1.5-Approximation for Treewidth of Graphs Excluding a Graph with One Crossing as a Minor

摘要

We give polynomial-time constant-factor approximation algorithms for the treewidth and branchwidth of any H-minor-free graph for a given graph H with crossing number at most 1. The approximation factors are 1.5 for treewidth and 2.25 for branchwidth. In particular, our result directly applies to classes of nonplanar graphs such as K_5-minor-free graphs and K_(3,3)-minor-free graphs. Along the way, we present a polynomial-time algorithm to decompose H-minor-free graphs into planar graphs and graphs of treewidth at most c_(H) (a constant dependent on H) using clique sums. This result has several applications in designing fully polynomial-time approximation schemes and fixed-parameter algorithms for many NP-complete problems on these graphs.