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, unused_macros, dead_code)]
use std::f64::*;
use std::cmp::*;
use std::collections::*;
use std::ops::*;

macro_rules! dump{
    ($($a:expr),*) => {
        #[cfg(debug_assertions)]
        eprintln!(
            concat!("{}:{}:{}: ",$(stringify!($a), " = {:?}, "),*),
            file!(), line!(), column!(), $($a),*
        );
        #[cfg(not(debug_assertions))]
        {};
    }
}

const INF: i64 = 0x3f3f3f3f3f3f3f3f;
const MOD: i64 = 1000000007;
const NOTHING: usize = std::usize::MAX;
const DX: [usize; 4] = [1, 0, std::usize::MAX, 0];
const DY: [usize; 4] = [0, 1, 0, std::usize::MAX];

fn solve<R: std::io::Read, W: std::io::Write>(io: &mut IO<R, W>) {
    let (n, m): (usize, usize) = io.read();
    let mut g: Vec<Vec<bool>> = vec![vec![true;n];n];
    for i in 0..n {
        g[i][i] = false;
    }

    for _ in 0..m {
        let (x, y): (usize, usize) = io.read();
        let x = x - 1;
        let y = y - 1;
        g[x][y] = false;
        g[y][x] = false;
    }

    let mut bg = BipartiteGraph::new(n);
    for i in 0..n {
        for j in i + 1..n {
            if g[i][j] {
                bg.add_edge(i, j);
            }
        }
    }

    if !bg.is_bipartite() {
        io.writeln("-1");
        return;
    }
    let components = bg.components().unwrap();

    let mut set: Vec<bool> = vec![false;n+1];
    set[0] = true;
    for component in components {
        let s0 = component.color0.len();
        let s1 = component.color1.len();
        let mut new = vec![false;n+1];
        for i in (0..n + 1).rev() {
            if s0 <= i {
                new[i] |= set[i - s0];
            }
            if s1 <= i {
                new[i] |= set[i - s1];
            }
        }
        std::mem::swap(&mut set, &mut new);
    }


    let mut res = m;
    for s in 1..n {
        if set[s] {
            res = min(res, s * (s - 1) / 2 + (n - s) * (n - s - 1) / 2);
        }
    }

    io.writeln(res)

}

#[derive(Debug)]
pub struct BipartiteComponent {
    color0: Vec<usize>,
    color1: Vec<usize>,
}

pub struct BipartiteGraph {
    num_v: usize,
    uf: UnionFind,
    is_bipartite_cache: bool,
}

impl BipartiteGraph {
    pub fn new(num_v: usize) -> BipartiteGraph {
        BipartiteGraph {
            num_v: num_v,
            uf: UnionFind::new(2 * num_v),
            is_bipartite_cache: true,
        }
    }

    pub fn add_edge(&mut self, x: usize, y: usize) {
        debug_assert!(x < self.num_v && y < self.num_v);
        self.uf.unite(2 * x, 2 * y + 1);
        self.uf.unite(2 * y, 2 * x + 1);
    }

    pub fn is_bipartite(&mut self) -> bool {
        if !self.is_bipartite_cache {
            return false;
        }

        for i in 0..self.num_v {
            if self.uf.equiv(2 * i, 2 * i + 1) {
                self.is_bipartite_cache = false;
                return false;
            }
        }
        true
    }

    pub fn components(&mut self) -> Option<Vec<BipartiteComponent>> {
        if !self.is_bipartite_cache {
            return None;
        }
        let mut reprs = HashSet::new();
        for i in 0..self.num_v {
            let p0 = self.uf.find(2 * i);
            let p1 = self.uf.find(2 * i + 1);
            if p0 == p1 {
                self.is_bipartite_cache = false;
                return None;
            } else if !reprs.contains(&p0) && !reprs.contains(&p1) {
                reprs.insert(p0);
            }
        }

        Some(self.uf
            .groups()
            .iter()
            .filter(|&(repr, _)| reprs.contains(repr))
            .map(|(_, group)| {
                let (color0, color1): (Vec<usize>, Vec<usize>) = group.into_iter()
                    .partition(|&x| x & 1 == 0);
                BipartiteComponent {
                    color0: color0.iter().map(|&x| x >> 1).collect(),
                    color1: color1.iter().map(|&x| x >> 1).collect(),
                }
            })
            .collect())
    }
}

pub struct UnionFind {
    parent: Vec<usize>,
    rank: Vec<usize>,
    size: Vec<usize>,
}

impl UnionFind {
    pub fn new(n: usize) -> UnionFind {
        UnionFind {
            parent: vec![NOTHING; n],
            rank: vec![0; n],
            size: vec![1; n],
        }
    }

    pub fn iter(&self) -> Range<usize> {
        0..self.parent.len()
    }

    pub fn find(&mut self, x: usize) -> usize {
        if self.parent[x] == NOTHING {
            x
        } else {
            let px = self.parent[x];
            let root = self.find(px);
            self.parent[x] = root;
            root
        }
    }

    pub fn unite(&mut self, x: usize, y: usize) {
        let px = self.find(x);
        let py = self.find(y);

        if px == py {
            return;
        }

        if self.rank[px] < self.rank[py] {
            self.parent[px] = py;
            self.size[py] += self.size[px];
        } else {
            self.parent[py] = px;
            self.size[px] += self.size[py];
            if self.rank[px] == self.rank[py] {
                self.rank[px] += 1;
            }
        }

    }

    pub fn equiv(&mut self, x: usize, y: usize) -> bool {
        self.find(x) == self.find(y)
    }

    pub fn group_size(&mut self, x: usize) -> usize {
        let px = self.find(x);
        self.size[px]
    }

    pub fn groups(&mut self) -> HashMap<usize, Vec<usize>> {
        let mut res = HashMap::new();
        for i in self.iter() {
            let p = self.find(i);
            let ref mut gp = res.entry(p).or_insert(vec![]);
            gp.push(i);
        }
        res
    }

    pub fn count_group(&self) -> usize {
        self.parent.iter().filter(|&p| *p == NOTHING).collect::<Vec<_>>().len()
    }
}

fn main() {
    use std::io::*;
    std::thread::Builder::new()
        .stack_size(32 * 1024 * 1024)
        .spawn(|| {
            let stdin = stdin();
            let stdout = stdout();
            let mut io = IO::new(BufReader::new(stdin.lock()), BufWriter::new(stdout.lock()));
            solve(&mut io);
            io.flush();
        })
        .unwrap()
        .join()
        .unwrap();
}

pub struct IO<R, W> {
    reader: std::io::Bytes<R>,
    writer: W,
}

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> IO<R, W> {
        IO {
            reader: r.bytes(),
            writer: w,
        }
    }

    pub fn read<T: Read>(&mut self) -> T {
        T::read(&mut self.reader)
    }

    pub fn reads<T: Read>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| T::read(&mut self.reader)).collect()
    }

    pub fn write<T: std::fmt::Display>(&mut self, x: T) {
        self.writer.write_fmt(format_args!("{}", x)).unwrap();
    }
    pub fn writeln<T: std::fmt::Display>(&mut self, x: T) {
        self.writer.write_fmt(format_args!("{}\n", x)).unwrap();
    }

    pub fn writes<T: std::fmt::Display>(&mut self, xs: &Vec<T>, sep: &str) {
        let mut it = xs.iter();
        for &x in it.next().iter() {
            self.write(x);
        }
        for x in it {
            self.writer.write(sep.as_bytes()).unwrap();
            self.write(x);
        }
    }

    pub fn writesln<T: std::fmt::Display>(&mut self, xs: &Vec<T>, sep: &str) {
        self.writes(xs, sep);
        self.writer.write(b"\n").unwrap();
    }

    pub fn flush(&mut self) {
        self.writer.flush().unwrap()
    }
}

pub trait Read {
    fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self;
}

#[inline(always)]
pub fn is_whitespace(x: u8) -> bool {
    x < 33u8 || 126u8 < x
}

macro_rules! read_int_impl{
    ($($t:ty)*) => ($(
        impl Read for $t {
            fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
                let head: u8 = bytes
                    .map(|x| x.unwrap())
                    .skip_while(|&x| is_whitespace(x))
                    .next()
                    .unwrap();
                if head == 45u8 {
                    bytes.map(|x| x.unwrap())
                        .take_while(|&x| !is_whitespace(x))
                        .fold(0, |a, x| 10 * a - x as $t + 48)
                } else {
                    bytes.map(|x| x.unwrap())
                        .take_while(|&x| !is_whitespace(x))
                        .fold((head - 48) as $t, |a, x| 10 * a + x as $t - 48)
                }
            }
        }
    )*)
}
read_int_impl! { isize i8 i16 i32 i64 usize u8 u16 u32 u64}

macro_rules! read_float_impl{
    ($($t:ty)*) => ($(
        impl Read for $t {
            fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
                String::read(bytes).as_str().parse::<$t>().unwrap()
            }
        }
    )*)
}
read_float_impl! { f32 f64 }

impl Read for char {
    fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
        bytes.map(|x| x.unwrap())
            .skip_while(|&x| is_whitespace(x))
            .next()
            .unwrap() as char
    }
}

impl Read for String {
    fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
        bytes.map(|x| x.unwrap())
            .skip_while(|&x| is_whitespace(x))
            .take_while(|&x| !is_whitespace(x))
            .map(|x| x as char)
            .collect()
    }
}

impl Read for Vec<char> {
    fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
        bytes.map(|x| x.unwrap())
            .skip_while(|&x| is_whitespace(x))
            .take_while(|&x| !is_whitespace(x))
            .map(|x| x as char)
            .collect()
    }
}

macro_rules! read_tuple_impl{
    ($($t:ident),+) => {
        impl<$($t: Read),*> Read for ($($t),*) {
            fn read<R: std::io::Read>(bytes: &mut std::io::Bytes<R>) -> Self {
                ($($t::read(bytes)),+)
            }
        }
    }
}
read_tuple_impl!(A, B);
read_tuple_impl!(A, B, C);
read_tuple_impl!(A, B, C, D);
read_tuple_impl!(A, B, C, D, E);