def reduction_from_kcol_to_3col(G, k):
# add the main gadget
T = lambda *x: "dummy(" + ",".join(map(str, x)) + ')'
V = lambda v, c: str(v) + '(' + str(c) + ')'
if k == 3: return G
H = Graph()
# Create the color blue
H.add_named_vertex('c1')
H.add_named_vertex('c2')
H.add_named_vertex('c3')
H.add_edge('c1', 'c2')
H.add_edge('c1', 'c3')
H.add_edge('c2', 'c3')
# add vertices
for v in G:
H.add_named_vertex(V(v, 0))
H.add_edge(V(v, 0), 'c1')
H.add_edge(V(v, 0), 'c3')
H.add_named_vertex(V(v, k))
H.add_edge(V(v, k), 'c2')
H.add_edge(V(v, k), 'c3')
for i in range(1, k):
H.add_named_vertex(V(v, i))
H.add_edge('c3', V(v, i))
for u, v in G.edges:
for i in range(1, k + 1):
H.add_named_vertex(T(u, v, u, i))
H.add_edge(V(u, i - 1), T(u, v, u, i))
H.add_edge(V(u, i), T(u, v, u, i))
H.add_named_vertex(T(u, v, v, i))
H.add_edge(V(v, i - 1), T(u, v, v, i))
H.add_edge(V(v, i), T(u, v, v, i))
H.add_edge(T(u, v, u, i), T(u, v, v, i))
H.set_vertex_index()
return H
def load_from_edge_list_named(strFileName, G = None):
with open(strFileName, 'r') as f:
lines = f.readlines()
if G is None: G = Graph()
for line in lines:
if len(line) == 0 or line == '\n': continue
fields = line.split()
if not len(fields): continue
if fields[0] == 'e':
u, v = nvertex(fields[1]), nvertex(fields[2])
if u not in V(G):
G.add_named_vertex(u)
if v not in V(G):
G.add_named_vertex(v)
G.add_edge(u, v)
G.set_vertex_index()
return G
def reduce_3sat_to_3col(instance, G = None):
"""reduces a 3sat instance to a graph 3-coloring instance
receives a graph G
for each clause (a,b,c)
gadget:
(-a)----(a)---(g1)
| \
| (g3)---(g4) (X)
| / | \ / |
(-b)----(b)---(g2) | (T) |
| / \ |
(-c)----(c)-------------(g5) (F)
X is adjacent to all variables
"""
G = Graph() if G is None else G
# add common gadget
G.add_named_vertex('T')
G.add_named_vertex('F')
G.add_named_vertex('X')
G.add_edge('T', 'F')
G.add_edge('F', 'X')
G.add_edge('X', 'T')
# add gadget for variables
variables = sorted(set([abs(v) for clause in instance for v in clause]))
for v in variables:
G.add_named_vertex(v)
G.add_named_vertex(-v)
G.add_edge('X', v)
G.add_edge('X', -v)
G.add_edge(v, -v)
G.set_vertex_index(max(variables) + 1)
# add the clause gadgets
for a, b, c in instance:
g1, g2, g3, g4, g5 = [G.add_vertex() for _i in range(5)]
# triangle 1,2,3
G.add_edge(g1, g2) # 1
G.add_edge(g2, g3) # 2
G.add_edge(g3, g1) # 3
# bridge betwen triangle 1,2,3T and 4,5,T
G.add_edge(g3, g4) # 4
# triangle 3,4,5
G.add_edge(g4, g5) # 5
G.add_edge(g5, 'T') # 6
G.add_edge('T', g4) # 7
# edges for clause a,b,c
G.add_edge(a, g1) # 8
G.add_edge(b, g2) # 9
G.add_edge(c, g5) # 10
return G