## Abstract

We study the traveling salesman problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3-conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal value of a TSP instance and that of its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on \(n\) vertices a tour of length \(4n/3-2\) exists, which also implies the 4/3-conjecture, as an upper bound, for this class of graph-TSP. Recently, Mömke and Svensson presented an algorithm that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3-conjecture for this class of graph-TSP. The algorithm by Mömke and Svensson is initially randomized but the authors remark that derandomization is trivial. We will present a different way to derandomize their algorithm which leads to a faster running time. All of the latter also works for multigraphs.

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## Notes

To see that \(n\) is a lower bound for SER, sum all of the so-called “degree constraints” for SER. Dividing the result by 2 shows that the sum of the edge variables in any feasible SER solution equals \(n\).

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This research was partially supported by Tinbergen Institute, the Netherlands and the Natural Sciences and Engineering Research Council of Canada.

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Boyd, S., Sitters, R., van der Ster, S. *et al.* The traveling salesman problem on cubic and subcubic graphs.
*Math. Program.* **144**, 227–245 (2014). https://doi.org/10.1007/s10107-012-0620-1

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DOI: https://doi.org/10.1007/s10107-012-0620-1

### Keywords

- Traveling salesman problem
- Approximation algorithms
- Subtour elimination
- Matching polytope
- Polyhedral combinatorics