# Approximate TSP in Graphs with Forbidden Minors

@inproceedings{Grigni2000ApproximateTI,
title={Approximate TSP in Graphs with Forbidden Minors},
author={Michelangelo Grigni},
booktitle={ICALP},
year={2000}
}
• M. Grigni
• Published in ICALP 9 July 2000
• Mathematics
Given as input an edge-weighted graph, we analyze two algorithms for finding subgraphs with low total edge weight. The first algorithm finds a separator subgraph with a small number of components, and is analyzed for graphs with an arbitrary excluded minor. The second algorithm finds a spanner with small stretch factor, and is analyzed for graphs in a hereditary family G(k). These results imply (i) a QPTAS (quasi-polynomial time approximation scheme) for the TSP (traveling salesperson problem…
29 Citations

### A subset spanner for Planar graphs,: with application to subset TSP

• P. Klein
• Mathematics, Computer Science
STOC '06
• 2006
This paper presents a PTAS for finding a TSP among a given subset of nodes of a planar graph such that distances in H between nodes in S are at most 1+ε times the corresponding distances in G.

### A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

• P. Klein
• Computer Science
SIAM J. Comput.
• 2008
An algorithm to find an $\epsilon$-optimal traveling salesman tour in the shortest-path metric defined by an undirected planar graph with nonnegative edge-lengths is given.

### Grigni: [21] Well-Connected Separators for Planar Graphs

Given an n-vertex weighted planar graph G, a separator is a subset S of vertices such that each component of G S has at most two-thirds of the original weight. We give an algorithm nding a separator

### A linear-time approximation scheme for planar weighted TSP

• P. Klein
• Computer Science
46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
• 2005
An algorithm requiring O(c/sup 1/c2/ n) time to find an /spl epsi/-optimal traveling salesman tour in the metric defined by a planar graph with nonnegative edge-lengths is given.

### On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs

• Computer Science, Mathematics
2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
• 2020
The paper designs the first FPT approximation scheme for bounded-capacity vehicle routing on bounded-treewidth graphs (parameterized by the treewidth) and shows the two following structural results for minor-free metrics.

### Grigni : [ 21 ] Well-Connected Separators for Planar

• Mathematics
• 2007
Given an n-vertex weighted planar graph G, a separator is a subset S of vertices such that each component of G ? S has at most two-thirds of the original weight. We give an algorithm nding a

### A PTAS for subset TSP in minor-free graphs

We give the first PTAS for the subset Traveling Salesperson Problem (TSP) in $H$-minor-free graphs. This resolves a long standing open problem in a long line of work on designing PTASes for TSP in

### Finding Light Spanners in Bounded Pathwidth Graphs

• Mathematics, Computer Science
ArXiv
• 2011
This paper shows that light spanners exist for graphs with bounded pathwidth, via the construction of light enough monotone spanning trees in such graphs.

### Approximation algorithms via contraction decomposition

• Mathematics
SODA '07
• 2007
The decomposition result is a powerful tool for obtaining PTASs for contraction-closed problems (whose optimal solution only improves under contraction) and is the only main difficulty in extending the results to general H-minor-free graphs.

### A linear-time approximation scheme for TSP in planar graphs with edge-weights

• Mathematics, Computer Science
• 2005
We give an algorithm requiringO(c 2 n) time to find an -optimal traveling salesman tour in the metric defined by a planar graph with nonnegative edge-lengt hs.

## References

SHOWING 1-10 OF 12 REFERENCES

### On sparse spanners of weighted graphs

• Mathematics, Computer Science
Discret. Comput. Geom.
• 1993
This paper gives a simple algorithm for constructing sparse spanners for arbitrary weighted graphs and applies this algorithm to obtain specific results for planar graphs and Euclidean graphs.

### A polynomial-time approximation scheme for weighted planar graph TSP

• Computer Science
SODA '98
• 1998
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.

### A separator theorem for graphs with an excluded minor and its applications

• Mathematics
STOC '90
• 1990
It follows that for any fixed graph H, given a graph G with n vertices and with no H-minor one can approximate the size of the maximum independent set of G up to a relative error of 1/ √ log n in polynomial time, find that size exactly and solve any sparse system of n linear equations in n unknowns in time O(n).

### A Separator Theorem for Planar Graphs

• Mathematics
• 1977
Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more

### An approximation scheme for planar graph TSP

• Computer Science, Mathematics
Proceedings of IEEE 36th Annual Foundations of Computer Science
• 1995
We consider the special case of the traveling salesman problem (TSP) in which the distance metric is the shortest-path metric of a planar unweighted graph. We present a polynomial-time approximation

### Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems

A consequence of the main theorem is a simple polynomial-time approximation scheme for geometric instances of several network optimization problems, including the Steiner minimum spanning tree, the traveling salesperson problem (TSP), and the k-MST problem.

### An extremal function for contractions of graphs

• A. Thomason
• Mathematics, Computer Science
Mathematical Proceedings of the Cambridge Philosophical Society
• 1984
A simple argument is presented showing c(p) ≤ 2.68p √(log2p)(l + ο(l)).

### Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems

The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-aproximation in polynomial time.

### The Traveling Salesman Problem with Distances One and Two

• Computer Science
Math. Oper. Res.
• 1993
We present a polynomial-time approximation algorithm with worst-case ratio 7/6 for the special case of the traveling salesman problem in which all distances are either one or two. We also show that