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 Rust (1.15.1) (judge verdict: Accepted)

#![allow(unused_imports)]
#![allow(dead_code)]
#![allow(unused_must_use)]
use std::io::{stdin, stdout, stderr, Write, BufWriter, BufReader, BufRead};
use std::collections::BinaryHeap;

use util::IterExt;
macro_rules! read_from_line {
    ($inbuf:ident => $($name:ident : $ty:ty),* ) => {
        let mut buf = String::new();
        $inbuf.read_line(&mut buf).unwrap();
        buf.pop();
        let mut words = buf.split_whitespace();
        $(
            let $name:$ty = words.next().unwrap().parse().unwrap();
        )*
    };
}

macro_rules! read_many {
    ($inbuf:ident => $name:ident : $ty:ty) => {
        let $name: Vec<$ty> = strio::read_many_from_line(&mut $inbuf);
    };
}

#[derive(PartialEq, Eq, Debug)]
struct RevOrd<T>(T);

impl<T> PartialOrd for RevOrd<T> where T: PartialOrd {
    fn partial_cmp(&self, other: &Self) -> Option<std::cmp::Ordering> {
        other.0.partial_cmp(&self.0)
    }
}

impl<T> Ord for RevOrd<T> where T: Ord {
    fn cmp(&self, other: &Self) -> std::cmp::Ordering {
        other.0.cmp(&self.0)
    }
}

#[derive(Debug)]
struct KV<K, V> {
    k: K,
    v: V
}
impl <K, V> KV<K, V> {
    fn new(k: K, v: V) -> Self {
        KV {k:k, v:v}
    }
}
impl <K, V> PartialEq for KV<K, V> where V: PartialEq {
    fn eq(&self, other: &Self) -> bool {
        self.v.eq(&other.v)
    }
}
impl <K, V> Eq for KV<K, V> where V: Eq { }
impl<K, V> PartialOrd for KV<K, V> where V: PartialOrd {
    fn partial_cmp(&self, other: &Self) -> Option<std::cmp::Ordering> {
        self.v.partial_cmp(&other.v)
    }
}
impl<K, V> Ord for KV<K, V> where V: Ord {
    fn cmp(&self, other: &Self) -> std::cmp::Ordering {
        self.v.cmp(&other.v)
    }
}

#[derive(Debug)]
struct UnionFind {
    tree: Vec<usize>,
}

impl UnionFind {
    fn new(n: usize) -> UnionFind {
        UnionFind { 
            tree: (0..n+1).collect()
        }
    }

    fn root(&mut self, i: usize) -> usize {
        let mut r = i;
        let mut stack = vec![];
        while self.tree[r] != r {
            stack.push(r);
            r = self.tree[r];
        }

        for x in stack {
            self.tree[x] = r
        }
        r
    }

    fn unify(&mut self, i: usize, j: usize) {
        let i = self.root(i);
        let j = self.root(j);
        self.tree[std::cmp::max(i, j)] = std::cmp::min(i, j);
    }

}


fn solve() {
    let cout = stdout();
    let mut cout = BufWriter::new(cout.lock());
    let cin = stdin();
    let mut cin = BufReader::new(cin.lock());

    read_from_line!(cin => n: usize, m: usize);
    let es = {
        let mut es = vec![vec![false; n+1]; n+1];
        for i in 1..m+1 {
            read_from_line!(cin => a: usize, b: usize);
            es[a][b] = true;
            es[b][a] = true;
        }
        es
    };
    let mut uf = UnionFind::new(n+1);
    for i in 1..n+1 {
        for j in i..n+1 {
            if !es[i][j] {
                uf.unify(i, j);
            }
        }
    }

    let mut check_bi = vec![None; n+1];
    let mut counts = vec![];
    let mut contradiction = false;
    'mainloop: for i in 1..n+1 {
        if (uf.root(i) != i) { 
            continue;
        }
        let mut stack = vec![i];
        let mut t_count = 1;
        let mut f_count = 0;
        check_bi[i] = Some(true);
        while let Some(v) = stack.pop() {
            let v_sig = check_bi[v].unwrap();
            for u in 1..n+1 {
                if es[v][u] || v == u {
                    continue;
                }
                match check_bi[u] {
                    None => {
                        check_bi[u] = Some(!v_sig);
                        if v_sig {
                            f_count += 1;
                        } else {
                            t_count += 1
                        }
                        stack.push(u);
                    },
                    Some(s) => 
                        if s == !v_sig {
                            continue
                        } else {
                            contradiction = true;
                            break 'mainloop;
                        },
                }
            }
            
        }
        counts.push((t_count, f_count));
    }
    if contradiction {
        writeln!(cout, "-1");
        return;
    }
    let mut possible_sum = vec![false; n+1];
    possible_sum[0] = true;
    for (x, y) in counts {
        let mut next = vec![false; n+1];
        for i in 0..n+1 {
            if (i >= x) {
                next[i] |= possible_sum[i-x];
            }
            if (i >= y) {
                next[i] |= possible_sum[i-y];
            }
        }
        possible_sum = next;
    }
    let ans = (0..n+1).filter(|i| possible_sum[*i]).map(|x| x * (x - 1) / 2 + (n - x) * (n - x - 1) / 2).min().unwrap();
    writeln!(cout, "{}", ans);
    

}

fn main() {
    let stack_size = 100 * 1024 * 1024;
    let thread = std::thread::Builder::new().stack_size(stack_size);
    thread.spawn(|| solve()).unwrap().join().unwrap();
}

mod util {
    use std;
    use std::ops::{Deref, AddAssign};
    use std::iter::{Once, Chain, Scan};
    use std::borrow::Borrow;

    pub trait IterExt where
        Self: Iterator
    {
        fn scan_init<'a, St : 'a, F : 'a> (self, initial_state: St, f: F) -> Box<Iterator<Item=St> + 'a> /*Chain<Once<St>, Scan<Self, St, F>>*/  where
            Self: Sized + 'a,
            F: FnMut(&mut St, Self::Item) -> Option<St>,
            St: Clone
        {
            Box::new(std::iter::once(initial_state.clone()).chain(self.scan(initial_state, f)))
        }

        fn accum<'a, T> (self, init: T) -> Box<Iterator<Item=T> + 'a> where
            Self: Sized + 'a,
            Self::Item: Borrow<T>,
            T: Clone + AddAssign + 'a,
        {
            Box::new(self.scan_init(init, |sum, x| { *sum += x.borrow().clone(); Some(sum.clone()) }))
        }
    }

    impl<T> IterExt for T where
        T: Iterator
    { }

    pub fn test()
    {
        let v:Vec<i32> = vec![1, 4 ,2];
        for x in v.iter().accum(0) {
            println!("{}", x);
        }
        for x in v.iter().accum(0).accum(0) {
            println!("{}", x);
        }

    }

}

mod strio {
    use std::str::FromStr;
    use std;

    pub fn read_from_line<T, R> (r: &mut R) -> T where
        T: FromStr,
        <T as std::str::FromStr>::Err : std::fmt::Debug,
        R: std::io::BufRead,
    {
        let mut buf = String::new();
        r.read_line(&mut buf).unwrap();
        buf.pop();
        buf.parse().unwrap()
    }

    pub fn read_many_from_line<T, R> (r: &mut R) -> Vec<T> where
        T: FromStr,
        <T as std::str::FromStr>::Err : std::fmt::Debug,
        R: std::io::BufRead,
    {
        let mut buf = String::new();
        r.read_line(&mut buf).unwrap();
        buf.pop();
        buf.split_whitespace().map(|x| x.parse().unwrap()).collect()
    }
}