Problem:
Score : 700 points

Problem Statement

In the State of Takahashi in AtCoderian Federation, there are N cities, numbered 1, 2, ..., N.
M bidirectional roads connect these cities.
The i-th road connects City A_i and City B_i.
Every road connects two distinct cities.
Also, for any two cities, there is at most one road that directly connects them.

One day, it was decided that the State of Takahashi would be divided into two states, Taka and Hashi.
After the division, each city in Takahashi would belong to either Taka or Hashi.
It is acceptable for all the cities to belong Taka, or for all the cities to belong Hashi.
Here, the following condition should be satisfied:

Any two cities in the same state, Taka or Hashi, are directly connected by a road.

Find the minimum possible number of roads whose endpoint cities belong to the same state.
If it is impossible to divide the cities into Taka and Hashi so that the condition is satisfied, print -1.

Constraints

2 \leq N \leq 700

0 \leq M \leq N(N-1)/2

1 \leq A_i \leq N

1 \leq B_i \leq N

A_i \neq B_i

If i \neq j, at least one of the following holds: A_i \neq A_j and B_i \neq B_j.

If i \neq j, at least one of the following holds: A_i \neq B_j and B_i \neq A_j.

Input

Input is given from Standard Input in the following format:

N M
A_1 B_1
A_2 B_2
:
A_M B_M

Output

Print the answer.

Sample Input 1

5 5
1 2
1 3
3 4
3 5
4 5

Sample Output 1

4

For example, if the cities 1, 2 belong to Taka and the cities 3, 4, 5 belong to Hashi, the condition is satisfied.
Here, the number of roads whose endpoint cities belong to the same state, is 4.

Sample Input 2

5 1
1 2

Sample Output 2

-1

In this sample, the condition cannot be satisfied regardless of which cities belong to each state.

Sample Input 3

4 3
1 2
1 3
2 3

Sample Output 3

3

Sample Input 4

10 39
7 2
7 1
5 6
5 8
9 10
2 8
8 7
3 10
10 1
8 10
2 3
7 4
3 9
4 10
3 4
6 1
6 7
9 5
9 7
6 9
9 4
4 6
7 5
8 3
2 5
9 2
10 7
8 6
8 9
7 3
5 3
4 5
6 3
2 10
5 10
4 2
6 2
8 4
10 6

Sample Output 4

21

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
#ifdef DEBUG
#include "_debug.cpp"
#include "callable.hpp/callable.hpp"
#endif
#define endl '\n'
using namespace std;
using ll = long long;
#define N 700
bool conn[N+1][N+1] = {};
vector<int> adj[N+1];
void link(int u, int v) {
    adj[u].push_back(v);
    adj[v].push_back(u);
}
bool vis[N+1] = {};
int p[N+1] = {};
int depth[N+1] = {};
int sz[N+1] = {};
template <typename PreFunction, typename PostFunction>
int dfs(int u, const PreFunction& pre, const PostFunction& post) {
#ifdef DEBUG
    using pre_actual_type = typename callable_traits<PreFunction>::function_type;
    using pre_expected_type = typename callable_traits<void(int)>::function_type;
    static_assert(is_same<pre_actual_type, pre_expected_type>::value, "dfs 'pre' must be void(int)");
    using post_actual_type = typename callable_traits<PostFunction>::function_type;
    using post_expected_type = typename callable_traits<void(int)>::function_type;
    static_assert(is_same<post_actual_type, post_expected_type>::value, "dfs 'post' must be void(int)");
#endif
    int ans = 1;
    vis[u] = true;
    depth[u] = depth[p[u]] + 1;
    sz[u] = 1;
    pre(u);
    for(int v : adj[u]) {
        if(not vis[v]) {
            p[v] = u;
            ans += dfs(v, pre, post);
            sz[u] += sz[v];
        }
    }
    post(u);
    return ans;
}
int dfs(int u) { return dfs(u, [](int u) {}, [](int u) {}); }
bool color[N+1];
vector<int> black_counts;
vector<int> white_counts;
vector<int> color_counts;
vector<int> color_counts_psum;
int mem[N+1][N+1];
int add_new_nodes(int x, int y, int new_x, int new_y) {
    return new_x * x + new_x * (new_x - 1) / 2
         + new_y * y + new_y * (new_y - 1) / 2;
}
int opt(int i, int x) {
    int& ans = mem[i][x];
    if(ans == -1) {
        if(i == color_counts.size()) {
            ans = 0;
        } else {
            int y = (i == 0 ? 0 : color_counts_psum[i - 1]) - x;
            ans = min(add_new_nodes(x, y, black_counts[i], white_counts[i]) + opt(i + 1, x + black_counts[i]),
                      add_new_nodes(x, y, white_counts[i], black_counts[i]) + opt(i + 1, x + white_counts[i]));
        }
    }
    return ans;
}
int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    int n, m;
    cin >> n >> m;
    for(int i = 0; i < m; i++) {
        int u, v;
        cin >> u >> v;
        conn[u][v] = conn[v][u] = true;
    }
    for(int i = 1; i <= n; i++) {
        for(int j = i + 1; j <= n; j++) {
            if(not conn[i][j]) {
                link(i, j);
            }
        }
    }
    fill(vis, vis + n + 1, false);
    bool can = true;
    for(int u = 1; u <= n and can; u++) {
        if(not vis[u]) {
            color[u] = true;
            int black = 0, white = 0;
            dfs(u, [&](int u) {
                color[u] = not color[p[u]];
                (color[u] ? black : white)++;
            }, [&](int u) {
                for(int v : adj[u]) {
                    can &= color[v] != color[u];
                }
            });
            black_counts.push_back(black);
            white_counts.push_back(white);
            color_counts.push_back(black + white);
        }
    }
    if(can) {
        color_counts_psum.resize(color_counts.size());
        partial_sum(color_counts.begin(), color_counts.end(), color_counts_psum.begin());
        memset(mem, -1, sizeof mem);
        cout << opt(0, 0) << endl;
    } else {
        cout << -1 << endl;
    }
    return 0;
}