Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast

TL;DR

This paper introduces new cycle-based constraints and polyhedral liftings to improve LP relaxations, enabling faster algorithms for the NP-hard Maximum Planar Subgraph problem by strengthening classical approaches.

Contribution

It proposes novel cycle-based constraints and polyhedral liftings that significantly enhance LP relaxations for the Maximum Planar Subgraph problem, leading to more efficient exact algorithms.

Findings

Stronger LP relaxations improve computational efficiency.

Cycle-based constraints enhance the polyhedral approach.

Practical algorithms outperform previous methods.

Abstract

The NP-hard Maximum Planar Subgraph problem asks for a planar subgraph $H$ of a given graph $G$ such that $H$ has maximum edge cardinality. For more than two decades, the only known non-trivial exact algorithm was based on integer linear programming and Kuratowski's famous planarity criterion. We build upon this approach and present new constraint classes, together with a lifting of the polyhedron, to obtain provably stronger LP-relaxations, and in turn faster algorithms in practice. The new constraints take Euler's polyhedron formula as a starting point and combine it with considering cycles in $G$. This paper discusses both the theoretical as well as the practical sides of this strengthening.