"""Functions for computing treewidth decomposition.
Treewidth of an undirected graph is a number associated with the graph.
It can be defined as the size of the largest vertex set (bag) in a tree
decomposition of the graph minus one.
`Wikipedia: Treewidth <https://en.wikipedia.org/wiki/Treewidth>`_
The notions of treewidth and tree decomposition have gained their
attractiveness partly because many graph and network problems that are
intractable (e.g., NP-hard) on arbitrary graphs become efficiently
solvable (e.g., with a linear time algorithm) when the treewidth of the
input graphs is bounded by a constant [1]_ [2]_.
There are two different functions for computing a tree decomposition:
:func:`treewidth_min_degree` and :func:`treewidth_min_fill_in`.
.. [1] Hans L. Bodlaender and Arie M. C. A. Koster. 2010. "Treewidth
computations I.Upper bounds". Inf. Comput. 208, 3 (March 2010),259-275.
http://dx.doi.org/10.1016/j.ic.2009.03.008
.. [2] Hans L. Bodlaender. "Discovering Treewidth". Institute of Information
and Computing Sciences, Utrecht University.
Technical Report UU-CS-2005-018.
http://www.cs.uu.nl
.. [3] K. Wang, Z. Lu, and J. Hicks *Treewidth*.
http://web.eecs.utk.edu/~cphillip/cs594_spring2015_projects/treewidth.pdf
"""
import sys
import networkx as nx
from networkx.utils import not_implemented_for
from heapq import heappush, heappop, heapify
import itertools
__all__ = ["treewidth_min_degree", "treewidth_min_fill_in"]
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def treewidth_min_degree(G):
""" Returns a treewidth decomposition using the Minimum Degree heuristic.
The heuristic chooses the nodes according to their degree, i.e., first
the node with the lowest degree is chosen, then the graph is updated
and the corresponding node is removed. Next, a new node with the lowest
degree is chosen, and so on.
Parameters
----------
G : NetworkX graph
Returns
-------
Treewidth decomposition : (int, Graph) tuple
2-tuple with treewidth and the corresponding decomposed tree.
"""
deg_heuristic = MinDegreeHeuristic(G)
return treewidth_decomp(G, lambda graph: deg_heuristic.best_node(graph))
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def treewidth_min_fill_in(G):
""" Returns a treewidth decomposition using the Minimum Fill-in heuristic.
The heuristic chooses a node from the graph, where the number of edges
added turning the neighbourhood of the chosen node into clique is as
small as possible.
Parameters
----------
G : NetworkX graph
Returns
-------
Treewidth decomposition : (int, Graph) tuple
2-tuple with treewidth and the corresponding decomposed tree.
"""
return treewidth_decomp(G, min_fill_in_heuristic)
class MinDegreeHeuristic:
""" Implements the Minimum Degree heuristic.
The heuristic chooses the nodes according to their degree
(number of neighbours), i.e., first the node with the lowest degree is
chosen, then the graph is updated and the corresponding node is
removed. Next, a new node with the lowest degree is chosen, and so on.
"""
def __init__(self, graph):
self._graph = graph
# nodes that have to be updated in the heap before each iteration
self._update_nodes = []
self._degreeq = [] # a heapq with 2-tuples (degree,node)
# build heap with initial degrees
for n in graph:
self._degreeq.append((len(graph[n]), n))
heapify(self._degreeq)
def best_node(self, graph):
# update nodes in self._update_nodes
for n in self._update_nodes:
# insert changed degrees into degreeq
heappush(self._degreeq, (len(graph[n]), n))
# get the next valid (minimum degree) node
while self._degreeq:
(min_degree, elim_node) = heappop(self._degreeq)
if elim_node not in graph or len(graph[elim_node]) != min_degree:
# outdated entry in degreeq
continue
elif min_degree == len(graph) - 1:
# fully connected: abort condition
return None
# remember to update nodes in the heap before getting the next node
self._update_nodes = graph[elim_node]
return elim_node
# the heap is empty: abort
return None
def min_fill_in_heuristic(graph):
""" Implements the Minimum Degree heuristic.
Returns the node from the graph, where the number of edges added when
turning the neighbourhood of the chosen node into clique is as small as
possible. This algorithm chooses the nodes using the Minimum Fill-In
heuristic. The running time of the algorithm is :math:`O(V^3)` and it uses
additional constant memory."""
if len(graph) == 0:
return None
min_fill_in_node = None
min_fill_in = sys.maxsize
# create sorted list of (degree, node)
degree_list = [(len(graph[node]), node) for node in graph]
degree_list.sort()
# abort condition
min_degree = degree_list[0][0]
if min_degree == len(graph) - 1:
return None
for (_, node) in degree_list:
num_fill_in = 0
nbrs = graph[node]
for nbr in nbrs:
# count how many nodes in nbrs current nbr is not connected to
# subtract 1 for the node itself
num_fill_in += len(nbrs - graph[nbr]) - 1
if num_fill_in >= 2 * min_fill_in:
break
num_fill_in /= 2 # divide by 2 because of double counting
if num_fill_in < min_fill_in: # update min-fill-in node
if num_fill_in == 0:
return node
min_fill_in = num_fill_in
min_fill_in_node = node
return min_fill_in_node
def treewidth_decomp(G, heuristic=min_fill_in_heuristic):
""" Returns a treewidth decomposition using the passed heuristic.
Parameters
----------
G : NetworkX graph
heuristic : heuristic function
Returns
-------
Treewidth decomposition : (int, Graph) tuple
2-tuple with treewidth and the corresponding decomposed tree.
"""
# make dict-of-sets structure
graph = {n: set(G[n]) - {n} for n in G}
# stack containing nodes and neighbors in the order from the heuristic
node_stack = []
# get first node from heuristic
elim_node = heuristic(graph)
while elim_node is not None:
# connect all neighbours with each other
nbrs = graph[elim_node]
for u, v in itertools.permutations(nbrs, 2):
if v not in graph[u]:
graph[u].add(v)
# push node and its current neighbors on stack
node_stack.append((elim_node, nbrs))
# remove node from graph
for u in graph[elim_node]:
graph[u].remove(elim_node)
del graph[elim_node]
elim_node = heuristic(graph)
# the abort condition is met; put all remaining nodes into one bag
decomp = nx.Graph()
first_bag = frozenset(graph.keys())
decomp.add_node(first_bag)
treewidth = len(first_bag) - 1
while node_stack:
# get node and its neighbors from the stack
(curr_node, nbrs) = node_stack.pop()
# find a bag all neighbors are in
old_bag = None
for bag in decomp.nodes:
if nbrs <= bag:
old_bag = bag
break
if old_bag is None:
# no old_bag was found: just connect to the first_bag
old_bag = first_bag
# create new node for decomposition
nbrs.add(curr_node)
new_bag = frozenset(nbrs)
# update treewidth
treewidth = max(treewidth, len(new_bag) - 1)
# add edge to decomposition (implicitly also adds the new node)
decomp.add_edge(old_bag, new_bag)
return treewidth, decomp