diff --git a/Kmn_2factor.sage b/Kmn_2factor.sage
new file mode 100644
index 0000000..21ff5fd
--- /dev/null
+++ b/Kmn_2factor.sage
@@ -0,0 +1,102 @@
+# cogirth-packing gap of projections of graphic matroids
+# see if gap(projection) <= 2 * gap(graph)
+
+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 base_hittingset_with_callback(M, integral=true):
+    # model
+    model = gp.Model("mip1",env=env)
+    groundset = M.groundset()
+    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,ub=1)
+    # there is no lazy constraint for LP...
+    for B in M.bases():
+        model.addConstr(gp.quicksum([x[e] for e in B])>=1)
+    model.setObjective(gp.quicksum([x[e] for e in groundset]), GRB.MINIMIZE)
+    # callback
+    def callback_func(model, where):
+        if integral and where == GRB.Callback.MIPSOL:
+            sol_values = {key: model.cbGetSolution(var) 
+                          for key, var in x.items()}
+            # find min weight base in the matroid
+            base = frozenset()
+            minweight=0
+            for e in sorted(groundset, key=lambda ee: sol_values[ee]):
+                if M.is_independent(base.union([e])):
+                    base=base.union([e])
+                    minweight=minweight+sol_values[e]
+            if minweight < 1:
+                model.cbLazy(gp.quicksum([x[e] for e in base])>=1)
+    # solve
+    if integral:
+        model.Params.Heuristics = 0
+        model.Params.LazyConstraints = 1
+        model.optimize(callback_func)
+    else:
+        model.optimize()
+    return model.ObjVal
+
+
+cnt=0   # actual number of instances tested
+maxfactor=0
+for N in range(2,10):
+    for M in range(N,10):
+        g=graphs.CompleteBipartiteGraph(N,M)
+        MG=Matroid(g)
+        mincut=base_hittingset_with_callback(MG)
+        treepacking=base_hittingset_with_callback(MG,integral=false)
+        g_gap=mincut/treepacking
+        A=g.incidence_matrix()
+        n,m = A.dimensions()
+        # enumerate all vectors in F_2^n with even number of 1s
+        m_gap=0
+        for v in representative_vectors(N,M):
+            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
+            cogirth = base_hittingset_with_callback(MM,integral=true)
+            strength = base_hittingset_with_callback(MM,integral=false)
+            gap = cogirth/strength
+            m_gap=max(m_gap,gap)
+            # maxgap=max(gap,maxgap)
+            # if gap > maxgap:
+            #     maxgap = gap
+            #     print(f"find a large gap: {gap}")
+            #     with open("projection.out", "a") as file:
+            #         file.write("##################################\n"
+            #                    +str(gap)+"\n"+str(A_t)
+            #                    +"\n##################################\n")
+        factor=m_gap/g_gap
+        fr = lambda xx:  str(Fraction(xx).limit_denominator(m))
+        maxfactor=max(maxfactor,factor)
+        print(f"K_{{{N},{M}}}    graph gap = {fr(g_gap):8}   projection gap = {fr(m_gap):8}  factor={fr(factor):8}   max={maxfactor}")
+        if factor >= 2:
+            with open("Kmn_2factor.out", "a") as file:
+                file.write(f"K_{{{N},{M}}}    graph gap = {fr(g_gap):8}   projection gap = {fr(m_gap):8}  factor={fr(factor):8}   max={maxfactor}\n"
+                            +str(A_t)+"\n")
+        if factor > 2:
+            print("conjecture is wrong")
+            print(str(A_t))
+            exit(1)
\ No newline at end of file
diff --git a/cogirth_callback.sage b/cogirth_callback.sage
new file mode 100644
index 0000000..506f13f
--- /dev/null
+++ b/cogirth_callback.sage
@@ -0,0 +1,73 @@
+from sage.all import *
+from sage.matroids.all import *
+from sage.graphs.all import *
+import gurobipy as gp
+from gurobipy import GRB
+
+env = gp.Env(empty=True)
+env.setParam("OutputFlag",0)
+env.start()
+
+def base_hittingset_with_callback(M, integral=true):
+    # model
+    model = gp.Model("mip1",env=env)
+    groundset = M.groundset()
+    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,ub=1)
+    # there is no lazy constraint for LP...
+    for B in M.bases():
+        model.addConstr(gp.quicksum([x[e] for e in B])>=1)
+    model.setObjective(gp.quicksum([x[e] for e in groundset]), GRB.MINIMIZE)
+    # callback
+    def callback_func(model, where):
+        if integral and where == GRB.Callback.MIPSOL:
+            sol_values = {key: model.cbGetSolution(var) 
+                          for key, var in x.items()}
+            # find min weight base in the matroid
+            base = frozenset()
+            minweight=0
+            for e in sorted(groundset, key=lambda ee: sol_values[ee]):
+                if M.is_independent(base.union([e])):
+                    base=base.union([e])
+                    minweight=minweight+sol_values[e]
+            if minweight < 1:
+                model.cbLazy(gp.quicksum([x[e] for e in base])>=1)
+    # solve
+    if integral:
+        model.Params.Heuristics = 0
+        model.Params.LazyConstraints = 1
+        model.optimize(callback_func)
+    else:
+        model.optimize()
+    return model.ObjVal
+
+# tests
+if __name__ == "__main__":
+    def cogirthip(bases, integral=true):
+        model = gp.Model("mip1",env=env)
+        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
+    f = lambda g: g.is_connected()
+    for N in range(4,10):
+        for g in filter(f, graphs(N)):
+            M=Matroid(g)
+            cb=base_hittingset_with_callback(M)
+            bf=cogirthip(M.bases())
+            if randint(0,100)==1:
+                print(f"callback:{cb}, bf:{bf}")
+            if abs(cb-bf) > 0.01:
+                print(f"callback:{cb}, bf:{bf}")
+                exit(1)
\ No newline at end of file
diff --git a/projection.out b/projection.out
index 2108156..6afb851 100644
--- a/projection.out
+++ b/projection.out
@@ -1,108 +1,8 @@
-##################################
-1.0
-[1 0 0 1]
-[0 1 0 1]
-[0 0 1 0]
-[1 1 1 0]
-##################################
-##################################
-1.3333333333333333
-[1 0 0 1]
-[0 1 0 1]
-[0 0 1 1]
-[1 1 1 1]
-##################################
-##################################
-1.4999999999999998
-[1 0 0 0 1]
-[0 1 0 0 1]
-[0 0 1 0 1]
-[0 0 0 1 1]
-[1 1 1 1 0]
-##################################
-##################################
-1.5
-[1 1 0 0 0 0 1]
-[0 0 1 1 0 0 1]
-[0 0 0 0 1 1 1]
-[1 0 1 0 1 0 0]
-[0 1 0 1 0 1 1]
-##################################
-##################################
-1.7142857142857137
-[1 1 0 0 0 0 0 1]
-[0 0 1 1 0 0 0 1]
-[0 0 0 0 1 1 0 1]
-[1 0 1 0 1 0 1 1]
-[0 1 0 1 0 1 1 0]
-##################################
-##################################
-1.7142857142857142
-[1 1 0 0 0 0 0 1]
-[0 0 1 1 0 0 0 1]
-[0 0 0 0 1 1 0 1]
-[0 0 0 0 0 0 1 1]
-[1 0 1 0 1 0 0 0]
-[0 1 0 1 0 1 1 0]
-##################################
-##################################
-2.0
-[1 1 0 0 0 0 0 0 1]
-[0 0 1 1 0 0 0 0 1]
-[0 0 0 0 1 1 0 0 1]
-[0 0 0 0 0 0 1 1 1]
-[1 0 1 0 1 0 1 0 0]
-[0 1 0 1 0 1 0 1 0]
-##################################
-##################################
-2.1333333333333337
-[1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1]
-[1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1]
-[0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 1]
-[0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1]
-[0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1]
-[0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1]
-##################################
-##################################
-2.1428571428571423
-[1 1 1 0 0 0 0 0 0 0 0 0 0 0 1]
-[0 0 0 1 1 1 0 0 0 0 0 0 0 0 1]
-[0 0 0 0 0 0 1 1 1 0 0 0 0 0 1]
-[0 0 0 0 0 0 0 0 0 1 1 1 0 0 1]
-[1 0 0 1 0 0 1 0 0 1 0 0 1 0 1]
-[0 1 0 0 1 0 0 1 0 0 1 0 0 1 1]
-[0 0 1 0 0 1 0 0 1 0 0 1 1 1 0]
-##################################
-##################################
-2.1818181818181817
-[1 1 0 0 0 0 0 0 0 0 0 1]
-[0 0 1 1 0 0 0 0 0 0 0 1]
-[0 0 0 0 1 1 0 0 0 0 0 1]
-[0 0 0 0 0 0 1 1 0 0 0 1]
-[0 0 0 0 0 0 0 0 1 1 0 1]
-[1 0 1 0 1 0 0 0 0 0 1 1]
-[0 1 0 1 0 0 1 0 1 0 0 0]
-[0 0 0 0 0 1 0 1 0 1 1 0]
-##################################
-##################################
-2.181818181818182
-[1 1 0 0 0 0 0 0 0 0 0 1]
-[0 0 1 1 0 0 0 0 0 0 0 1]
-[0 0 0 0 1 1 0 0 0 0 0 1]
-[0 0 0 0 0 0 1 1 0 0 0 1]
-[0 0 0 0 0 0 0 0 1 1 0 1]
-[1 0 1 0 1 0 0 0 0 0 1 0]
-[0 1 0 1 0 0 1 0 1 0 0 1]
-[0 0 0 0 0 1 0 1 0 1 1 0]
-##################################
-##################################
-2.4000000000000004
-[1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1]
-[0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1]
-[0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1]
-[0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1]
-[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]
-[1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1]
-[0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1]
-[0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1]
-##################################
+n=7 , m=10      graph gap = 1          projection gap = 2         factor=2          max=2.0
+[1 1 1 0 0 0 0 0 0 0 0]
+[0 0 0 1 1 1 0 0 0 0 1]
+[0 0 0 0 0 0 1 1 1 0 1]
+[0 0 0 0 0 0 0 0 0 1 1]
+[1 0 0 1 0 0 1 0 0 0 1]
+[0 1 0 0 1 0 0 1 0 0 1]
+[0 0 1 0 0 1 0 0 1 1 1]
diff --git a/projection.sage b/projection.sage
index a15d080..e12b9c1 100644
--- a/projection.sage
+++ b/projection.sage
@@ -1,41 +1,67 @@
 # cogirth-packing gap of projections of graphic matroids
+# see if gap(projection) <= 2 * gap(graph)
 
 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 cogirthip(bases, integral=true):
+def base_hittingset_with_callback(M, integral=true):
+    # model
     model = gp.Model("mip1",env=env)
-    # model.Params.LogToConsole = 0
-    groundset=frozenset()
-    for B in bases: groundset=groundset|frozenset(B)
+    groundset = M.groundset()
     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:
+        for e in groundset: x[e]=model.addVar(vtype=GRB.CONTINUOUS,lb=0,ub=1)
+    # there is no lazy constraint for LP...
+    for B in M.bases():
         model.addConstr(gp.quicksum([x[e] for e in B])>=1)
-    model.optimize()
+    model.setObjective(gp.quicksum([x[e] for e in groundset]), GRB.MINIMIZE)
+    # callback
+    def callback_func(model, where):
+        if integral and where == GRB.Callback.MIPSOL:
+            sol_values = {key: model.cbGetSolution(var) 
+                          for key, var in x.items()}
+            # find min weight base in the matroid
+            base = frozenset()
+            minweight=0
+            for e in sorted(groundset, key=lambda ee: sol_values[ee]):
+                if M.is_independent(base.union([e])):
+                    base=base.union([e])
+                    minweight=minweight+sol_values[e]
+            if minweight < 1:
+                model.cbLazy(gp.quicksum([x[e] for e in base])>=1)
+    # solve
+    if integral:
+        model.Params.Heuristics = 0
+        model.Params.LazyConstraints = 1
+        model.optimize(callback_func)
+    else:
+        model.optimize()
     return model.ObjVal
 
 
 cnt=0   # actual number of instances tested
-maxgap=0
+maxfactor=0
 f = lambda g: g.is_connected()
 for N in range(4,10):
     for g in filter(f, graphs(N)):
+        MG=Matroid(g)
+        mincut=base_hittingset_with_callback(MG)
+        treepacking=base_hittingset_with_callback(MG,integral=false)
+        g_gap=mincut/treepacking
         A=g.incidence_matrix()
         n,m = A.dimensions()
         # enumerate all vectors in F_2^n with even number of 1s
+        m_gap=0
         V = VectorSpace(GF(2), N)
         for v in filter(lambda v: v.hamming_weight() % 2 == 0 and v!=0, V):
             cnt=cnt+1
@@ -44,17 +70,27 @@ for N in range(4,10):
             # print(A_t)
             M=Matroid(matrix=A_t,field=GF(2))/m     #contract the last element
             # print(M)
-            bases=M.bases()
-            strength=cogirthip(bases,integral=false)
-            cogirth =cogirthip(bases,integral=true)
+            cogirth = base_hittingset_with_callback(M,integral=true)
+            strength = base_hittingset_with_callback(M,integral=false)
             gap = cogirth/strength
+            m_gap=max(m_gap,gap)
             # maxgap=max(gap,maxgap)
-            if gap > maxgap:
-                maxgap = gap
-                print(f"find a large gap: {gap}")
-                with open("projection.out", "a") as file:
-                    file.write("##################################\n"
-                               +str(gap)+"\n"+str(A_t)
-                               +"\n##################################\n")
-            if cnt%100==0:
-                print(f"#{cnt}, n={n}, max gap = {maxgap}")
\ No newline at end of file
+            # if gap > maxgap:
+            #     maxgap = gap
+            #     print(f"find a large gap: {gap}")
+            #     with open("projection.out", "a") as file:
+            #         file.write("##################################\n"
+            #                    +str(gap)+"\n"+str(A_t)
+            #                    +"\n##################################\n")
+        factor=m_gap/g_gap
+        fr = lambda xx:  str(Fraction(xx).limit_denominator(m))
+        maxfactor=max(maxfactor,factor)
+        print(f"n={n:<2}, m={m:<4}    graph gap = {fr(g_gap):8}   projection gap = {fr(m_gap):8}  factor={fr(factor):8}   max={maxfactor}")
+        if factor >= 2:
+            with open("projection.out", "a") as file:
+                file.write(f"n={n:<2}, m={m:<4}    graph gap = {fr(g_gap):8}   projection gap = {fr(m_gap):8}  factor={fr(factor):8}   max={maxfactor}\n"
+                            +str(A_t)+"\n")
+        if factor > 2:
+            print("conjecture is wrong")
+            print(str(A_t))
+            exit(1)
\ No newline at end of file