本文整理汇总了Python中cnfformula.cnf.CNF.loose_majority_constraint方法的典型用法代码示例。如果您正苦于以下问题：Python CNF.loose_majority_constraint方法的具体用法？Python CNF.loose_majority_constraint怎么用？Python CNF.loose_majority_constraint使用的例子？那么恭喜您, 这里精选的方法代码示例或许可以为您提供帮助。您也可以进一步了解该方法所在类cnfformula.cnf.CNF的用法示例。
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示例1: SubsetCardinalityFormula
# 需要导入模块: from cnfformula.cnf import CNF [as 别名]
# 或者: from cnfformula.cnf.CNF import loose_majority_constraint [as 别名]
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)
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:
for cls in CNF.exactly_half_ceil(edge_vars):
ssc.add_clause(cls, strict=True)
else:
for cls in CNF.loose_majority_constraint(edge_vars):
ssc.add_clause(cls, strict=True)
for v in Right:
edge_vars = [var_name(u, v) for u in neighbors(B, v)]
if equalities:
for cls in CNF.exactly_half_floor(edge_vars):
ssc.add_clause(cls, strict=True)
else:
#.........这里部分代码省略.........