Package {momst}

Type: Package
Title: Multi-Objective Minimum Spanning Tree via NSGA-II with Local Search
Version: 0.1.1
Description: Solves the Multi-Criteria Minimum Spanning Tree (mc-MST) problem on complete weighted graphs by combining the Non-dominated Sorting Genetic Algorithm II (NSGA-II) with optional Pareto local search operators. Chromosomes are represented as Prufer sequences so that every random individual decodes to a valid spanning tree (Cayley's theorem), avoiding repair operators. Four solver variants are provided: pure NSGA-II ("base"), Path Relinking ("PR"), Pareto Local Search ("PLS"), and Tabu Search ("TS"). The package supports 2 and 3 objective formulations and provides convenience functions to plot Pareto fronts and best-compromise spanning trees. This package is the reference implementation of the method described in Parraga-Alava, Inostroza-Ponta and Dorn (2017) <doi:10.1109/CEC.2017.7969432>.
License: GPL (≥ 3)
URL: https://github.com/jorgeklz/momst, https://jorgeklz.github.io/momst/
BugReports: https://github.com/jorgeklz/momst/issues
Encoding: UTF-8
Depends: R (≥ 4.0)
Imports: stats, utils, graphics
Suggests: igraph, knitr, rmarkdown, testthat (≥ 3.0.0)
VignetteBuilder: knitr
Config/testthat/edition: 3
Config/roxygen2/version: 8.0.0
NeedsCompilation: no
Packaged: 2026-06-16 23:17:12 UTC; jorge
Author: Jorge A. Parraga-Alava ORCID iD [aut, cre]
Maintainer: Jorge A. Parraga-Alava <jorge.parraga@utm.edu.ec>
Repository: CRAN
Date/Publication: 2026-06-22 14:40:15 UTC

momst: Multi-Objective Minimum Spanning Tree via NSGA-II with Local Search

Description

The momst package implements the Multi-Criteria Minimum Spanning Tree (mc-MST) algorithm using the NSGA-II (Non-dominated Sorting Genetic Algorithm II). Four variants are supported depending on the local search operator applied after each generation:

"base"

Pure NSGA-II without local search.

"PR"

NSGA-II plus Path Relinking.

"PLS"

NSGA-II plus Pareto Local Search.

"TS"

NSGA-II plus Tabu Search.

Details

Chromosomes are encoded as Prufer sequences, taking advantage of Cayley's theorem (every sequence of length n-2 with values in {1,...,n} decodes to a unique spanning tree of n nodes). This bijection makes every random chromosome a valid solution, avoiding repair operators.

The main entry point is run_momst.

Reference

This package is the reference implementation of the method described in:

Parraga-Alava, J., Inostroza-Ponta, M., & Dorn, M. (2017). Using local search strategies to improve the performance of NSGA-II for the Multi-Criteria Minimum Spanning Tree problem. In 2017 IEEE Congress on Evolutionary Computation (CEC) (pp. 1818-1825). IEEE. doi:10.1109/CEC.2017.7969432

Author(s)

Jorge A. Parraga-Alava jorge.parraga@utm.edu.ec

See Also

Useful links:

Description

Apply the Configured Local-Search Variant

Usage

apply_local_search(
  instance,
  pareto_pop,
  num_obj,
  n,
  variant,
  pop_size,
  lookup = NULL,
  verbose = FALSE
)

Arguments

instance

Edge-list data.frame.

pareto_pop

Integer matrix.

num_obj

Integer.

n

Integer.

variant

One of "base", "PR", "PLS", "TS".

pop_size

Integer.

lookup

Optional lookup.

verbose

Logical.

Value

data.frame.

Pre-build Edge-Weight Lookup Matrices

Description

Constructs one n x n matrix per objective so that the weight of any edge (i, j) can be queried in O(1) via standard matrix indexing.

Usage

build_weight_lookup(instance, n, num_obj)

Arguments

instance

data.frame returned by generate_instance or one with columns from, to, weight_1, weight_2 (and optionally weight_3).

n

Integer. Number of nodes.

num_obj

Integer. Number of objectives (2 or 3).

Value

A list of length num_obj; element k is the symmetric n x n weight matrix of objective k.

Examples

inst <- generate_instance(10, 2, seed = 1)
L <- build_weight_lookup(inst, 10, 2)
L[[1]][1, 2]

Compute Multi-Objective Costs for a Population

Description

Compute Multi-Objective Costs for a Population

Usage

compute_objectives(instance, chromosomes, num_obj, lookup = NULL)

Arguments

instance

Edge-list data.frame (only used when lookup = NULL).

chromosomes

Numeric matrix [pop_size x (n - 2)].

num_obj

Integer.

lookup

Optional list returned by build_weight_lookup.

Value

Numeric matrix [pop_size x (n - 2 + num_obj)].

Examples

inst <- generate_instance(10, 2, seed = 1)
lk   <- build_weight_lookup(inst, 10, 2)
pop  <- generate_prufer_population(10, 5)
compute_objectives(inst, pop, 2, lk)

Decode a Prufer Sequence to its Spanning Tree (Linear Time)

Description

Returns the edge list of the spanning tree encoded by a Prufer sequence of length n - 2. Uses Wang's pointer-based linear-time algorithm, replacing the classic O(n^2) "search smallest leaf each step" approach.

Usage

decode_prufer(seq_prufer, n)

Arguments

seq_prufer

Integer vector of length n - 2 with values in {1, ..., n}.

n

Integer. Number of nodes of the underlying graph.

Value

Integer matrix of dimensions (n - 1) x 2; each row is an undirected edge (from, to).

Examples

decode_prufer(c(3, 1, 5), n = 5)

Generate a Complete-Graph Instance for MO-MST

Description

Produces a complete undirected weighted graph with random multi-objective edge weights. Each edge gets two or three independent uniform weights, one per objective.

Usage

generate_instance(
  n,
  num_obj,
  range_a = c(10, 100),
  range_b = c(10, 50),
  range_c = c(30, 200),
  seed = NULL
)

Arguments

n

Integer. Number of nodes of the graph (must be at least 3).

num_obj

Integer in {2, 3}. Number of objectives.

range_a

Numeric vector c(min, max) for weights of objective 1.

range_b

Numeric vector c(min, max) for weights of objective 2.

range_c

Numeric vector c(min, max) for weights of objective 3. Ignored when num_obj == 2.

seed

Optional integer. If supplied, the RNG seed is fixed before sampling and restored afterwards, leaving the global RNG untouched.

Value

A data.frame with n*(n-1)/2 rows and columns from, to, weight_1, weight_2 (and weight_3 when num_obj == 3).

Examples

inst <- generate_instance(n = 10, num_obj = 2, seed = 12345)
head(inst)

Generate an Initial Prufer-Encoded Population

Description

Generate an Initial Prufer-Encoded Population

Usage

generate_prufer_population(n, pop_size)

Arguments

n

Integer. Number of nodes.

pop_size

Integer. Number of individuals to generate.

Value

Integer matrix of dimensions pop_size x (n - 2).

Examples

generate_prufer_population(n = 6, pop_size = 4)

Assign Pareto Rank and Crowding Distance

Description

Assign Pareto Rank and Crowding Distance

Usage

non_dominated_crowding(population, num_obj)

Arguments

population

Numeric matrix [N x (vars + num_obj)].

num_obj

Integer.

Value

Matrix with extra columns rankingIndex and density.

Description

Pareto Local Search

Usage

pareto_local_search(
  instance,
  pareto_pop,
  num_obj,
  n,
  neighbour_frac = 0.1,
  pop_size,
  lookup = NULL,
  verbose = FALSE
)

Arguments

instance

Edge-list data.frame.

pareto_pop

Integer matrix [k x (n - 2)].

num_obj

Integer.

n

Integer.

neighbour_frac

Numeric.

pop_size

Integer.

lookup

Optional lookup.

verbose

Logical.

Value

data.frame of chromosomes.

Path Relinking on the Current Pareto Front

Description

Path Relinking on the Current Pareto Front

Usage

path_relinking(
  instance,
  pareto_pop,
  num_obj,
  n,
  pop_size,
  lookup = NULL,
  verbose = FALSE
)

Arguments

instance

Edge-list data.frame.

pareto_pop

Integer matrix [k x (n - 2)].

num_obj

Integer.

n

Integer.

pop_size

Integer.

lookup

Optional lookup.

verbose

Logical.

Value

data.frame of non-dominated chromosomes.

Plot the Best-Compromise Spanning Tree

Description

Plot the Best-Compromise Spanning Tree

Usage

plot_best_tree(result, n)

Arguments

result

List returned by run_momst.

n

Integer.

Value

Invisible NULL.

Plot a Pareto Front (2-objective case)

Description

Plot a Pareto Front (2-objective case)

Usage

plot_pareto_front(result, show_dominated = FALSE)

Arguments

result

List returned by run_momst.

show_dominated

Logical.

Value

Invisible NULL.

Random Mutation on Prufer Sequences

Description

Random Mutation on Prufer Sequences

Usage

random_mutation(population, pop_size, mut_rate)

Arguments

population

Numeric matrix [pop_size x (n - 2)].

pop_size

Integer.

mut_rate

Numeric in \[0, 1\].

Value

Integer matrix [pop_size x (n - 2)].

Run the MO-MST NSGA-II Solver

Description

Single entry point that replaces the original main.R script.

Usage

run_momst(
  instance = NULL,
  instance_file = NULL,
  n = 10L,
  num_obj = 2L,
  variant = c("base", "PR", "PLS", "TS"),
  iterations = 10L,
  pop_size = 50L,
  tour_size = 2L,
  cross_rate = 0.8,
  mut_rate = 0.05,
  max_generations = 100L,
  convergence_window = 10L,
  range_a = c(10, 100),
  range_b = c(10, 50),
  range_c = c(30, 200),
  save_dir = NULL,
  verbose = TRUE,
  seed = NULL
)

Arguments

instance

Optional data.frame.

instance_file

Optional path.

n

Integer.

num_obj

Integer in {2, 3}.

variant

One of "base", "PR", "PLS", "TS".

iterations

Integer or two-length integer c(min_iter, max_iter).

pop_size

Integer (must be even).

tour_size

Integer.

cross_rate

Numeric in \[0, 1\].

mut_rate

Numeric in \[0, 1\].

max_generations

Integer.

convergence_window

Integer.

range_a, range_b, range_c

Weight ranges for instance generation.

save_dir

Optional directory for per-iteration result files.

verbose

Logical.

seed

Optional integer.

Value

Invisible list with the solution data.

References

Parraga-Alava, J., Inostroza-Ponta, M., & Dorn, M. (2017). Using local search strategies to improve the performance of NSGA-II for the Multi-Criteria Minimum Spanning Tree problem. In 2017 IEEE Congress on Evolutionary Computation (CEC) (pp. 1818-1825). IEEE. doi:10.1109/CEC.2017.7969432

Examples


res <- run_momst(n = 10, num_obj = 2, iterations = 3,
                 pop_size = 20, max_generations = 30,
                 variant = "base", seed = 1)
head(res$global_pareto)

Description

Tabu Search on the Current Pareto Front

Usage

tabu_search(
  instance,
  pareto_pop,
  num_obj,
  n,
  neighbour_frac = 0.05,
  pop_size,
  lookup = NULL,
  verbose = FALSE
)

Arguments

instance

Edge-list data.frame.

pareto_pop

Integer matrix [k x (n - 2)].

num_obj

Integer.

n

Integer.

neighbour_frac

Numeric.

pop_size

Integer.

lookup

Optional lookup.

verbose

Logical.

Value

data.frame of non-dominated chromosomes.

Tournament Selection

Description

Tournament Selection

Usage

tournament_selection(population, pop_size, tour_size)

Arguments

population

Matrix with last two columns being rankingIndex and density.

pop_size

Integer.

tour_size

Integer.

Value

Selected subpopulation matrix.

Uniform Crossover for Prufer Sequences

Description

Uniform Crossover for Prufer Sequences

Usage

uniform_crossover(pool, pop_size, cross_rate)

Arguments

pool

Numeric matrix [pop_size x (n - 2)].

pop_size

Integer (must be even).

cross_rate

Numeric in \[0, 1\].

Value

Integer matrix [pop_size x (n - 2)].