def EvenColoringFormula(G):
"""Even coloring formula
The formula is defined on a graph :math:`G` and claims that it is
possible to split the edges of the graph in two parts, so that
each vertex has an equal number of incident edges in each part.
The formula is defined on graphs where all vertices have even
degree. The formula is satisfiable only on those graphs with an
even number of vertices in each connected component [1]_.
Arguments
---------
G : networkx.Graph
a simple undirected graph where all vertices have even degree
Raises
------
ValueError
if the graph in input has a vertex with odd degree
Returns
-------
CNF object
References
----------
.. [1] Locality and Hard SAT-instances, Klas Markstrom
Journal on Satisfiability, Boolean Modeling and Computation 2 (2006) 221-228
"""
F = CNF()
F.mode_strict()
F.header = "Even coloring formula on graph " + G.name + "\n" + F.header
def var_name(u, v):
if u <= v:
return 'x_{{{0},{1}}}'.format(u, v)
else:
return 'x_{{{0},{1}}}'.format(v, u)
for (u, v) in enumerate_edges(G):
F.add_variable(var_name(u, v))
# Defined on both side
for v in enumerate_vertices(G):
if G.degree(v) % 2 == 1:
raise ValueError(
"Markstrom formulas requires all vertices to have even degree."
)
edge_vars = [var_name(u, v) for u in neighbors(G, v)]
# F.add_exactly_half_floor would work the same
F.add_exactly_half_ceil(edge_vars)
return F
def EvenColoringFormula(G):
"""Even coloring formula
The formula is defined on a graph :math:`G` and claims that it is
possible to split the edges of the graph in two parts, so that
each vertex has an equal number of incident edges in each part.
The formula is defined on graphs where all vertices have even
degree. The formula is satisfiable only on those graphs with an
even number of vertices in each connected component [1]_.
Arguments
---------
G : networkx.Graph
a simple undirected graph where all vertices have even degree
Raises
------
ValueError
if the graph in input has a vertex with odd degree
Returns
-------
CNF object
References
----------
.. [1] Locality and Hard SAT-instances, Klas Markstrom
Journal on Satisfiability, Boolean Modeling and Computation 2 (2006) 221-228
"""
F = CNF()
F.mode_strict()
F.header = "Even coloring formula on graph " + G.name + "\n" + F.header
def var_name(u,v):
if u<=v:
return 'x_{{{0},{1}}}'.format(u,v)
else:
return 'x_{{{0},{1}}}'.format(v,u)
for (u, v) in enumerate_edges(G):
F.add_variable(var_name(u, v))
# Defined on both side
for v in enumerate_vertices(G):
if G.degree(v) % 2 == 1:
raise ValueError("Markstrom formulas requires all vertices to have even degree.")
edge_vars = [ var_name(u, v) for u in neighbors(G, v) ]
F.add_exactly_half_ceil(edge_vars) # F.add_exactly_half_floor would work the same
return F
def SubsetCardinalityFormula(B, equalities=False):
r"""SubsetCardinalityFormula
Consider a bipartite graph :math:`B`. The CNF claims that at least half
of the edges incident to each of the vertices on left side of :math:`B`
must be zero, while at least half of the edges incident to each
vertex on the left side must be one.
Variants of these formula on specific families of bipartite graphs
have been studied in [1]_, [2]_ and [3]_, and turned out to be
difficult for resolution based SAT-solvers.
Each variable of the formula is denoted as :math:`x_{i,j}` where
:math:`\{i,j\}` is an edge of the bipartite graph. The clauses of
the CNF encode the following constraints on the edge variables.
For every left vertex i with neighborhood :math:`\Gamma(i)`
.. math::
\sum_{j \in \Gamma(i)} x_{i,j} \geq \frac{|\Gamma(i)|}{2}
For every right vertex j with neighborhood :math:`\Gamma(j)`
.. math::
\sum_{i \in \Gamma(j)} x_{i,j} \leq \frac{|\Gamma(j)|}{2}.
If the ``equalities`` flag is true, the constraints are instead
represented by equations.
.. math::
\sum_{j \in \Gamma(i)} x_{i,j} = \left\lceil \frac{|\Gamma(i)|}{2} \right\rceil
.. math::
\sum_{i \in \Gamma(j)} x_{i,j} = \left\lfloor \frac{|\Gamma(j)|}{2} \right\rfloor .
Parameters
----------
B : networkx.Graph
the graph vertices must have the 'bipartite' attribute
set. Left vertices must have it set to 0 and the right ones to 1.
A KeyException is raised otherwise.
equalities : boolean
use equations instead of inequalities to express the
cardinality constraints. (default: False)
Returns
-------
A CNF object
References
----------
.. [1] Mladen Miksa and Jakob Nordstrom
Long proofs of (seemingly) simple formulas
Theory and Applications of Satisfiability Testing--SAT 2014 (2014)
.. [2] Ivor Spence
sgen1: A generator of small but difficult satisfiability benchmarks
Journal of Experimental Algorithmics (2010)
.. [3] Allen Van Gelder and Ivor Spence
Zero-One Designs Produce Small Hard SAT Instances
Theory and Applications of Satisfiability Testing--SAT 2010(2010)
"""
Left, Right = bipartite_sets(B)
ssc = CNF()
ssc.header = "Subset cardinality formula for graph {0}\n".format(B.name)
ssc.mode_strict()
def var_name(u, v):
"""Compute the variable names."""
if u <= v:
return 'x_{{{0},{1}}}'.format(u, v)
else:
return 'x_{{{0},{1}}}'.format(v, u)
for u in Left:
for v in neighbors(B, u):
ssc.add_variable(var_name(u, v))
for u in Left:
edge_vars = [var_name(u, v) for v in neighbors(B, u)]
if equalities:
ssc.add_exactly_half_ceil(edge_vars)
else:
ssc.add_loose_majority(edge_vars)
for v in Right:
edge_vars = [var_name(u, v) for u in neighbors(B, v)]
if equalities:
ssc.add_exactly_half_floor(edge_vars)
else:
ssc.add_loose_minority(edge_vars)
return ssc
