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Minimum Dominating Set API Reference

Data

Data model for Minimum Dominating Set use case.

MdsData

Bases: UcData

Data for the Minimum Dominating Set use case.

Finds a minimum-size set of nodes such that every node is either in the set or adjacent to a member of the set.

Attributes:

Name Type Description
name Literal['minimum_dominating_set']

Identifier for this data type.
adjacency_matrix BinAdjMatrix

Symmetric binary adjacency matrix.
node_names list[int | str]

Node identifiers.

plot(*, ax: Axes | None = None) -> Axes

Plot the Minimum Dominating Set graph instance.

Parameters:

Name Type Description Default
ax Axes | None

Matplotlib axes to draw on. Creates a new figure if None.

 None

Returns:

Type Description
Axes

The axes with the plot.

to_string() -> str

Format the data as a human-readable string.

Returns:

Type Description
str

String representation of the data.

from_adjacency_matrix(adjacency_matrix: np.ndarray, node_names: list[int | str]) -> MdsData staticmethod

Create MdsData from an adjacency matrix.

Parameters:

Name Type Description Default
adjacency_matrix ndarray

Symmetric binary adjacency matrix.

 required
node_names list[int | str]

List of node identifiers.

 required

Returns:

Type Description
MdsData

The Minimum Dominating Set data instance.

Raises:

Type Description
ValueError

If the node_names length doesn't match the matrix, or if node_names contains duplicates.

from_graph(graph: nx.Graph) -> MdsData staticmethod

Create MdsData from a NetworkX graph.

Parameters:

Name Type Description Default
graph Graph

A NetworkX graph.

 required

Returns:

Type Description
MdsData

The Minimum Dominating Set data instance.

generate_random(n_nodes: int = 6, edge_prob: float = 0.4, seed: int | None = None) -> MdsData staticmethod

Generate a random Minimum Dominating Set instance.

Parameters:

Name Type Description Default
n_nodes int

Number of nodes, by default 6.

 6
edge_prob float

Probability of an edge between any two nodes, by default 0.4.

 0.4
seed int | None

Random seed for reproducibility, by default None.

 None

Returns:

Type Description
MdsData

A randomly generated data instance.

Examples:

>>> data = MdsData.generate_random(n_nodes=8, seed=42)

Formulation

Formulation for Minimum Dominating Set use case.

MdsFormulation

Bases: UcFormulation[MdsData, MdsSolution]

Constraint-based formulation for Minimum Dominating Set.

Mathematical Formulation 
Symbols:
    n -- number of nodes in the graph.
    N(i) -- the set of neighbours of node i (nodes j with an edge to i).

Decision Variables:
    x_i in {0, 1} -- 1 if node i is in the dominating set, 0 otherwise.

Objective:
    minimize sum_i x_i

Constraints:
    For each node i: x_i + sum_{j in N(i)} x_j >= 1
    (every node must be selected or adjacent to a selected node).

to_string(data: MdsData) -> str staticmethod

Format the formulation as a string.

Parameters:

Name Type Description Default
data MdsData

The problem data.

 required

Returns:

Type Description
str

Formatted description of the formulation.

formulate(data: MdsData) -> Model staticmethod

Formulate the Minimum Dominating Set problem as a constraint model.

Parameters:

Name Type Description Default
data MdsData

The problem data containing the graph structure.

 required

Returns:

Type Description
Model

A Luna Model ready to be solved.

interpret(solution: Solution, data: MdsData) -> MdsSolution staticmethod

Extract a Minimum Dominating Set solution from the solver result.

Parameters:

Name Type Description Default
solution Solution

The solver solution.

 required
data MdsData

The original problem data.

 required

Returns:

Type Description
MdsSolution

Structured solution with the dominating set and validity.

Raises:

Type Description
NoSolutionFoundError

If the solver did not find any solution.

Solution

Solution model for Minimum Dominating Set use case.

MdsSolution

Bases: UcSolution

Solution for the Minimum Dominating Set use case.

Attributes:

Name Type Description
name Literal['minimum_dominating_set']

Identifier.
dominating_set list[int | str]

Nodes in the dominating set.
set_size int

Size of the dominating set.
is_valid bool

Whether every node is selected or adjacent to a selected node.

plot(data: MdsData | None = None, *, ax: Axes | None = None) -> Axes

Plot the Minimum Dominating Set solution on the problem graph.

Selected nodes are highlighted in green; other nodes are grey.

Parameters:

Name Type Description Default
data MdsData | None

Problem data used to reconstruct the graph. Required -- a ValueError is raised when None.

 None
ax Axes | None

Matplotlib axes to draw on. Creates a new figure if None.

 None

Returns:

Type Description
Axes

The axes with the plot.

Raises:

Type Description
ValueError

If data is None.

to_string() -> str

Format the solution as a human-readable string.

Returns:

Type Description
str

String representation of the solution.

Instance

Instance model for Minimum Dominating Set use case.

MdsInstance

Bases: UcInstance[MdsData, MdsFormulation, MdsSolution]

Instance combining data and formulation for Minimum Dominating Set.

Collection

Collection of Minimum Dominating Set instances.

MdsCollection

Bases: UcInstanceCollection[MdsInstance]

Collection of Minimum Dominating Set instances.

This collection provides methods to generate benchmark instances with various characteristics for testing and evaluation.

from_random(min_nodes: int, max_nodes: int, edge_prob: float = 0.4, num_instances: int = 1, *, seed: int | None = None) -> MdsCollection classmethod

Generate random Minimum Dominating Set instances.

Parameters:

Name Type Description Default
min_nodes int

Minimum number of nodes.

 required
max_nodes int

Maximum number of nodes.

 required
edge_prob float

Edge probability, by default 0.4.

 0.4
num_instances int

Number of instances per size, by default 1.

 1
seed int | None

Random seed for reproducibility, by default None.

 None

Returns:

Type Description
MdsCollection

Collection containing generated instances.

Examples:

>>> collection = MdsCollection.from_random(
...     min_nodes=5,
...     max_nodes=10,
...     num_instances=3,
...     seed=42,
... )

filter_infeasible(max_runtime: float = 3600, *, quiet: bool = True) -> list[bool]

Drop the instances of this collection that have no feasible solution.

Every instance is formulated and handed to SCIP, which stops as soon as it finds the first feasible solution. An instance is removed from the collection when SCIP proves the model infeasible, when no solution turns up within max_runtime, or when formulating it fails altogether. This keeps randomly generated instances from breaking a downstream pipeline.

Parameters:

Name Type Description Default
max_runtime float

SCIP time limit per instance in seconds. Must be positive. Defaults to 3600 seconds.

 3600
quiet bool

Suppress the SCIP solver output.

 True

Returns:

Type Description
list[bool]

Feasibility mask over the instances as they were before filtering, in that order: True where the instance was kept, False where it was removed.

Raises:

Type Description
ValueError

If max_runtime is not positive.