[Paper Review] A Heuristic for Magic and Antimagic Graph Labellings

François Bertault, Mirka Miller+3 more|arXiv (Cornell University)|May 8, 2013

Graph Labeling and Dimension Problems4 references3 citations

TL;DR

This paper proposes a general heuristic algorithm for finding magic and antimagic graph labellings by formulating the labelling problem as a combinatorial optimization task. Using simulated annealing, the method efficiently discovers new vertex-antimagic edge labellings for small graphs of the form $P_2^r \times P_3^s$, where no general construction was previously known.

ABSTRACT

Graph labellings have been a very fruitful area of research in the last four decades. However, despite the staggering number of papers published in the field (over 1000), few general results are available, and most papers deal with particular classes of graphs and methods. Here we approach the problem from the computational viewpoint, and in a quite general way. We present the existence problem of a particular labelling as a combinatorial optimization problem, then we discuss the possible strategies to solve it, and finally we present a heuristic for finding different classes of labellings, like vertex-, edge-, or face-magic, and $(a, d)$-antimagic $(v, e, f)$-labellings. The algorithm has been implemented in C++ and MATLAB, and with its aid we have been able to derive new results for some classes of graphs, in particular, vertex-antimagic edge labellings for small graphs of the type $P_2^r imes P_3^s$, for which no general construction is known so far.

Motivation & Objective

• To address the lack of general computational methods for finding magic and antimagic graph labellings across diverse graph classes.
• To develop a unified algorithm applicable to multiple labelling types, including vertex-, edge-, face-magic, and $(a,d)$-antimagic labellings.
• To overcome the limitations of exhaustive search methods by introducing a heuristic approach that balances efficiency and effectiveness.
• To enable the discovery of new labellings for graphs where no known constructive methods exist, such as $P_2^r \times P_3^s$.
• To compare the performance of heuristic methods like simulated annealing with exhaustive search strategies, highlighting trade-offs in completeness and speed.

Proposed method

• Formulate the graph labelling existence problem as a combinatorial optimization task, where the objective function is defined based on the desired labelling type (e.g., magic or antimagic).
• Define vertex, edge, and face weights using a labelling function $\lambda$ that maps vertices, edges, and faces to integers, with weights computed as sums of incident labels.
• Implement a simulated annealing heuristic to explore the search space of possible label assignments, minimizing the variance of weights across vertices, edges, or faces.
• Use a dynamic temperature schedule to control the acceptance of worse solutions, reducing the risk of getting trapped in local minima.
• Integrate a pruning mechanism to avoid redundant or invalid label assignments, improving search efficiency.
• Apply the algorithm to various graph families, including complete graphs $K_n$ and Cartesian products $P_3^k$, to evaluate performance and scalability.

Experimental results

Research questions

• RQ1 Can a single heuristic framework be effectively applied to a broad range of magic and antimagic graph labelling problems?
• RQ2 How does the performance of simulated annealing compare to exhaustive search methods in terms of runtime and success rate for magic labellings?
• RQ3 What is the computational complexity of finding antimagic versus magic labellings, and can this be quantified empirically?
• RQ4 Can the heuristic discover new labellings for graph families where no general construction is known, such as $P_2^r \times P_3^s$?
• RQ5 How do different metaheuristics, such as simulated annealing versus evolutionary algorithms, compare in solving the same labelling optimization problem?

Key findings

• The simulated annealing heuristic successfully discovered new vertex-antimagic edge labellings for small graphs of the form $P_2^r \times P_3^s$, which lack known general constructions.
• The number of iterations required to find vertex-magic edge superlabellings in $K_n$ (for $n \geq 6$, $n \neq 8,12$) grows exponentially, with a best-fit function of $y = 357.8e^{0.4811x}$, suggesting NP-hardness.
• In contrast, finding vertex-antimagic edge labellings for $P_3^k$ required significantly fewer iterations, approximated by a linear function $y = 0.653x - 57.08$, indicating lower computational complexity.
• The algorithm's performance is highly dependent on the labelling type, with antimagic-type labellings generally being easier to find than magic ones.
• The heuristic approach is efficient and effective for exploring large search spaces, enabling the discovery of new labellings where traditional constructive methods fail.
• The authors note that while the algorithm is not guaranteed to find a solution, it provides a practical alternative to exhaustive search, especially for complex or poorly understood graph families.