The traveling salesman problem with distances one and two
Mathematics of Operations Research
On sparse spanners of weighted graphs
Discrete & Computational Geometry
A polynomial-time approximation scheme for weighted planar graph TSP
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
An approximation scheme for planar graph TSP
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Approximation schemes for minimum 2-edge-connected and biconnected subgraphs in planar graphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Bidimensionality: new connections between FPT algorithms and PTASs
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
A linear-time approximation scheme for planar weighted TSP
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Approximation algorithms via contraction decomposition
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Contraction decomposition in h-minor-free graphs and algorithmic applications
Proceedings of the forty-third annual ACM symposium on Theory of computing
Light spanners in bounded pathwidth graphs
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
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Given an edge weighted graph G with n vertices and no Kτ-minor and a small positive constant ε, we show that a simple greedy algorithm [1] finds a spanning subgraph approximating all shortest-path distances within a factor of 1 + ε, and with total edge weight at most O((τ√logτ · logn)/ε) times the weight of a minimum spanning tree. This result implies a quasi-polynomial time approximation scheme (QPTAS) for the traveling salesman problem (TSP) in such graphs, with running time nO((τ4√logτ·log n·log log n)/ε2).Our analysis shows that a graph with detour gap number [5] Ω(τ√logτ · log n) has a Kτ-minor. We also show that this dependence on n is nearly tight, by exhibiting graphs with no K6-minor (apex graphs) and detour gap number Ω((log n)/log log n).As a step towards eliminating the log n factors the first paragraph, we propose a generalized detour gap number, now depending on ε, and we show that it remains bounded for apex graphs and some similar graph families.