Approximation algorithms via structural results for apex-minor-free graphs
Author(s)Demaine, Erik D.; Hajiaghayi, Mohammad Taghi; Kawarabayashi, Ken-ichi
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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  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 apex-minor-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 . 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  and bounded-genus graphs . We also highlight the difficulties in extending our results to general H-minor-free graphs.
DepartmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Automata, Languages and Programming, 36th International Coloquium, ICALP 2009
Demaine, Erik, Mohammadtaghi Hajiaghayi, and Ken-ichi Kawarabayashi. “Approximation Algorithms via Structural Results for Apex-Minor-Free Graphs.” Automata, Languages and Programming. Springer Berlin/Heidelberg, 2009. 316-327. (Lecture Notes in Computer Science, 2009, Volume 5555/2009).
Author's final manuscript