf("%d%d", &s, &t); 84 int b, c; 85 for(int i = 1; i <= m; i++){ 86 scanf("%d%d%d%d", &_u, &_v, &b, &c); 87 add(_u,_v,c-b); add(_v,_u,0); 88 M[_u] -= b; M[_v] += b; 89 } 90 int sum = 0; 91 for(int i = 1; i <= n; i++){ 92 if(M[i] > 0) add(ss, i, M[i]), add(i, ss, 0), sum += M[i]; 93 if(M[i] < 0) add(i, tt, -M[i]), add(tt, i, 0); 94 } 95 add(t,s,INF); 96 add(s,t,0); 97 int ans = Dinic(ss, tt); 98 if(ans == sum){ 99 ans = w[--tot]; 100 w[tot] = 0; w[--tot] = 0; 101 ans += Dinic(s, t); 102 printf("%d ", ans); 103 } 104 else puts("please go home to sleep"); 105 } 106 return 0; 107 }
有源汇有上下界最小流
1 #include<bits/stdc++.h> 2 using namespace std; 3 #define Fopen freopen("_in.txt","r",stdin); freopen("_out.txt","w",stdout); 4 #define LL long long 5 #define ULL unsigned LL 6 #define fi first 7 #define se second 8 #define pb push_back 9 #define lson l,m,rt<<1 10 #define rson m+1,r,rt<<1|1 11 #define max3(a,b,c) max(a,max(b,c)) 12 #define min3(a,b,c) min(a,min(b,c)) 13 #define _S(X) cout << x << ' '; 14 #define __S(x) cout << x << endl; 15 typedef pair<int,int> pll; 16 const int INF = 0x3f3f3f3f; 17 const LL mod = (int)1e9+7; 18 const int N = 50100; 19 const int _M = 350010; 20 int head[N]; 21 int M[N]; 22 int w[_M], to[_M], nx[_M]; 23 int n, m, _u, _v, _w; 24 int tot, s, t, ss, tt; 25 int deep[N], cur[N]; 26 void add(int u, int v, int val){ 27 w[tot] = val; 28 to[tot] = v; 29 nx[tot] = head[u]; 30 head[u] = tot++; 31 } 32 33 void init(){ 34 memset(head, -1, sizeof(int)*(n+3)); 35 memset(M, 0, sizeof(int)*(n+3)); 36 tot = 0; 37 ss = n + 1; 38 tt = n + 2; 39 } 40 41 int bfs(int s, int t){ 42 queue<int> q; 43 memset(deep, 0, sizeof(int)*(n+3)); 44 q.push(s); 45 deep[s] = 1; 46 47 while(!q.empty()){ 48 int u = q.front(); 49 q.pop(); 50 for(int i = head[u]; ~i; i = nx[i]){ 51 if(w[i] > 0 && deep[to[i]] == 0){ 52 deep[to[i]] = deep[u] + 1; 53 q.push(to[i]); 54 } 55 } 56 } 57 if(deep[t] > 0) return 1; 58 return 0; 59 } 60 int Dfs(int u, int t, int flow){ 61 if(u == t) return flow; 62 for(int &i = cur[u]; ~i; i = nx[i]){ 63 if(deep[u]+1 == deep[to[i]] && w[i] > 0){ 64 int di = Dfs(to[i], t, min(w[i], flow)); 65 if(di > 0){ 66 w[i] -= di, w[i^1] += di; 67 return di; 68 } 69 } 70 } 71 return 0; 72 } 73 int Dinic(int s, int t){ 74 int ans = 0, tmp; 75 while(bfs(s,t)){ 76 for(int i = 1; i <= n+2; i++) cur[i] = head[i]; 77 while(tmp = Dfs(s, t, INF)) ans += tmp; 78 } 79 return ans; 80 } 81 int main(){ 82 while(~scanf("%d%d", &n, &m)){ 83 init(); 84 scanf("%d%d", &s, &t); 85 int b, c; 86 for(int i = 1; i <= m; i++){ 87 scanf("%d%d%d%d", &_u, &_v, &b, &c); 88 add(_u,_v,c-b); add(_v,_u,0); 89 M[_u] -= b; M[_v] += b; 90 } 91 int sum = 0; 92 for(int i = 1; i <= n; i++){ 93 if(M[i] > 0) add(ss, i, M[i]), add(i, ss, 0), sum += M[i]; 94 if(M[i] < 0) add(i, tt, -M[i]), add(tt, i, 0); 95 } 96 add(t,s,INF); 97 add(s,t,0); 98 int ans = Dinic(ss, tt); 99 if(ans == sum){ 100 ans = w[--tot]; 101 w[tot] = 0; w[--tot] = 0; 102 ans -= Dinic(t, s); 103 printf("%d ", ans); 104 } 105 else puts("please go home to sleep"); 106 } 107 return 0; 108 }
5.最大密度子图
（可选出合法的节点）
1 #include<cstdio> 2 #include<cstring> 3 #include<queue> 4 using namespace std; 5 #define Fopen freopen("_in.txt","r",stdin); freopen("_out.txt","w",stdout); 6 #define LL long long 7 #define ULL unsigned LL 8 #define fi first 9 #define se second 10 #define pb push_back 11 #define lson l,m,rt<<1 12 #define rson m+1,r,rt<<1|1 13 #define max3(a,b,c) max(a,max(b,c)) 14 #define min3(a,b,c) min(a,min(b,c)) 15 typedef pair<int,int> pll; 16 const LL INF = 0x3f3f3f3f3f3f3f3f; 17 const int inf = 0x3f3f3f3f; 18 const LL mod = (int)1e9+7; 19 const int N = 120; 20 const int M = 10000; 21 double eps = 1e-6; 22 int head[N], deep[N], cur[N]; 23 int u[M], v[M]; 24 int vis[N], d[N]; 25 double w[M]; int to[M], nx[M]; 26 int n, m, tot; 27 void add(int u, int v,double val){ 28 w[tot] = val; to[tot] = v; 29 nx[tot] = head[u]; head[u] = tot++; 30 w[tot] = 0; to[tot] = u; 31 nx[tot] = head[v]; head[v] = tot++; 32 } 33 int bfs(int s, int t){ 34 queue<int> q; 35 memset(deep, 0, sizeof(int)*(n+3)); 36 q.push(s); 37 deep[s] = 1; 38 while(!q.empty()){ 39 int u = q.front(); 40 q.pop(); 41 for(int i = head[u]; ~i; i = nx[i]){ 42 if(w[i] > 0 && deep[to[i]] == 0){ 43 deep[to[i]] = deep[u] + 1; 44 q.push(to[i]); 45 } 46 } 47 } 48 if(deep[t] > 0) return 1; 49 return 0; 50 } 51 double Dfs(int u, int t, double flow){ 52 if(u == t) return flow; 53 for(int &i = cur[u]; ~i; i = nx[i]){ 54 if(deep[u] + 1 == deep[to[i]] && w[i] > 0){ 55 double di = Dfs(to[i], t, min(w[i], flow)); 56 if(di > 0){ 57 w[i] -= di, w[i^1] += di; 58 return di; 59 } 60 } 61 } 62 return 0; 63 } 64 65 int Dinic(int s, int t){ 66 double ans = 0, tmp; 67 while(bfs(s, t)){ 68 for(int i = 0; i <= n+1; i++) cur[i] = head[i]; 69 while(tmp = Dfs(s, t, INF)) ans += tmp; 70 } 71 return ans; 72 } 73 74 bool check(double x){ 75 memset(head, -1, sizeof(int) * (n+2)); 76 tot = 0; 77 int s = 0, t = n + 1; 78 for(int i = 1; i <= m; i++){ 79 add(u[i], v[i], 1); 80 add(v[i], u[i], 1); 81 } 82 for(int i = 1; i <= n; i++){ 83 add(s, i, m); 84 add(i, t, m+2*x-d[i]); 85 } 86 return (m*n-Dinic(s,t))/2.0 >= 1e-6; 87 } 88 89 void Find(int x){ 90 vis[x] = 1; 91 for(int i = head[x]; ~i; i = nx[i]) 92 if(w[i] > 0 && !vis[to[i]]) 93 Find(to[i]); 94 } 95 96 int main(){ 97 while(~scanf("%d%d", &n, &m)){ 98 if(m == 0){ 99 printf("1 1 "); 100 continue; 101 } 102 memset(d, 0, sizeof(int)*(n+1)); 103 for(int i = 1; i <= m; i++){ 104 scanf("%d%d", &u[i], &v[i]); 105 d[u[i]]++; d[v[i]]++; 106 } 107 double l = 0, r = m, mid; 108 while(r - l >= 1.0/n/n){ 109 mid = (l+r)/2; 110 if(check(mid)) l = mid; 111 else r = mid; 112 } 113 check(l); 114 memset(vis, 0, sizeof(vis)); 115 Find(0); 116 int ans = 0; 117 for(int i = 1; i <= n; i++) ans += vis[i]; 118 printf("%d ", ans); 119 for(int i = 1; i <= n; i++) 120 if(vis[i]) printf("%d ", i); 121 } 122 return 0; 123 }