# Approximate TSP in Graphs with Forbidden Minors

@inproceedings{Grigni2000ApproximateTI, title={Approximate TSP in Graphs with Forbidden Minors}, author={Michelangelo Grigni}, booktitle={ICALP}, year={2000} }

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

- Mathematics, Computer ScienceSTOC '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

- Computer ScienceSIAM 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

- Mathematics
- 2004

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

- Computer Science46th 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, Mathematics2020 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

- MathematicsSODA
- 2020

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 ScienceArXiv
- 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

- MathematicsSODA '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.

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