# cogirth-packing gap of projections of graphic matroids

from sage.all import *
from sage.matroids.all import *
from sage.graphs.all import *
import gurobipy as gp
from gurobipy import GRB
from fractions import Fraction

env = gp.Env(empty=True)
env.setParam("OutputFlag",0)
env.start()

def representative_vectors(m, n):
    for w1 in range(m+1):
        for w2 in range(n+1):
            v = [0]*(m+n)
            v[:w1] = [1]*w1
            v[m:m+w2] = [1]*w2
            yield tuple(v)

def cogirthip(bases, integral=true):
    model = gp.Model("mip1",env=env)
    # model.Params.LogToConsole = 0
    groundset=frozenset()
    for B in bases: groundset=groundset|frozenset(B)
    x = dict()
    if integral:
        for e in groundset: x[e]=model.addVar(vtype=GRB.BINARY)
    else:
        for e in groundset: x[e]=model.addVar(vtype=GRB.CONTINUOUS,lb=0)
    model.setObjective(gp.quicksum([x[e] for e in groundset]), GRB.MINIMIZE)
    for B in bases:
        model.addConstr(gp.quicksum([x[e] for e in B])>=1)
    model.optimize()
    return model.ObjVal


# cnt=0   # actual number of instances tested
# f = lambda g: g.is_connected()
for N in range(3,11):
    g=graphs.CompleteGraph(N)
    A=g.incidence_matrix()
    n,m = A.dimensions()
    maxgap = 0
    for v in representative_vectors(N,0):
        # cnt=cnt+1
        v_col=matrix(v).transpose()
        A_t=A.augment(v_col)
        # print(A_t)
        MM=Matroid(matrix=A_t,field=GF(2))/m     #contract the last element
        # print(M)
        bases=MM.bases()
        strength=cogirthip(bases,integral=false)
        cogirth =cogirthip(bases,integral=true)
        gap = cogirth/strength
        if gap > maxgap:
            maxgap = gap
            maxcol = v
    frac=str(Fraction(maxgap).limit_denominator(m))
    print(f"K_{N}   gap ≈ {frac:<8} column = {maxcol}")
