85 lines
3.1 KiB
Python
85 lines
3.1 KiB
Python
'''
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# cogirth-packing gap of projections of graphic matroids
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K_{2,2} gap ≈ 3/2 column = (0, 0, 0, 0)
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K_{2,3} gap ≈ 3/2 column = (1, 0, 1, 1, 1)
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K_{2,4} gap ≈ 2 column = (0, 0, 1, 1, 1, 1)
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K_{2,5} gap ≈ 2 column = (1, 0, 1, 1, 1, 1, 1)
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K_{2,6} gap ≈ 2 column = (1, 1, 1, 1, 1, 1, 1, 1)
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K_{2,7} gap ≈ 2 column = (1, 0, 1, 1, 1, 1, 1, 1, 1)
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K_{2,8} gap ≈ 2 column = (0, 0, 1, 1, 1, 1, 1, 1, 1, 1)
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K_{2,9} gap ≈ 2 column = (1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1)
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K_{3,3} gap ≈ 16/9 column = (1, 1, 1, 1, 1, 1)
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K_{3,4} gap ≈ 5/3 column = (0, 0, 0, 1, 1, 1, 1)
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K_{3,5} gap ≈ 12/5 column = (1, 1, 1, 1, 1, 1, 1, 1)
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K_{3,6} gap ≈ 7/3 column = (0, 0, 0, 1, 1, 1, 1, 1, 1)
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K_{3,7} gap ≈ 16/7 column = (1, 0, 0, 1, 1, 1, 1, 1, 1, 1)
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K_{3,8} gap ≈ 9/4 column = (1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1)
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K_{3,9} gap ≈ 20/9 column = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
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K_{4,4} gap ≈ 9/4 column = (1, 1, 1, 1, 1, 1, 1, 1)
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K_{4,5} gap ≈ 7/4 column = (1, 0, 0, 0, 1, 1, 1, 1, 1)
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K_{4,6} gap ≈ 8/3 column = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
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K_{4,7} gap ≈ 9/4 column = (1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1)
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'''
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from sage.all import *
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from sage.matroids.all import *
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from sage.graphs.all import *
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import gurobipy as gp
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from gurobipy import GRB
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from fractions import Fraction
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env = gp.Env(empty=True)
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env.setParam("OutputFlag",0)
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env.start()
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def representative_vectors(m, n):
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for w1 in range(m+1):
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for w2 in range(n+1):
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v = [0]*(m+n)
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v[:w1] = [1]*w1
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v[m:m+w2] = [1]*w2
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yield tuple(v)
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def cogirthip(bases, integral=true):
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model = gp.Model("mip1",env=env)
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# model.Params.LogToConsole = 0
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groundset=frozenset()
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for B in bases: groundset=groundset|frozenset(B)
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x = dict()
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if integral:
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for e in groundset: x[e]=model.addVar(vtype=GRB.BINARY)
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else:
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for e in groundset: x[e]=model.addVar(vtype=GRB.CONTINUOUS,lb=0)
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model.setObjective(gp.quicksum([x[e] for e in groundset]), GRB.MINIMIZE)
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for B in bases:
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model.addConstr(gp.quicksum([x[e] for e in B])>=1)
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model.optimize()
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return model.ObjVal
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# cnt=0 # actual number of instances tested
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# f = lambda g: g.is_connected()
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for N in range(2,10):
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for M in range(N,10):
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g=graphs.CompleteBipartiteGraph(N,M)
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A=g.incidence_matrix()
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n,m = A.dimensions()
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maxgap = 0
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for v in representative_vectors(N,M):
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# cnt=cnt+1
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v_col=matrix(v).transpose()
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A_t=A.augment(v_col)
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# print(A_t)
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MM=Matroid(matrix=A_t,field=GF(2))/m #contract the last element
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# print(M)
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bases=MM.bases()
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strength=cogirthip(bases,integral=false)
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cogirth =cogirthip(bases,integral=true)
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gap = cogirth/strength
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if gap > maxgap:
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maxgap = gap
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maxcol = v
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frac=str(Fraction(maxgap).limit_denominator(m))
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print(f"K_{{{N},{M}}} gap ≈ {frac:<8} column = {maxcol}")
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