Algorithms
GraphsOptim — ModuleGraphsOptimA package for graph optimization algorithms that rely on mathematical programming.
Flow
GraphsOptim.min_cost_flow — Functionmin_cost_flow(
+Algorithms · GraphsOptim.jl Algorithms
GraphsOptim — ModuleGraphsOptim
A package for graph optimization algorithms that rely on mathematical programming.
sourceFlow
GraphsOptim.min_cost_flow — Functionmin_cost_flow(
g, vertex_demand, edge_cost, edge_min_capacity, edge_max_capacity;
integer, optimizer
-)
Compute a minimum cost flow over a directed graph.
Arguments
g::Graphs.AbstractGraph: a directed graph G = (V, E)vertex_demand::AbstractVector: a vector in Rⱽ giving the flow requested by each vertex (should be positive for sinks, negative for sources and zero elsewhere)edge_cost::AbstractMatrix: a vector in Rᴱ giving the cost of a unit of flow on each edgeedge_min_capacity::AbstractMatrix: a vector in Rᴱ giving the minimum flow allowed on each edgeedge_max_capacity::AbstractMatrix: a vector in Rᴱ giving the maximum flow allowed on each edge
Keyword arguments
integer::Bool: whether the flow should be integer-valued or real-valuedoptimizer: JuMP-compatible solver (default is HiGHS.Optimizer)
sourceGraphsOptim.min_cost_flow! — Functionmin_cost_flow!(
+)
Compute a minimum cost flow over a directed graph.
Arguments
g::Graphs.AbstractGraph: a directed graph G = (V, E)vertex_demand::AbstractVector: a vector in Rⱽ giving the flow requested by each vertex (should be positive for sinks, negative for sources and zero elsewhere)edge_cost::AbstractMatrix: a vector in Rᴱ giving the cost of a unit of flow on each edgeedge_min_capacity::AbstractMatrix: a vector in Rᴱ giving the minimum flow allowed on each edgeedge_max_capacity::AbstractMatrix: a vector in Rᴱ giving the maximum flow allowed on each edge
Keyword arguments
integer::Bool: whether the flow should be integer-valued or real-valuedoptimizer: JuMP-compatible solver (default is HiGHS.Optimizer)
sourceGraphsOptim.min_cost_flow! — Functionmin_cost_flow!(
model,
g, vertex_demand, edge_cost, edge_min_capacity, edge_max_capacity;
var_name, integer
-)
Modify a JuMP model by adding the variable, constraints and objective necessary to compute a minimum cost flow over a directed graph.
The flow variable will be named var_name, see min_cost_flow for details on the other arguments.
sourceWe denote by:
- $f$ the edge flow variable
- $c$ the edge cost
- $a$ and $b$ the min and max edge capacity
- $d$ the vertex demand
The objective function is
\[\min_{f \in \mathbb{R}^E} \sum_{(u, v) \in E} c(u, v) f(u, v)\]
The edge capacity constraint dictates that for all $(u, v) \in E$,
\[a(u, v) \leq f(u, v) \leq b(u, v)\]
The flow conservation constraint with node demand dictates that for all $v \in V$,
\[f^-(v) = d(v) + f^+(v)\]
where the incoming flow $f^-(v)$ and outgoing flow $f^+(v)$ are defined as
\[f^-(v) = \sum_{u \in N^-(v)} f(u, v) \quad \text{and} \quad f^+(v) = \sum_{w \in N^+(v)} f(v, w)\]
Shortest Path
GraphsOptim.shortest_path — Functionshortest_path(
+)
Modify a JuMP model by adding the variable, constraints and objective necessary to compute a minimum cost flow over a directed graph.
The flow variable will be named var_name, see min_cost_flow for details on the other arguments.
sourceWe denote by:
- $f$ the edge flow variable
- $c$ the edge cost
- $a$ and $b$ the min and max edge capacity
- $d$ the vertex demand
The objective function is
\[\min_{f \in \mathbb{R}^E} \sum_{(u, v) \in E} c(u, v) f(u, v)\]
The edge capacity constraint dictates that for all $(u, v) \in E$,
\[a(u, v) \leq f(u, v) \leq b(u, v)\]
The flow conservation constraint with node demand dictates that for all $v \in V$,
\[f^-(v) = d(v) + f^+(v)\]
where the incoming flow $f^-(v)$ and outgoing flow $f^+(v)$ are defined as
\[f^-(v) = \sum_{u \in N^-(v)} f(u, v) \quad \text{and} \quad f^+(v) = \sum_{w \in N^+(v)} f(v, w)\]
Shortest Path
GraphsOptim.shortest_path — Functionshortest_path(
g, source, target, edge_cost;
integer, optimizer
-)
Compute the shortest path from a source to a target vertex over a graph. Returns a sequence of vertices from source to destination.
Arguments
g::Graphs.AbstractGraph: a directed or undirected graph G = (V, E)source::Int: source vertex of the pathtarget::Int: target vertex of the pathedge_cost::AbstractMatrix: a vector in Rᴱ giving the cost of traversing the edge
Keyword arguments
integer::Bool: whether the path should be integer-valued or real-valued (default is true)optimizer: JuMP-compatible solver (default is HiGHS.Optimizer)
sourceGraphsOptim.shortest_path! — Functionshortest_path!(
+)
Compute the shortest path from a source to a target vertex over a graph. Returns a sequence of vertices from source to destination.
Arguments
g::Graphs.AbstractGraph: a directed or undirected graph G = (V, E)source::Int: source vertex of the pathtarget::Int: target vertex of the pathedge_cost::AbstractMatrix: a vector in Rᴱ giving the cost of traversing the edge
Keyword arguments
integer::Bool: whether the path should be integer-valued or real-valued (default is true)optimizer: JuMP-compatible solver (default is HiGHS.Optimizer)
sourceGraphsOptim.shortest_path! — Functionshortest_path!(
model,
g, source, target, edge_cost;
var_name, integer
-)
Modify a JuMP model by adding the variable, constraints and objective necessary to compute the shortest path between 2 vertices over a graph.
The edge selection variable will be named var_name, see shortest_path for details on the other arguments.
sourceA special case of minimum cost flow without edge capacities, and where vertex demands are $0$ everywhere except at the source ($-1$) and target ($+1$).
Assignment
Work in progress Come back later!
GraphsOptim.min_cost_assignment — Functionmin_cost_assignment(
+)
Modify a JuMP model by adding the variable, constraints and objective necessary to compute the shortest path between 2 vertices over a graph.
The edge selection variable will be named var_name, see shortest_path for details on the other arguments.
sourceA special case of minimum cost flow without edge capacities, and where vertex demands are $0$ everywhere except at the source ($-1$) and target ($+1$).
Assignment
Work in progress Come back later!
GraphsOptim.min_cost_assignment — Functionmin_cost_assignment(
edge_cost;
optimizer
-)
Compute a minimum cost assignment over a bipartite graph.
Arguments
edge_cost::AbstractMatrix: a matrix in Rᵁˣⱽ giving the cost of matching u ∈ U to v ∈ V
Keyword arguments
integer::Bool: whether the flow should be integer-valued or real-valuedoptimizer: JuMP-compatible solver (default is HiGHS.Optimizer)
sourceGraphsOptim.min_cost_assignment! — Functionmin_cost_assignment!(
+)
Compute a minimum cost assignment over a bipartite graph.
Arguments
edge_cost::AbstractMatrix: a matrix in Rᵁˣⱽ giving the cost of matching u ∈ U to v ∈ V
Keyword arguments
integer::Bool: whether the flow should be integer-valued or real-valuedoptimizer: JuMP-compatible solver (default is HiGHS.Optimizer)
sourceGraphsOptim.min_cost_assignment! — Functionmin_cost_assignment!(
model,
edge_cost;
var_name, integer
-)
Modify a JuMP model by adding the variable, constraints and objective necessary to compute a minimum cost assignment over a bipartite graph.
The assignment variable will be named var_name, see min_cost_assignment for details on the other arguments.
sourceMinimum Vertex Cover
GraphsOptim.min_vertex_cover — Functionmin_vertex_cover(
+)
Modify a JuMP model by adding the variable, constraints and objective necessary to compute a minimum cost assignment over a bipartite graph.
The assignment variable will be named var_name, see min_cost_assignment for details on the other arguments.
sourceMinimum Vertex Cover
GraphsOptim.min_vertex_cover — Functionmin_vertex_cover(
g, optimizer
-)
Compute a minimum vertex cover of an undirected graph.
Arguments
g::Graphs.AbstractGraph: an undirected graph G = (V, E)
Keyword arguments
optimizer: JuMP-compatible solver (default is HiGHS.Optimizer)
sourceGraphsOptim.min_vertex_cover! — Functionmin_vertex_cover!(
+)
Compute a minimum vertex cover of an undirected graph.
Arguments
g::Graphs.AbstractGraph: an undirected graph G = (V, E)
Keyword arguments
optimizer: JuMP-compatible solver (default is HiGHS.Optimizer)
sourceGraphsOptim.min_vertex_cover! — Functionmin_vertex_cover!(
model, g;
var_name
-)
Modify a JuMP model by adding the variable, constraints and objective necessary to compute a minimum vertex cove of an undirected graph.
The vertex cover indicator variable will be named var_name
sourceFinds a subset $S \subset V$ of vertices of an undirected graph $G = (V,E)$ such that $\forall (u,v) \in E: u \in S \lor v \in S$
Maximum Weight Independent Set
GraphsOptim.maximum_weight_independent_set — Functionmaximum_weight_independent_set(g; optimizer, binary, vertex_weights)
Computes a maximum-weighted independent set or stable set of g.
sourceGraphsOptim.maximum_weight_independent_set! — Functionmaximum_independent_set!(model, g; var_name)
Computes in-place in the JuMP model a maximum-weighted independent set of g. An optional vertex_weights vector can be passed to the graph, defaulting to uniform weights (computing a maximum size independent set).
sourceFinds a subset $S \subset V$ of vertices of maximal weight of an undirected graph $G = (V,E)$ such that $\forall (u,v) \in E: u \notin S \lor v \notin S$.
Graph matching
Work in progress Come back later!
GraphsOptim.graph_matching — Functiongraph_matching(
+)
Modify a JuMP model by adding the variable, constraints and objective necessary to compute a minimum vertex cove of an undirected graph.
The vertex cover indicator variable will be named var_name
sourceFinds a subset $S \subset V$ of vertices of an undirected graph $G = (V,E)$ such that $\forall (u,v) \in E: u \in S \lor v \in S$
Maximum weight clique
GraphsOptim.maximum_weight_clique — Functionmaximum_weight_clique(g; optimizer, binary, vertex_weights)
Computes a maximum-weighted clique of g.
sourceGraphsOptim.maximum_weight_clique! — Functionmaximum_weight_clique!(model, g; var_name)
Computes in-place in the JuMP model a maximum-weighted clique of g. An optional vertex_weights vector can be passed to the graph, defaulting to uniform weights (computing a maximum size clique).
sourceA clique is a subset $S \subset V$ of vertices of an undirected graph $G = (V,E)$ such that $\forall (u,v) \in S: (u, v) \in E$. We search for the clique maximizing the total weight of selected vertices.
Maximum Weight Independent Set
GraphsOptim.maximum_weight_independent_set — Functionmaximum_weight_independent_set(g; optimizer, binary, vertex_weights)
Computes a maximum-weighted independent set or stable set of g.
sourceGraphsOptim.maximum_weight_independent_set! — Functionmaximum_independent_set!(model, g; var_name)
Computes in-place in the JuMP model a maximum-weighted independent set of g. An optional vertex_weights vector can be passed to the graph, defaulting to uniform weights (computing a maximum size independent set).
sourceFinds a subset $S \subset V$ of vertices of maximal weight of an undirected graph $G = (V,E)$ such that $\forall (u,v) \in E: u \notin S \lor v \notin S$.
Graph matching
Work in progress Come back later!
GraphsOptim.graph_matching — Functiongraph_matching(
algo, A, B;
optimizer, P_init, max_iter, tol,
max_iter_sinkhorn, regularizer
-)
Compute an approximately optimal alignment between two graphs using one of two variants of Frank-Wolfe (FAQ or GOAT)
The output is a tuple that contains:
- the permutation matrix
P defining the alignment; - the distance between the permuted graphs;
- a boolean indicating if the algorithm converged.
Arguments
algo: allows dispatch based on the choice of algorithm, either FAQ() or GOAT()A: the adjacency matrix of the first graphB: the adjacency matrix of the second graph
Keyword arguments
For both algorithms:
optimizer: JuMP-compatible solver (default is HiGHS.Optimizer)P_init: initialization matrix (default value is a flat doubly stochastic matrix).max_iter: maximum iterations of the Frank-Wolfe method (default value is 30).tol: tolerance for the convergence (default value is 0.1).
Only for GOAT:
regularization: penalty coefficient in the Sinkhorn algorithm (default value is 100.0).max_iter_sinkhorn: maximum iterations of the Sinkhorn algorithm (default value is 500).
References
- FAQ: Algorithm 1 of https://arxiv.org/pdf/2111.05366.pdf
- GOAT: Algorithm 3 of https://arxiv.org/pdf/2111.05366.pdf
sourceGraphsOptim.graph_matching_step_size — Functiongraph_matching_step_size(A, B, P, Q)
Given the adjacency matrices A and B, the doubly stochastic matrix P and the direction matrix Q, return the step size of the Frank-Wolfe method for graph matching algorithms.
sourceColoring
GraphsOptim.fractional_chromatic_number — Functionfractional_chromatic_number(g; optimizer)
Compute the fractional chromatic number of a graph. Gives the same result as fractional_clique_number, though one function may run faster than the other. Beware: this can run very slowly for graphs of any substantial size.
Keyword arguments
optimizer: JuMP-compatible solver (default is HiGHS.Optimizer)
References
- https://mathworld.wolfram.com/FractionalChromaticNumber.html
sourceGraphsOptim.fractional_clique_number — Functionfractional_clique_number(g; optimizer)
Compute the fractional clique number of a graph. Gives the same result as fractional_chromatic_number, though one function may run faster than the other. Beware: this can run very slowly for graphs of any substantial size.
Keyword arguments
optimizer: JuMP-compatible solver (default is HiGHS.Optimizer)
References
- https://mathworld.wolfram.com/FractionalCliqueNumber.html
sourceUtils
GraphsOptim.is_binary — Functionis_binary(x)
Check if a value is equal to 0 or 1 of its type.
sourceGraphsOptim.is_square — Functionis_square(M)
Check if a matrix is square.
sourceGraphsOptim.is_stochastic — Functionis_stochastic(S)
Check if a matrix is row-stochastic.
sourceGraphsOptim.is_doubly_stochastic — Functionis_doubly_stochastic(D)
Check if a matrix is row- and column-stochastic.
sourceGraphsOptim.is_permutation_matrix — Functionis_permutation_matrix(P)
Check if a matrix is a permutation matrix.
sourceGraphsOptim.flat_doubly_stochastic — Functionflat_doubly_stochastic(n)
Return the barycenter of doubly stochastic matrices J = 𝟏 * 𝟏ᵀ / n.
sourceGraphsOptim.indvec — Functionindvec(s, n)
Return a vector of length n with ones at indices specified by s.
sourceSettings
This document was generated with Documenter.jl version 1.1.2 on Friday 26 April 2024. Using Julia version 1.10.2.
Compute an approximately optimal alignment between two graphs using one of two variants of Frank-Wolfe (FAQ or GOAT)
The output is a tuple that contains:
- the permutation matrix
Pdefining the alignment; - the distance between the permuted graphs;
- a boolean indicating if the algorithm converged.
Arguments
algo: allows dispatch based on the choice of algorithm, eitherFAQ()orGOAT()A: the adjacency matrix of the first graphB: the adjacency matrix of the second graph
Keyword arguments
For both algorithms:
optimizer: JuMP-compatible solver (default isHiGHS.Optimizer)P_init: initialization matrix (default value is a flat doubly stochastic matrix).max_iter: maximum iterations of the Frank-Wolfe method (default value is 30).tol: tolerance for the convergence (default value is 0.1).
Only for GOAT:
regularization: penalty coefficient in the Sinkhorn algorithm (default value is 100.0).max_iter_sinkhorn: maximum iterations of the Sinkhorn algorithm (default value is 500).
References
- FAQ: Algorithm 1 of https://arxiv.org/pdf/2111.05366.pdf
- GOAT: Algorithm 3 of https://arxiv.org/pdf/2111.05366.pdf
GraphsOptim.graph_matching_step_size — Functiongraph_matching_step_size(A, B, P, Q)Given the adjacency matrices A and B, the doubly stochastic matrix P and the direction matrix Q, return the step size of the Frank-Wolfe method for graph matching algorithms.
Coloring
GraphsOptim.fractional_chromatic_number — Functionfractional_chromatic_number(g; optimizer)Compute the fractional chromatic number of a graph. Gives the same result as fractional_clique_number, though one function may run faster than the other. Beware: this can run very slowly for graphs of any substantial size.
Keyword arguments
optimizer: JuMP-compatible solver (default isHiGHS.Optimizer)
References
- https://mathworld.wolfram.com/FractionalChromaticNumber.html
GraphsOptim.fractional_clique_number — Functionfractional_clique_number(g; optimizer)Compute the fractional clique number of a graph. Gives the same result as fractional_chromatic_number, though one function may run faster than the other. Beware: this can run very slowly for graphs of any substantial size.
Keyword arguments
optimizer: JuMP-compatible solver (default isHiGHS.Optimizer)
References
- https://mathworld.wolfram.com/FractionalCliqueNumber.html
Utils
GraphsOptim.is_binary — Functionis_binary(x)Check if a value is equal to 0 or 1 of its type.
GraphsOptim.is_square — Functionis_square(M)Check if a matrix is square.
GraphsOptim.is_stochastic — Functionis_stochastic(S)Check if a matrix is row-stochastic.
GraphsOptim.is_doubly_stochastic — Functionis_doubly_stochastic(D)Check if a matrix is row- and column-stochastic.
GraphsOptim.is_permutation_matrix — Functionis_permutation_matrix(P)Check if a matrix is a permutation matrix.
GraphsOptim.flat_doubly_stochastic — Functionflat_doubly_stochastic(n)Return the barycenter of doubly stochastic matrices J = 𝟏 * 𝟏ᵀ / n.
GraphsOptim.indvec — Functionindvec(s, n)Return a vector of length n with ones at indices specified by s.
