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The connected Grundy coloring problem: Formulations and a local-search enhanced biased random-key genetic algorithm

Mateus C. Silva, Rafael A. Melo, Mauricio G. C. Resende, Marcio C. Santos, Rodrigo F. Toso

TL;DR

Two integer programming formulations and a local-search enhanced biased random-key genetic algorithm (BRKGA) for the connected Grundy coloring problem, which follows the standard way of partitioning the vertices into color classes while the second relies on the idea of representatives in an attempt to break symmetries.

Abstract

Given a graph G=(V,E), a connected Grundy coloring is a proper vertex coloring that can be obtained by a first-fit heuristic on a connected vertex sequence. A first-fit coloring heuristic is one that attributes to each vertex in a sequence the lowest-index color not used for its preceding neighbors. A connected vertex sequence is one in which each element, except for the first one, is connected to at least one element preceding it. The connected Grundy coloring problem consists of obtaining a connected Grundy coloring maximizing the number of colors. In this paper, we propose two integer programming (IP) formulations and a local-search enhanced biased random-key genetic algorithm (BRKGA) for the connected Grundy coloring problem. The first formulation follows the standard way of partitioning the vertices into color classes while the second one relies on the idea of representatives in an attempt to break symmetries. The BRKGA encompasses a local search procedure using a newly proposed neighborhood. A theoretical neighborhood analysis is also presented. Extensive computational experiments indicate that the problem is computationally demanding for the proposed IP formulations. Nonetheless, the formulation by representatives outperforms the standard one for the considered benchmark instances. Additionally, our BRKGA can find high-quality solutions in low computational times for considerably large instances, showing improved performance when enhanced with local search and a reset mechanism. Moreover we show that our BRKGA can be easily extended to successfully tackle the Grundy coloring problem, i.e., the one without the connectivity requirements.

The connected Grundy coloring problem: Formulations and a local-search enhanced biased random-key genetic algorithm

TL;DR

Two integer programming formulations and a local-search enhanced biased random-key genetic algorithm (BRKGA) for the connected Grundy coloring problem, which follows the standard way of partitioning the vertices into color classes while the second relies on the idea of representatives in an attempt to break symmetries.

Abstract

Given a graph G=(V,E), a connected Grundy coloring is a proper vertex coloring that can be obtained by a first-fit heuristic on a connected vertex sequence. A first-fit coloring heuristic is one that attributes to each vertex in a sequence the lowest-index color not used for its preceding neighbors. A connected vertex sequence is one in which each element, except for the first one, is connected to at least one element preceding it. The connected Grundy coloring problem consists of obtaining a connected Grundy coloring maximizing the number of colors. In this paper, we propose two integer programming (IP) formulations and a local-search enhanced biased random-key genetic algorithm (BRKGA) for the connected Grundy coloring problem. The first formulation follows the standard way of partitioning the vertices into color classes while the second one relies on the idea of representatives in an attempt to break symmetries. The BRKGA encompasses a local search procedure using a newly proposed neighborhood. A theoretical neighborhood analysis is also presented. Extensive computational experiments indicate that the problem is computationally demanding for the proposed IP formulations. Nonetheless, the formulation by representatives outperforms the standard one for the considered benchmark instances. Additionally, our BRKGA can find high-quality solutions in low computational times for considerably large instances, showing improved performance when enhanced with local search and a reset mechanism. Moreover we show that our BRKGA can be easily extended to successfully tackle the Grundy coloring problem, i.e., the one without the connectivity requirements.

Paper Structure

This paper contains 28 sections, 11 theorems, 6 equations, 11 figures, 16 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Related works
  4. Main contributions and organization
  5. Integer programming formulations
  6. Standard formulation
  7. Formulation by representatives
  8. Biased random-key genetic algorithm
  9. Solution encoding
  10. Decoder
  11. Local search
  12. Neighborhood
  13. Neighborhood analysis
  14. Local search algorithm
  15. Computational experiments
  16. ...and 13 more sections

Key Result

Proposition 1

Algorithm alg:decoder_con_grundy can be implemented to run in $O(|V|\log |V| + |E|)$.

Figures (11)

  • Figure 1: Difference in the resulting colorings when using a non-connected and a connected sequence with the first-fit heuristic.
  • Figure 2: Coloring of a graph corresponding to the solution with the only non-zero values $Z_{111} = Z_{222} = Z_{333} = Z_{144} = Z_{155} = Z_{266} = Z_{277} = Z_{388} = 1$, $Y_{12} = Y_{13} = Y_{23} = 1$. The external label represents the vertex index, while the internal label represents the color assigned to that vertex.
  • Figure 3: Framework of the BRKGA enhanced with local search.
  • Figure 4: Population evolution between two generations of a BRKGA.
  • Figure 5: Example of a random-key vector and the connected sequence generated by the decoder. On the right, the resulting coloring of the graph.
  • ...and 6 more figures

Theorems & Definitions (19)

  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Corollary 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • ...and 9 more