Finding Triangles for Maximum Planar Subgraphs
Abstract
In the Maximum Planar Subgraph (Mps) problem, we are given a graph G, and our goal is to find a planar subgraph H with maximum number of edges. Besides being a basic problem in graph theory, Mps has many applications including, for instance, circuit design, factory layout, and graph drawing, so it has received a lot of attention from both theoretical and empirical literature. Since the problem is NPhard, past research has focused on approximation algorithms. The current best known approximation ratio is \(\frac{4}{9}\) obtained two decades ago (Călinescu et al. SODA 1996) based on computing as many edgedisjoint triangles in an input graph as possible. The factor of \(\frac{4}{9}\) is also the limit of this “disjoint triangles” approach.

Mpt is NPhard, giving a simplified NPhardness proof for Mps as a byproduct.

We propose a natural class of greedy algorithms that captures all known greedy algorithms that have appeared in the literature. We show that a very simple greedy rule gives better approximation ratio than all known greedy algorithms (but still worse than \(\frac{4}{9}\)).
Our greedy results, despite not improving the approximation factor, illustrate the advantage of overlapping triangles in the context of greedy algorithms. The Mpt viewpoint offers various new angles that might be useful in designing a better approximation algorithm for Mps.
Notes
Acknowledgement
We are grateful to an anonymous reviewer, whose detailed suggestions contributed to a clearer presentation of this work.
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