PGM results and bipartite code

bipartite.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
+
+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(2,10):
+    for M in range(N,10):
+        g=graphs.CompleteBipartiteGraph(N,M)
+        A=g.incidence_matrix()
+        n,m = A.dimensions()
+        maxgap = 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
+            # 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
+
+        print(f"K_{{{N},{M}}}, gap = {maxgap}, column = {v}")

projection.out

@@ -0,0 +1,108 @@
+##################################
+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]
+##################################

projection.sage

@@ -29,8 +29,9 @@ def cogirthip(bases, integral=true):
 
 
 cnt=0   # actual number of instances tested
+maxgap=0
 f = lambda g: g.is_connected()
-for N in range(8,20):
+for N in range(4,10):
     for g in filter(f, graphs(N)):
         A=g.incidence_matrix()
         n,m = A.dimensions()
@@ -47,9 +48,13 @@ for N in range(8,20):
             strength=cogirthip(bases,integral=false)
             cogirth =cogirthip(bases,integral=true)
             gap = cogirth/strength
-            if gap > 3.01:
+            # maxgap=max(gap,maxgap)
-                print(f"bad example! gap={gap}")
+            if gap > maxgap:
+                maxgap = gap
+                print(f"find a large gap: {gap}")
                 with open("projection.out", "a") as file:
-                    file.write(str(gap)+"\n"+str(A_t)+"\n")
+                    file.write("##################################\n"
+                               +str(gap)+"\n"+str(A_t)
+                               +"\n##################################\n")
             if cnt%100==0:
-                print(f"#{cnt},n={n},m={m}")
+                print(f"#{cnt}, n={n}, max gap = {maxgap}")