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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using static System.Console;
using static System.Convert;
using static System.Math;
using Pi = Pair<int, int>;
using PL = Pair<long, int>;
//using static System.Globalization.CultureInfo;
using System.Text;
class Program
{
    public static bool chmin<T>(ref T num, T val) where T : IComparable<T>
    { if (num.CompareTo(val) == 1) { num = val; return true; } return false; }
    public static bool chmax<T>(ref T num, T val) where T : IComparable<T>
    { if (num.CompareTo(val) == -1) { num = val; return true; } return false; }
    public static void swap<T>(ref T v1, ref T v2)
    { var t = v2; v2 = v1; v1 = t; }
    public static void Fail() => Fail("No");
    public static void Fail<T>(T s) { WriteLine(s); Console.Out.Close(); Environment.Exit(0); }
    static void Main(string[] args)
    {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        Console.SetOut(sw);
        var p = new Program();
        for (var i = 1; i > 0; --i) 
            p.Solve();
        Console.Out.Flush();
    }
    int[] color;
    List<int>[] edge;
    int ct1, s;
    void Solve()
    {
        int num, m;
        Input.Make(out num, out m);
        var e = Input.create(num, _ => new bool[num]);
        for (var i = 0; i < m; i++)
        {
            int a, b;
            Input.Make(out a, out b);
            a--; b--;
            e[a][b] = e[b][a] = true;
        }
        edge = Input.edge<int>(num);
        for (var i = 0; i < num; i++)
            for (var j = 0; j < num; j++)
                if (i != j && !e[i][j])
                    edge[i].Add(j);
        color = new int[num];
        var t = true;
        var list = new List<Pi>();
        for (var i = 0; i < num; i++)
            if (color[i] == 0)
            {
                s = ct1 = 0;
                t &= IsBipartite(i);
                list.Add(new Pi(ct1, s - ct1));
            }
        if (!t) Fail(-1);
        var dp = new bool[num << 1]; dp[0] = true;
        foreach (var ar in list)
            for (var j = num; j >= 0; j--)
            {
                if (!dp[j]) continue;
                dp[j] = false;
                dp[j + ar.v1] = dp[j + ar.v2] = true;
            }
        var ct = -1;
        for (var i = 0; i <= num; i++)
            if (dp[i] && Abs(num / 2 - ct) > Abs(num / 2 - i))
                ct = i;
        WriteLine(ct * (ct - 1) / 2 + (num - ct) * (num - ct - 1) / 2);
    }
    public bool IsBipartite(int index, int c = 1)
    {
        if (color[index] != 0) return color[index] == c;
        s++; color[index] = c;if (c == 1) ct1++;
        var t = true;
        foreach (var ad in edge[index])
            if (t)
                t &= IsBipartite(ad, 3 - c);
            else break;
        return t;
    }
}

public class Input
{
    public static string read => ReadLine().Trim();
    public static int[] ar => read.Split(' ').Select(int.Parse).ToArray();
    public static int num => ToInt32(read);
    public static long[] arL => read.Split(' ').Select(long.Parse).ToArray();
    public static long numL => ToInt64(read);
    public static T[] create<T>(int n, Func<int, T> f)
        => Enumerable.Repeat(0, n).Select(f).ToArray();
    public static char[][] grid(int h)
        => create(h, _ => read.ToCharArray());
    public static int[] ar1D(int n)
        => create(n, _ => num);
    public static long[] arL1D(int n)
        => create(n, _ => numL);
    public static string[] strs(int n)
        => create(n, _ => read);
    public static int[][] ar2D(int n)
        => create(n, _ => ar);
    public static long[][] arL2D(int n)
        => create(n, _ => arL);
    public static List<T>[] edge<T>(int n)
        => create(n, _ => new List<T>());
    public static T GetValue<T>(string g)
    {
        var t = typeof(T);
        if (t == typeof(int))
            return (T)(object)int.Parse(g);
        if (t == typeof(long))
            return (T)(object)long.Parse(g);
        if (t == typeof(string))
            return (T)(object)g;
        if (t == typeof(char))
            return (T)(object)char.Parse(g);
        if (t == typeof(double))
            return (T)(object)double.Parse(g);
        if (t == typeof(bool))
            return (T)(object)bool.Parse(g);
        return default(T);
    }
    public static void Make<T1, T2>(out T1 v1, out T2 v2)
    {
        v1 = Next<T1>();
        v2 = Next<T2>();
    }
    public static void Make<T1, T2, T3>(out T1 v1, out T2 v2, out T3 v3)
    {
        Make(out v1, out v2);
        v3 = Next<T3>();
    }
    public static void Make<T1, T2, T3, T4>(out T1 v1, out T2 v2, out T3 v3, out T4 v4)
    {
        Make(out v1, out v2, out v3);
        v4 = Next<T4>();
    }
    public static void Make<T1, T2, T3, T4, T5>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5)
    {
        Make(out v1, out v2, out v3, out v4);
        v5 = Next<T5>();
    }
    public static void Make<T1, T2, T3, T4, T5, T6>(out T1 v1, out T2 v2, out T3 v3, out T4 v4, out T5 v5, out T6 v6)
    {
        Make(out v1, out v2, out v3, out v4, out v5);
        v6 = Next<T6>();
    }
    private static Queue<string> sc;
    public static T Next<T>() { sc = sc ?? new Queue<string>(); if (sc.Count == 0) foreach (var item in read.Split(' ')) sc.Enqueue(item); return GetValue<T>(sc.Dequeue()); }
    public static void Next<T>(ref T val) => val = Next<T>();
    public const long Inf = (long)1e18;
    public const double eps = 1e-6;
    public const string Alfa = "abcdefghijklmnopqrstuvwxyz";
    public const int MOD = 1000000007;
}

public class Pair<T1, T2> : IComparable<Pair<T1, T2>>
{
    public T1 v1 { get; set; }
    public T2 v2 { get; set; }
    public Pair() { v1 = Input.Next<T1>(); v2 = Input.Next<T2>(); }
    public Pair(T1 v1, T2 v2)
    { this.v1 = v1; this.v2 = v2; }

    public int CompareTo(Pair<T1, T2> p)
    {
        var c = Comparer<T1>.Default.Compare(v1, p.v1);
        if (c == 0)
            c = Comparer<T2>.Default.Compare(v2, p.v2);
        return c;
    }
    public override string ToString()
        => $"{v1.ToString()} {v2.ToString()}";
    public override bool Equals(object obj)
        => this == (Pair<T1, T2>)obj;
    public override int GetHashCode()
        => v1.GetHashCode() ^ v2.GetHashCode();
    public static bool operator ==(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) == 0;
    public static bool operator !=(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) != 0;
    public static bool operator >(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) == 1;
    public static bool operator >=(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) != -1;
    public static bool operator <(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) == -1;
    public static bool operator <=(Pair<T1, T2> p1, Pair<T1, T2> p2)
        => p1.CompareTo(p2) != 1;
}

public class Pair<T1, T2, T3> : Pair<T1, T2>, IComparable<Pair<T1, T2, T3>>
{
    public T3 v3 { get; set; }
    public Pair() : base() { v3 = Input.Next<T3>(); }
    public Pair(T1 v1, T2 v2, T3 v3) : base(v1, v2)
    { this.v3 = v3; }

    public int CompareTo(Pair<T1, T2, T3> p)
    {
        var c = base.CompareTo(p);
        if (c == 0)
            c = Comparer<T3>.Default.Compare(v3, p.v3);
        return c;
    }
    public override string ToString()
        => $"{base.ToString()} {v3.ToString()}";
}