Results 1 
2 of
2
Approximation Hardness of Graphic TSP on Cubic Graphs
, 2013
"... We prove explicit approximation hardness results for the Graphic TSP on cubic and subcubic graphs as well as the new inapproximability bounds for the corresponding instances of the (1,2)TSP. The proof technique uses new modular constructions of simulating gadgets for the restricted cubic and subcub ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
We prove explicit approximation hardness results for the Graphic TSP on cubic and subcubic graphs as well as the new inapproximability bounds for the corresponding instances of the (1,2)TSP. The proof technique uses new modular constructions of simulating gadgets for the restricted cubic and subcubic instances. The modular constructions used in the paper could be also of independent interest.
Short Tours through Large Linear Forests
"... Abstract. A tour in a graph is a connected walk that visits every vertex at least once, and returns to the starting vertex. Vishnoi (2012) proved that every connected dregular graph with n vertices has a tour of length at most (1 + o(1))n, where the o(1) term (slowly) tends to 0 as d grows. His pr ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. A tour in a graph is a connected walk that visits every vertex at least once, and returns to the starting vertex. Vishnoi (2012) proved that every connected dregular graph with n vertices has a tour of length at most (1 + o(1))n, where the o(1) term (slowly) tends to 0 as d grows. His proof is based on vanderWarden’s conjecture (proved independently by Egorychev (1981) and by Falikman (1981)) regarding the permanent of doubly stochastic matrices. We provide an exponential improvement in the rate of decrease of the o(1) term (thus increasing the range of d for which the upper bound on the tour length is nontrivial). Our proof does not use the vanderWarden conjecture, and instead is related to the linear arboricity conjecture of Akiyama, Exoo and Harary (1981), or alternatively, to a conjecture of Magnant and Martin (2009) regarding the path cover number of regular graphs. More generally, for arbitrary connected graphs, our techniques provide an upper bound on the minimum tour length, expressed as a function of their maximum, average, and minimum degrees. Our bound is best possible up to a term that tends to 0 as the minimum degree grows. 1