Vertex separation¶

This module implements several algorithms to compute the vertex separation of a digraph and the corresponding ordering of the vertices. Given an ordering $v_1, ..., v_n$ of the vertices of $V(G)$ , its cost is defined as:

$c(v_1, ..., v_n) = \max_{1\leq i \leq n} c'(\{v_1, ..., v_i\})$

Where

$c'(S) = |N^+_G(S)\backslash S|$

The vertex separation of a digraph $G$ is equal to the minimum cost of an ordering of its vertices.

Vertex separation and pathwidth

The vertex separation is defined on a digraph, but one can obtain from a graph $G$ a digraph $D$ with the same vertex set, and in which each edge $uv$ of $G$ is replaced by two edges $uv$ and $vu$ in $D$ . The vertex separation of $D$ is equal to the pathwidth of $G$ , and the corresponding ordering of the vertices of $D$ encodes an optimal path-decomposition of $G$ .

This is a result of Kinnersley [Kin92] and Bodlaender [Bod98].

References

[Kin92]

The vertex separation number of a graph equals its path-width, Nancy G. Kinnersley, Information Processing Letters, Volume 42, Issue 6, Pages 345-350, 24 July 1992

[Bod98]

A partial k-arboretum of graphs with bounded treewidth, Hans L. Bodlaender, Theoretical Computer Science, Volume 209, Issues 1-2, Pages 1-45, 6 December 1998

Authors

Nathann Cohen

Exponential algorithm for vertex separation¶

In order to find an optimal ordering of the vertices for the vertex separation, this algorithm tries to save time by computing the function $c'(S)$ at most once once for each of the sets $S\subseteq V(G)$ . These values are stored in an array of size $2^n$ where reading the value of $c'(S)$ or updating it can be done in constant (and small) time.

Assuming that we can compute the cost of a set $S$ and remember it, finding an optimal ordering is an easy task. Indeed, we can think of the sequence $v_1, ..., v_n$ of vertices as a sequence of sets $\{v_1\}, \{v_1,v_2\}, ..., \{v_1,...,v_n\}$ , whose cost is precisely $\max c'(\{v_1\}), c'(\{v_1,v_2\}), ... , c'(\{v_1,...,v_n\})$ . Hence, when considering the digraph on the $2^n$ sets $S\subseteq V(G)$ where there is an arc from $S$ to $S'$ if $S'=S\cap \{v\}$ for some $v$ (that is, if the sets $S$ and $S'$ can be consecutive in a sequence), an ordering of the vertices of $G$ corresponds to a path from $\emptyset$ to $\{v_1,...,v_n\}$ . In this setting, checking whether there exists a ordering of cost less than $k$ can be achieved by checking whether there exists a directed path $\emptyset$ to $\{v_1,...,v_n\}$ using only sets of cost less than $k$ . This is just a depth-first-search, for each $k$ .

Lazy evaluation of $c'$

In the previous algorithm, most of the time is actually spent on the computation of $c'(S)$ for each set $S\subseteq V(G)$ – i.e. $2^n$ computations of neighborhoods. This can be seen as a huge waste of time when noticing that it is useless to know that the value $c'(S)$ for a set $S$ is less than $k$ if all the paths leading to $S$ have a cost greater than $k$ . For this reason, the value of $c'(S)$ is computed lazily during the depth-first search. Explanation :

When the depth-first search discovers a set of size less than $k$ , the costs of its out-neighbors (the potential sets that could follow it in the optimal ordering) are evaluated. When an out-neighbor is found that has a cost smaller than $k$ , the depth-first search continues with this set, which is explored with the hope that it could lead to a path toward $\{v_1,...,v_n\}$ . On the other hand, if an out-neighbour has a cost larger than $k$ it is useless to attempt to build a cheap sequence going though this set, and the exploration stops there. This way, a large number of sets will never be evaluated and a lot of computational time is saved this way.

Besides, some improvement is also made by “improving” the values found by $c'$ . Indeed, $c'(S)$ is a lower bound on the cost of a sequence containing the set $S$ , but if all out-neighbors of $S$ have a cost of $c'(S) + 5$ then one knows that having $S$ in a sequence means a total cost of at least $c'(S) + 5$ . For this reason, for each set $S$ we store the value of $c'(S)$ , and replace it by $\max (c'(S), \min_{\text{next}})$ (where $\min_{\text{next}}$ is the minimum of the costs of the out-neighbors of $S$ ) once the costs of these out-neighbors have been evaluated by the algrithm.

Note

Because of its current implementation, this algorithm only works on graphs on less than 32 vertices. This can be changed to 64 if necessary, but 32 vertices already require 4GB of memory.

Lower bound on the vertex separation

One can obtain a lower bound on the vertex separation of a graph in exponential time but small memory by computing once the cost of each set $S$ . Indeed, the cost of a sequence $v_1, ..., v_n$ corresponding to sets $\{v_1\}, \{v_1,v_2\}, ..., \{v_1,...,v_n\}$ is

$\max c'(\{v_1\}),c'(\{v_1,v_2\}),...,c'(\{v_1,...,v_n\})\geq\max c'_1,...,c'_n$

where $c_i$ is the minimum cost of a set $S$ on $i$ vertices. Evaluating the $c_i$ can take time (and in particular more than the previous exact algorithm), but it does not need much memory to run.

Methods¶

class sage.graphs.graph_decompositions.vertex_separation.FastDigraph¶

Bases: object

x.__init__(...) initializes x; see help(type(x)) for signature

print_adjacency_matrix()¶

sage.graphs.graph_decompositions.vertex_separation.lower_bound(G)¶

Returns a lower bound on the vertex separation of $G$

INPUT:

G – a digraph

OUTPUT:

A lower bound on the vertex separation of $D$ (see the module’s documentation).

Note

This method runs in exponential time but has no memory constraint.

EXAMPLE:

On a cycle:

sage: from sage.graphs.graph_decompositions.vertex_separation import lower_bound
sage: g = digraphs.Circuit(6)
sage: lower_bound(g)
1

TEST:

Given anything else than a DiGraph:

sage: from sage.graphs.graph_decompositions.vertex_separation import lower_bound
sage: g = graphs.CycleGraph(5)
sage: lower_bound(g)
Traceback (most recent call last):
...
ValueError: The parameter must be a DiGraph.

sage.graphs.graph_decompositions.vertex_separation.path_decomposition(G, verbose=False)¶

Returns the pathwidth of the given graph and the ordering of the vertices resulting in a corresponding path decomposition.

INPUT:

G – a digraph
verbose (boolean) – whether to display information on the computations.

OUTPUT:

A pair (cost, ordering) representing the optimal ordering of the vertices and its cost.

Note

Because of its current implementation, this algorithm only works on graphs on less than 32 vertices. This can be changed to 54 if necessary, but 32 vertices already require 4GB of memory.

EXAMPLE:

The vertex separation of a circuit is equal to 2:

sage: from sage.graphs.graph_decompositions.vertex_separation import path_decomposition
sage: g = graphs.CycleGraph(6)
sage: path_decomposition(g)
(2, [0, 1, 2, 3, 4, 5])

TEST:

Given anything else than a Graph:

sage: from sage.graphs.graph_decompositions.vertex_separation import path_decomposition
sage: g = digraphs.Circuit(6)
sage: path_decomposition(g)
Traceback (most recent call last):
...
ValueError: The parameter must be a Graph.

sage.graphs.graph_decompositions.vertex_separation.test_popcount()¶

Correction test for popcount32.

EXAMPLE:

sage: from sage.graphs.graph_decompositions.vertex_separation import test_popcount
sage: test_popcount() # not tested

sage.graphs.graph_decompositions.vertex_separation.vertex_separation(G, verbose=False)¶

Returns an optimal ordering of the vertices and its cost for vertex-separation.

INPUT:

G – a digraph
verbose (boolean) – whether to display information on the computations.

OUTPUT:

A pair (cost, ordering) representing the optimal ordering of the vertices and its cost.

Note

Because of its current implementation, this algorithm only works on graphs on less than 32 vertices. This can be changed to 54 if necessary, but 32 vertices already require 4GB of memory.

EXAMPLE:

The vertex separation of a circuit is equal to 2:

sage: from sage.graphs.graph_decompositions.vertex_separation import vertex_separation
sage: g = digraphs.Circuit(6)
sage: vertex_separation(g)
(1, [0, 1, 2, 3, 4, 5])        

TEST:

Given anything else than a DiGraph:

sage: from sage.graphs.graph_decompositions.vertex_separation import lower_bound
sage: g = graphs.CycleGraph(5)
sage: lower_bound(g)
Traceback (most recent call last):
...
ValueError: The parameter must be a DiGraph.

Graphs with non-integer vertices:

sage: D=digraphs.DeBruijn(2,3)
sage: vertex_separation(D)
(2, ['000', '001', '100', '010', '101', '011', '110', '111'])

Vertex separation¶

Exponential algorithm for vertex separation¶

Methods¶

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Exponential algorithm for vertex separation¶

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