Approximation Algorithms via Structural Results for Apex-Minor-Free Graphs [chapter]

Erik D. Demaine, MohammadTaghi Hajiaghayi, Ken-ichi Kawarabayashi
2009 Lecture Notes in Computer Science  
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
more » ... 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 structure theorem is a powerful tool for developing algorithms on apex-minor-free graphs, including for the classic problems of coloring and TSP. In particular, we use this theorem to partition the edges of a graph into k pieces, for any k, such that contracting any piece results in a bounded-treewidth graph, generalizing previous similar results for planar graphs [24] and bounded-genus graphs [15] . We also highlight the difficulties in extending our results to general H-minor-free graphs.
doi:10.1007/978-3-642-02927-1_27 fatcat:en2uxhqyxraznbrc4cyucem64e