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TSP tours in cubic graphs: Beyond 4/3
, 2015
"... After a sequence of improvements Boyd et al. [TSP on cubic and subcubic graphs, Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 6655, Springer, Heidelberg, 2011, pp. 65–77] proved that any 2connected graph whose n vertices have degree 3, i.e., a cubic 2connected ..."
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After a sequence of improvements Boyd et al. [TSP on cubic and subcubic graphs, Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 6655, Springer, Heidelberg, 2011, pp. 65–77] proved that any 2connected graph whose n vertices have degree 3, i.e., a cubic 2connected graph, has a Hamiltonian tour of length at most (4/3)n, establishing in particular that the integrality gap of the subtour LP is at most 4/3 for cubic 2connected graphs and matching the conjectured value of the famous 4/3 conjecture. In this paper we improve upon this result by designing an algorithm that finds a tour of length (4/3−1/61236)n, implying that cubic 2connected graphs are among the few interesting classes of graphs for which the integrality gap of the subtour LP is strictly less than 4/3. With the previous result, and by considering an even smaller , we show that the integrality gap of the TSP relaxation is at most 4/3 − even if the graph is not 2connected (i.e., for cubic connected graphs), implying that the approximability threshold of the TSP in cubic graphs is strictly below 4/3. Finally, using similar techniques we show, as an additional result, that every Barnette graph admits a tour of length at most (4/3 − 1/18)n.
9Approximation for Graphic TSP
, 2012
"... © The Author(s) 2012. This article is published with open access at Springerlink.com Abstract The Travelling Salesman Problem is one of the fundamental and intensively studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christo ..."
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© The Author(s) 2012. This article is published with open access at Springerlink.com Abstract The Travelling Salesman Problem is one of the fundamental and intensively studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides’s algorithm with an approximation factor of 32, even though the socalled HeldKarp LP relaxation of the problem is conjectured to have the integrality gap of only 43. Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis Gharan et al. (FOCS, 550–559, 2011), and then by Mömke and Svensson (FOCS, 560–569, 2011). In this paper, we provide an improved analysis of the approach presented in Mömke and Svensson (FOCS, 560–569, 2011) yielding a bound of 139 on the approximation factor, as well as a bound of 19 12 + ε for any ε> 0 for a more general Travelling Salesman Path Problem in graphic metrics.