We show that every H-minor-free graph has a light (1 + )-spanner, resolving an open problem of Grigni and Sissokho [13] and proving a conjecture of Grigni and Hung [12]. Our lightness bound is O σH 3 log 1 where σH = |V (H)| log |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 [7] is an efficient PTAS whose running time is 2 O H 1 4 log 1 n O(1) where OH 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 [6].
Indeed, sparsity is repeatedly used in the analysis of light spanners for H-minor-free graphs REF .
Minor-Free Graphs Have Light Spanners
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
Computer Science Mathematics