uva12534 Binary Matrix 2(最小费用最大流)

http://blog.csdn.net/qq564690377/article/details/17082055

做的时候觉得明显是费用流，但是真的不知道怎么建图，看了上面的博客会稍微清晰一点。后面再补一点细节吧，然后发现这道题用自己平时的费用流模板是水不过去的，所以找了份别人AC的代码弄了个zkw最小费用流的模板上来，算是存下模板吧。

补充点个人的理解吧，个人的网络流做的题还是太少了，所以想不到怎么建模，其实感觉上还是比较直接的一个行列二分图建模。首先就是枚举最后有多少个1剩下来，假设当前的已经有cur个1了，然后我要达到tot，那么每一行应该有tot/n个，每一列应该有tot/m个，所以我们可以对每行建一个点，对每列建一个点，从源点到行的点限流tot/n，列的点到汇点限流tot/m，这样的话当满流的时候其实就能保证每一行的1相等，每一列的1也相等。然后建边的时候就是每个处于(i,j)的点和i行j列的点连一条边，假设原本是1就费用为0，原本是0就费用为1.那么按照这样跑一遍费用流，费用x就是图里面需要由0变成1的点的个数，我们只需要再求出y，即图里由1变成0的个数就可以了。那么y等于多少呢？ 不难发现图中多出来的1应该满足 x-y=tot-cur  所以y=x+cur-tot  所以最后的费用是x+y=2*x+cur-tot。

http://blog.sina.com.cn/s/blog_61034ad90100gwdw.html

上面的博客研究了spfa的网络流和zkw的网络流，可能zkw网络流会比较适用在二分图，稠密图上吧。

#pragma warning(disable:4996)

#include <iostream>

#include <cstring>

#include <string>

#include <vector>

#include <cmath>

#include <algorithm>

#include <cstdio>

#include <queue>

using namespace std;



#define ll long long

#define eps 1e-8

#define maxn 100

#define maxe 4000

#define inf 0x3f3f3f3f

using namespace std;





char b[50][50];

int n, m;



/*int siz;



struct Edge{

int u, v, nxt, cap, cost;

}edge[maxe];

int head[maxn];





struct MinCostMaxFlow

{

	queue<int> que;

	int add; // edges number

	int vn; // total vertex number

	int cost[maxn], in[maxn], pre[maxn];

	bool vis[maxn];

	void init(){

		add = 0; vn = siz + 10; memset(head, -1, sizeof(head));

		while (!que.empty()) que.pop();

	}

	void insert(int u, int v, int w, int c){

		edge[add].u = u; edge[add].v = v; edge[add].cap = w; edge[add].cost = c;

		edge[add].nxt = head[u]; head[u] = add++;

		edge[add].u = v; edge[add].v = u; edge[add].cap = 0; edge[add].cost = -c;

		edge[add].nxt = head[v]; head[v] = add++;

	}



	bool spfa(int s, int e){

		memset(cost, 0x3f3f3f3f, sizeof(cost));

		memset(in, 0, sizeof(in));

		memset(vis, 0, sizeof(vis));

		cost[s] = 0; pre[s] = -1;

		que.push(s); vis[s] = true; in[s]++;

		while (!que.empty()){

			int u = que.front(); que.pop();

			vis[u] = false;

			for (int i = head[u]; i != -1; i = edge[i].nxt){

				int v = edge[i].v;

				if (edge[i].cap > 0 && cost[v] > cost[u] + edge[i].cost){

					cost[v] = cost[u] + edge[i].cost; pre[v] = i;

					if (!vis[v]){

						que.push(v); vis[v] = true; in[v]++;

						if (in[v] > vn) return false;

					}

				}

			}

		}

		if (cost[e] < inf) return true;

		else return false;

	}

	int mincostmaxflow(int s, int e){

		int mincost = 0, maxflow = 0;

		while (spfa(s, e)){

			int flow = inf;

			for (int i = pre[e]; i != -1; i = pre[edge[i].u]){

				flow = min(flow, edge[i].cap);

			}

			maxflow += flow;

			for (int i = pre[e]; i != -1; i = pre[edge[i].u]){

				edge[i].cap -= flow;

				edge[i ^ 1].cap += flow;

			}

			mincost += cost[e] * flow;

		}

		return mincost;

	}

}net;

*/





struct Edge

{

	int u, v, cap, cost, nxt;

	Edge(int _u, int _v, int _cap, int _cost, int _nxt) :

		u(_u), v(_v), cap(_cap), cost(_cost), nxt(_nxt){}

	Edge(){};

}edge[maxe];

int head[maxn];



struct ZKW_MinCostMaxFlow {

	int add;

	int cur[maxn];

	int dis[maxn];

	bool inq[maxn];

	queue<int> q;

	bool vis[maxn];



	int ss, tt, n;

	int min_cost, max_flow;



	void init() {

		memset(head, -1, sizeof(head));

		add = 0;

	}



	void insert(int u, int v, int cp, int ct) {

		edge[add] = Edge(u, v, cp, ct, head[u]);

		head[u] = add++;

		edge[add] = Edge(v, u, 0, -ct, head[v]);

		head[v] = add++;

	}



	int aug(int u, int flow) {

		if (u == tt) return flow;

		vis[u] = true;

		for (int i = cur[u]; i != -1; i = edge[i].nxt) {

			int v = edge[i].v;

			if (edge[i].cap && !vis[v] && dis[u] == dis[v] + edge[i].cost) {

				int tmp = aug(v, min(flow, edge[i].cap));

				edge[i].cap -= tmp;

				edge[i ^ 1].cap += tmp;

				cur[u] = i;

				if (tmp) return tmp;

			}

		}

		return 0;

	}

	bool modify_label() {

		int d = inf;

		for (int u = 0; u < n; u++) if (vis[u])

		for (int i = head[u]; i != -1; i = edge[i].nxt) {

			int v = edge[i].v;

			if (edge[i].cap && !vis[v])

				d = min(d, dis[v] + edge[i].cost - dis[u]);

		}

		if (d == inf) return false;

		for (int i = 0; i < n; ++i) if (vis[i]) {

			vis[i] = false;

			dis[i] += d;

		}

		return true;

	}



	pair<int, int> mincostmaxflow(int s, int t, int _n) {

		ss = s, tt = t, n = _n;

		min_cost = max_flow = 0;

		for (int i = 0; i < n; i++) dis[i] = 0;

		while (true) {

			for (int i = 0; i < n; i++) cur[i] = head[i];

			while (true) {

				for (int i = 0; i < n; i++) vis[i] = 0;

				int tmp = aug(s, inf);

				if (tmp == 0) break;

				max_flow += tmp;

				min_cost += tmp * dis[ss];

			}

			if (!modify_label()) break;

		}

		return make_pair(min_cost, max_flow);

	}

}net;



int main()

{

	int T; cin >> T; int ca = 0;

	while (T--){

		scanf("%d%d", &n, &m);

		for (int i = 0; i < n; i++) scanf("%s", b[i]);

		int tot = 0;

		for (int i = 0; i < n; ++i){

			for (int j = 0; j < m; ++j){

				if (b[i][j] == '1') tot++;

			}

		}

		int ans = min(n*m - tot, tot);

		for (int i = 1; i <= n*m; ++i){

			if (i%n != 0 || i%m != 0) continue;

			if (abs(tot - i) >= ans) continue;

			net.init();

			int src = n + m, sink = src + 1;

			for (int k = 0; k < n; ++k){

				net.insert(src, k, i / n, 0);

			}

			for (int k = n; k < n + m; k++){

				net.insert(k, sink, i / m, 0);

			}

			for (int x = 0; x < n; x++){

				for (int j = 0; j < m; j++){

					if (b[x][j] == '1') net.insert(x, j + n, 1, 0);

					else net.insert(x, j + n, 1, 1);

				}

			}

			ans = min(ans, net.mincostmaxflow(src, sink,sink+1).first * 2 + tot - i);

		}

		printf("Case %d: %d\n", ++ca, ans);

	}

	return 0;

}