# The traveling salesman problem on cubic and subcubic graphs

@article{Boyd2014TheTS, title={The traveling salesman problem on cubic and subcubic graphs}, author={Sylvia C. Boyd and Ren{\'e} Sitters and Suzanne van der Ster and Leen Stougie}, journal={Mathematical Programming}, year={2014}, volume={144}, pages={227-245} }

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…

## 37 Citations

### On the integrality gap of the subtour LP for the 1,2-TSP

- MathematicsMath. Program.
- 2015

It is shown computationally that the integrality gap is at most 10/9 for all instances with at most 12 cities and under a weaker assumption, which is an analog of a recent conjecture by Schalekamp et al., it is shown that this conjecture is true when the optimal subtour LP solution has a certain structure.

### Improved approximations for cubic bipartite and cubic TSP

- Mathematics, Computer ScienceIPCO
- 2016

Improved approximation guarantees for the traveling salesman problem on cubic bipartite graphs and cubic graphs are shown and the techniques of Mömke and Svensson can be combined with the technique of Correa, Larré and Soto to obtain a tour of length at most.

### Applications and Constructions of Cut-Covering Decompositions for Connectivity Problems

- Mathematics, Computer Science
- 2017

This work presents a procedure to decompose an optimal solution for the subtour linear program into spanning, connected subgraphs that cover each 2-edge cut an even number of times, and designs a simple-approximation algorithm for 2EC on subcubic, node-weighted graphs.

### An improved approximation algorithm for the traveling salesman problem with relaxed triangle inequality

- Mathematics, Computer ScienceInf. Process. Lett.
- 2015

### 2-Matchings, the Traveling Salesman Problem, and the Subtour LP: A Proof of the Boyd-Carr Conjecture

- MathematicsMath. Oper. Res.
- 2014

It is proved that an optimal 2-matching has cost at most 10/9 times the cost of the fractional 2- Matching, the Boyd-Carr conjecture.

### On Applying Methods for Graph-TSP to Metric TSP

- Computer Science
- 2016

A new heuristic for metric TSP is developed based on extending ideas successfully used by Mömke and Svensson for the special case of graph-TSP to the more general case of metric T SP, and the exact value of the ratio between the cost of the optimal TSP tour and the optimal subtour linear programming relaxation is found, which was previously unknown.

### Improved Approximations for Cubic and Cubic Bipartite TSP

- Mathematics, Computer ScienceArXiv
- 2015

Improved approximation guarantees for the traveling salesman problem on cubic graphs, and cubic bipartite graphs, are shown and a simple "local improvement" algorithm is given that finds a tour of length at most 5/4 n - 2.

### Approximating TSP walks in subcubic graphs

- MathematicsJournal of Combinatorial Theory, Series B
- 2021

### An Improved Analysis of the Mömke-Svensson Algorithm for Graph-TSP on Subquartic Graphs

- Computer Science, MathematicsESA
- 2014

It is shown that the approximation guarantee holds for all graphs that have an optimal solution to a standard linear programming relaxation of graph-TSP with subquartic support and that Momke and Svensson's algorithm has an approximation guarantee of at most $25/18$ on sub Quartic graphs.

### Cover and Conquer: Augmenting Decompositions for Connectivity Problems

- Computer Science, MathematicsArXiv
- 2017

A procedure to decompose an optimal solution for the subtour linear program into spanning, connected subgraphs that cover each 2-edge cut an even number of times is presented.

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It is proved that, as an upper bound, the 4/3 conjecture is true for this problem on cubic graphs, and a polynomial-time 7/5-approximation algorithm and a 7/ 5 bound on the integrality gap are obtained.

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This paper designs an algorithm that finds a tour of length (4/3−1/61236)n, implying that cubic graphs are among the few interesting classes for which the integrality gap of the subtour LP is strictly less than 4/3.

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