Results for the maximum weight planar subgraph problem

TL;DR

This paper proves that local moves of only edge substitution are sufficient to transform any feasible solution into any other in the maximum-weight planar subgraph problem, settling a longstanding conjecture and clarifying solution structure.

Contribution

It confirms that only edge substitution moves are needed for solution transformations and shows maximal spanning trees are not always part of optimal solutions.

Findings

Edge substitution moves suffice for solution transformations.

Maximal spanning trees are not necessarily part of optimal solutions.

Settles a longstanding conjecture in the problem's solution space.

Abstract

The problem of finding the maximum-weight, planar subgraph of a finite, simple graph with nonnegative real edge weights is well known in industrial and electrical engineering, systems biology, sociology and finance. As the problem is known to be NP-hard, much research effort has been devoted over the years to attempt to improve a given approximate solution to the problem by using local moves applied to a planar embedding of the solution. It has long been established that any feasible solution to the problem, a maximal planar graph, can be transformed into any other (having the same vertex set) in a finite sequence of local moves of based on: (i) edge substitution and (ii) vertex relocation and it has been conjectured that moves of only type (i) are sufficient. In this note we settle this conjecture in the affirmative. Furthermore, contrary to recent supposition, we demonstrate that any maximal spanning tree of the original graph is not necessarily a part of any optimal solution to the problem. We hope these results will be useful in the design of future approximate methods for the problem.