Approximation Algorithms via Structural Results for Apex-Minor-Free Graphs

We develop new structural results for apex-minor-free graphs and show their power by developing two new approximation algorithms. The first is an additive approximation for coloring within 2 of the optimal chromatic number, which is essentially best possible, and generalizes the seminal result by Thomassen [32] for bounded-genus graphs. This result also improves our understanding from an algorithmic point of view of the venerable Hadwiger conjecture about coloring H-minor-free graphs. The second approximation result is a PTAS for unweighted TSP in apexminor-free graphs, which generalizes PTASs for TSP in planar graphs and bounded-genus graphs [20,2,24,15]. We strengthen the structural results from the seminal Graph Minor Theory of Robertson and Seymour in the case of apex-minor-free graphs, showing that apices can be made adjacent only to vortices if we generalize the notion of vortices to "quasivortices" of bounded treewidth, proving a conjecture from [10]. We show that this...

• Type: Conference
• Year: 2009
• Where: ICALP
• Authors: Erik D. Demaine, MohammadTaghi Hajiaghayi, Ken-ichi Kawarabayashi
• Keywords: Apex-minor-free Graphs, Bounded-genus Graphs, ICALP 2009, Planar Graphs, Theoretical Computer Science