The traveling salesman problem on cubic and subcubic graphs

  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},
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… 

Applications of Circulations and Removable Pairings to TSP and 2ECSS

An approximation algorithm is presented that gives a 2-edge-connected spanning subgraph with the number of edges at most n+k−1 − k−2 k−1 with a novel use of circulations, which improves both the approximation ratio and the simplicity of the proof compared to i a result by Huh in 2004.

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

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

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

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.

On Applying Methods for Graph-TSP to Metric TSP

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

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.

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

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.

Shorter tours and longer detours: uniform covers and a bit beyond

The results show that if the everywhere 2 3 vector is an optimal solution for the subtour elimination linear programming relaxation for TSP, then a tour with weight at most 27 19 times that of an optimal tour can be found efficiently.

Approximating the Regular Graphic TSP in Near Linear Time

The key ingredient of the algorithm is a technique that uses edge-coloring algorithms to sample a cycle cover with $O(n/\log k)$ cycles with high probability, in near linear time.



Graph Properties that Facilitate Travelling

It is proved that, given a graph and a partition of the vertices into cliques, optimal solutions for both Graph TSP and TSP(1,2) can be found in time n O(2), and it is conjecture that the time complexity can be improved to n O() for any partition of a graph into clique.

TSP Tours in Cubic Graphs: Beyond 4/3

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.

A Randomized Rounding Approach to the Traveling Salesman Problem

This work gives a (3/2-\eps_0)-approximation algorithm that finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree.

Approximating Graphic TSP by Matchings

  • Tobias MömkeO. Svensson
  • Computer Science, Mathematics
    2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
  • 2011
A framework for approximating the metric TSP based on a novel use of matchings that allows for generalizations in a natural way and also leads to a 1.586-approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.

Shorter tours by nicer ears: 7/5-approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs

The key new ingredient of all algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs that provides the lower bounds that are used to deduce the approximation ratios.

Analyzing the Held-Karp TSP Bound: A Monotonicity Property with Application

Approximability of dense and sparse instances of minimum 2-connectivity, TSP and path problems

It is proved that 2-EC, 2-VC and TSP (1,2) are Max SNP-hard even on 3-regular graphs, and provide explicit hardness constants, under P ≠ NP, which are the first explicit hardness results on sparse and dense graphs for these problems.

13/9-approximation for Graphic TSP

This paper provides an improved analysis of the approach used by Momke and Svensson, yielding a bound of 13/9 on the approximation factor, as well as a Bound of 19/12+epsilon for any epsilon>0 for a more general Travelling Salesman Path Problem in graphic metrics.

An improved upper bound for the TSP in cubic 3-edge-connected graphs

A polynomial-time approximation scheme for weighted planar graph TSP

This work finds a salesman tour of total cost at most (1 + E) times optimal in time n for any E > 6, and presents a quasi-polynomial time algorithm for the Steiner version of this problem.