gap of projection of complete graphs is not so large??

complete.sage

@@ -0,0 +1,61 @@
+# 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}")