Minor-free graphs have light spanners
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We show that every H-minor-free graph has a light (1+ϵ)-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung. Our lightness bound is O(σ_H/ϵ^31/ϵ) where σ_H = |V(H)|√( |V(H)|) is the sparsity coefficient of H-minor-free graphs. That is, it has a practical dependency on the size of the minor H. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in H-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi is an efficient PTAS whose running time is 2^O_H(1/ϵ^41/ϵ)n^O(1) where O_H ignores dependencies on the size of H. Our techniques significantly deviate from existing lines of research on spanners for H-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs.
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