Graph-TSP from Steiner Cycles

@article{Iwata2014GraphTSPFS,
  title={Graph-TSP from Steiner Cycles},
  author={Satoru Iwata and Alantha Newman and Ramamoorthi Ravi},
  journal={ArXiv},
  year={2014},
  volume={abs/1407.2844}
}
We present an approach for the traveling salesman problem with graph metric based on Steiner cycles. A Steiner cycle is a cycle that is required to contain some specified subset of vertices. For a graph \(G\), if we can find a spanning tree \(T\) and a simple cycle that contains the vertices with odd-degree in \(T\), then we show how to combine the classic “double spanning tree” algorithm with Christofides’ algorithm to obtain a TSP tour of length at most \(\frac{4n}{3}\). We use this approach… 
Approximation Algorithms for the Minimum 2-edge Connected Spanning Subgraph Problem and the Graph-TSP in Regular Bipartite Graphs via Restricted 2-factors
TLDR
A 7 =6-approximation algorithm for the minimum 2-edge connected spanning subgraph problem in 3-edgeconnected cubic bipartite graphs, which improves upon the previous best ratio and makes use of bipartiteness to attain a better ratio.
An Improved Discrete Firefly Algorithm Used for Traveling Salesman Problem
TLDR
The comparison experiment results show that the novel algorithm can search perfect solution within a short time, and greatly improve the effectiveness of solving the traveling salesman problem.
Modified Discrete Firefly Algorithm Combining Genetic Algorithm for Traveling Salesman Problem
TLDR
The comparison experiment results show that the novel algorithm can search perfect solution within a short time, and greatly improve the effectiveness of solving the traveling salesman problem, it also significantly improves computing speed and reduces iteration number.

References

SHOWING 1-10 OF 43 REFERENCES
Approximability of the Minimum Steiner Cycle Problem
TLDR
This paper shows that, if P NP, there is no approximation algorithm for SCP on directed graphs with an approximation ratio polynomial in the input size, and shows that SCP on undirected graphs with constant number of terminals and edge costs satisfying the beta-relaxed triangle inequality is approximable with the ratio beta^2+beta.
The traveling salesman problem on a graph and some related integer polyhedra
TLDR
Some facet inducing inequalities of the convex hull of the solutions to the Graphical Traveling Salesman Problem are given and the so-called comb inequalities of Grötschel and Padberg are generalized.
A Permanent Approach to the Traveling Salesman Problem
  • Nisheeth K. Vishnoi
  • Mathematics
    2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
  • 2012
TLDR
The techniques in this paper suggest new permanent-based approaches for TSP which could be useful in attacking other interesting cases of TSP.
Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem
TLDR
An O(n3) heuristic algorithm is described for solving d-city travelling salesman problems (TSP) whose cost matrix satisfies the triangularity condition and a worst-case analysis of this heuristic shows that the ratio of the answer obtained to the optimum TSP solution is strictly less than 3/2.
Approximating Graphic TSP by Matchings
  • Tobias Mömke, O. Svensson
  • Computer Science, Mathematics
    2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
  • 2011
TLDR
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.
A Randomized Rounding Approach to the Traveling Salesman Problem
TLDR
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.
Analysis of Christofides ' heuristic : Some paths are more difficult than cycles
For the traveling salesman problem in which the distances satisfy the triangle inequality, Christofides' heuristic produces a tour whose length is guaranteed to be less than ~ times the optimum tour
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
TLDR
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.
TSP on Cubic and Subcubic Graphs
TLDR
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.
...
...