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# Approximating Graphic TSP by Matchings

foundations of computer science, (2011): 560-569

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Abstract

We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), the approach yie...More

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Introduction

- The traveling salesman problem in metric graphs is one of most fundamental NP-hard optimization problems.
- While the problem is known to be APX-hard and NP-hard to approximate with a ratio better than 220/219 [20], the best upper bound is still the 1.5-approximation algorithm obtained by Christofides [4] more than three decades ago.
- The best lower bound on its integrality gap is 4/3 and conjectured to be tight [11].
- The best known analysis [22, 23] is based on Christofides’ algorithm and gives an upper bound on the integrality gap of 1.5

Highlights

- The traveling salesman problem in metric graphs is one of most fundamental NP-hard optimization problems
- While the problem is known to be APX-hard and NP-hard to approximate with a ratio better than 220/219 [20], the best upper bound is still the 1.5-approximation algorithm obtained by Christofides [4] more than three decades ago
- We propose an alternative framework for approximating the metric TSP and use it to obtain an improved approximation algorithm for graph-TSP
- The result implies an upper bound on the integrality gap of the Held-Karp relaxation for graph
- We show how to use our framework to obtain an improved approximation algorithm for general graphs
- We have introduced a framework of removable pairings to find Eulerian multigraphs. This framework proved to be useful to obtain an approximation algorithm for graph-TSP with an approximation ratio smaller than 1.461 and to obtain a tight upper bound on the integrality gap of the Held-Karp relaxation for a restricted class of graphs that contains degree three bounded and clawfree graphs

Results

**The authors' Results and Overview of Techniques**

The authors propose an alternative framework for approximating the metric TSP and use it to obtain an improved approximation algorithm for graph-TSP.

Theorem 1.1 There√ is a polynomial time approximation algorithm for graph-TSP with performance guarantee 14·(√ 2−1) < 1.461. 12· 2−13

The result implies an upper bound on the integrality gap of the Held-Karp relaxation for graph-

TSP that matches the approximation ratio.- The result implies an upper bound on the integrality gap of the Held-Karp relaxation for graph-.
- E., each maximally 2-vertex-connected subgraph) is either claw-free or of degree at most 3, the authors use the framework to construct a polynomial time 4/3-approximation algorithm showing that the conjectured integrality gap of the Held-Karp relaxation is tight for those graphs.
- The techniques allow them to prove the tight result that any 2-vertex-connected graph of degree at most.
- The authors' framework is based on earlier works by Frederickson & Ja’ja’ [7] and Monma et al [18], who related the cost of an optimal tour to the size of a minimum 2-vertex-connected subgraph.

Conclusion

- The authors have introduced a framework of removable pairings to find Eulerian multigraphs. This framework proved to be useful to obtain an approximation algorithm for graph-TSP with an approximation ratio smaller than 1.461 and to obtain a tight upper bound on the integrality gap of the Held-Karp relaxation for a restricted class of graphs that contains degree three bounded and clawfree graphs.
- With the same techniques and a more detailed analysis, the result translates to the traveling salesman path problem on graphic metrics with prespecified start and end vertex
- In this way, one is guaranteed to obtain an approximation ratio smaller than 1.586 and, for the degree three bounded case, the approximation ratio gets arbitrarily close to 1.5

Funding

- ∗This research was supported by ERC Advanced investigator grant 226203

Reference

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- 2. Otherwise, let v be a cut vertex whose removal results in components C1, C2,..., Cl with l > 1. Recursively run A on the l sub-instances (G1, s1, t1),..., (Gl, sl, tl) and return the union of the obtained solutions, where Gi denotes the subgraph of G induced by Ci ∪ {v}, si =

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