https://github.com/bergio13/heuristic-optimization-mwccp
Construction Heuristics, Local Search, VND, GRASP, GVNS, GA and ACO applied to the Minimum Wieghted Crossings with Constraints Problem (MWCCP)
https://github.com/bergio13/heuristic-optimization-mwccp
aco ant-colony-optimization bipartite-matching genetic-algorithm grasp gvns heuristics local-search metaheuristics variable-neighbourhood-descent
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Construction Heuristics, Local Search, VND, GRASP, GVNS, GA and ACO applied to the Minimum Wieghted Crossings with Constraints Problem (MWCCP)
- Host: GitHub
- URL: https://github.com/bergio13/heuristic-optimization-mwccp
- Owner: bergio13
- Created: 2024-10-26T15:23:56.000Z (11 months ago)
- Default Branch: main
- Last Pushed: 2025-01-22T17:26:38.000Z (9 months ago)
- Last Synced: 2025-08-01T20:07:09.305Z (2 months ago)
- Topics: aco, ant-colony-optimization, bipartite-matching, genetic-algorithm, grasp, gvns, heuristics, local-search, metaheuristics, variable-neighbourhood-descent
- Language: Jupyter Notebook
- Homepage:
- Size: 21.1 MB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# Minimum Weighted Crossings with Constraints Problem (MWCCP)
The **Minimum Weighted Crossings with Constraints Problem (MWCCP)** is a generalization of the Minimum Crossings Problem.
## Problem Definition
We are given an **undirected weighted bipartite graph** $G = (U ∪ V, E)$ with:
- **Node sets**:
- `U = {1, ..., m}`: the first partition.
- `V = {m + 1, ..., n}`: the second partition.
- `U` and `V` are disjoint.
- **Edge set**: $E ⊆ U × V$, consisting of edges between the two partitions.
- **Edge weights**: $w_e = w_{u,v}$ for $e = (u, v) ∈ E$.
### Constraints
Precedence constraints $C$ are given as a set of ordered node pairs:
$C ⊆ V × V$,
where $(v, v') ∈ C$ means that **node $v$ must appear before node $v'$** in a feasible solution.
### Goal
The nodes of the graph $G$ are to be arranged in two layers:
1. The **first layer** contains all nodes in $U$, arranged in fixed label order `1, ..., m`.
2. The **second layer** contains all nodes in $V$, arranged in an order to be determined.
The goal is to find an ordering of the nodes in $V$ such that:
- The **weighted edge crossings** are minimized.
- All precedence constraints $C$ are satisfied.
---
## Solution Representation
A candidate solution is represented by a **permutation** $π = (π_{m+1}, ..., π_n)$ of the nodes in $V$.
### Feasibility
A solution is feasible if all precedence constraints $C$ are fulfilled:
$$
pos_π(v) < pos_π(v'), \forall (v, v') ∈ C
$$
where $pos_π(v)$ is the position of node $v$ in the permutation $π$.
---
## Objective Function
The objective is to minimize the following function:
$$
f(π) = ∑_{(u, v) ∈ E} ∑_{(u', v') ∈ E, u < u'}
(w_{u,v} + w_{u',v'}) · δ_π((u, v), (u', v'))
$$
### Crossing Indicator
The indicator function $δ_π((u, v), (u', v'))$ is defined as:
$δ_π((u, v), (u', v')) =$
- $1$, if $pos_π(v) > pos_π(v')$
- $0$, otherwise
This formulation ensures that the **weighted crossings** are minimized while satisfying all precedence constraints.
---
# Repository Content
- `source` is a folder containing the python code for the different heuristics used
- `report_1.pdf` contains a description of the Local Search, VND, GVNS, and GRASP heuristics used, as well as detailed results and analysis of how these techniques perform when applied to different instances of the MWCPP
- `report_2.pdf` contains a description of the Genetic Algorithm and Ant Colony Optimization metaheuristics used and results and comparisons between them
---
# Results
