Utilizing graph thickness heuristics on the Earth-moon Problem

This paper utilizes heuristic algorithms for determining graph thickness in order to attempt to find a 10-chromatic thickness-2 graph. Doing so would eliminate 9 colors as a potential solution to the Earth-moon Problem. An empirical analysis of the algorithms made by the author are provided. Additionally, the paper lists various graphs that may or nearly have a thickness of 2, which may be solutions if one can find two planar subgraphs that partition all of the graph’s edges

Two new approximation algorithms for the maximum planar subgraph problem

The maximum planar subgraph problem (MPS) is defined as follows: given a graph G, find a largest planar subgraph of G. The problem is NP-hard and it has applications in graph drawing and resource location optimization. Călinescu et al. [J. Alg. 27, 269-302 (1998)] presented the first approximation algorithms for MPS with nontrivial performance ratios. Two algorithms were given, a simple algorithm which runs in linear time for bounded-degree graphs with a ratio 7/18 and a more complicated algorithm with a ratio 4/9. Both algorithms produce outerplanar subgraphs. In this article we present two new versions of the simpler algorithm. The first new algorithm still runs in the same time, produces outerplanar subgraphs, has at least the same performance ratio as the original algorithm, but in practice it finds larger planar subgraphs than the original algorithm. The second new algorithm has similar properties to the first algorithm, but it produces only planar subgraphs. We conjecture that the performance ratios of our algorithms are at least 4/9 for MPS. We experimentally compare the new algorithms against the original simple algorithm. We also apply the new algorithms for approximating the thickness and outerthickness of a graph. Experiments show that the new algorithms produce clearly better approximations than the original simple algorithm by Călinescu et al

Graph Planarization Problem Optimization Based on Triple-Valued Gravitational Search Algorithm

This article presents a triple-valued gravitational search algorithm (TGSA) to tackle the graph planarization problem (GPP). GPP is one of the most important tasks in graph theory, and has proved to be an NP-hard problem. To solve it, TGSA uses a triple-valued encoding scheme and models the search space into a triangular hypercube quantitatively based on the well-known single-row routing representation method. The agents in TGSA, whose interactions are driven by the gravity law, move toward the global optimal position gradually. The position updating rule for each agent is based on two indices: one is a velocity index which is a function of the current velocity of the agent, and the other is a population index based on the cumulative information in the whole population. To verify the performance of the algorithm, 21 benchmark instances are tested. Experimental results indicate that TGSA can solve the GPP by finding its maximum planar subgraph and embedding the resulting edges into a plane simultaneously. Compared with traditional algorithms, a novelty of TGSA is that it can find multiple optimal solutions for the GPP. Comparative results also demonstrate that TGSA outperforms the traditional meta-heuristics in terms of the solution qualities within reasonable computational times. © 2013 Institute of Electrical Engineers of Japan