Oct 20, (1+sqrt(5))/2-approximation algorithm for the s-t path TSP for an that the natural variant of Christofides’ algorithm is a 5/3-approximation. If P ≠ NP, there is no ρ-approximation for TSP for any ρ ≥ 1. Proof (by contradiction). s. Suppose . a b c h d e f g a. TSP: Christofides Algorithm. Theorem. The Traveling Salesman Problem (TSP) is a challenge to the salesman who wants to visit every location . 4 Approximation Algorithm 2: Christofides’. Algorithm.
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After creating the minimum spanning tree, the next step in Christofides’ TSP algorithm is to find all the N vertices with odd degree and find a christotides weight perfect matching for these odd vertices. N is even, so a bipartite matching is possible. Does Christofides’ algorithm really need to run a min-weight bipartite matching for all of these possible partitions?
Or is there a better way? I realize there is an approximate solution, which is to greedily match each vertex with another vertex that is closest to it. However, if christofudes exact solution is to try all possible partitions, this seems inefficient.
There is the Blossom algorithm by Edmonds that determines a maximal matching for a weighted graph.
The blossom algorithm can be used to find a minimal matching of an arbitrary graph. It’s nicer to use than tep bipartite matching algorithm on all possible bipartitions, and will always find a minimal perfect matching in the TSP case.
Usually when we talk about approximation algorithms, we are considering only efficient polytime algorithms.
This one is no exception. The standard blossom algorithm is applicable to a non-weighted graph.
 Improving Christofides’ Algorithm for the s-t Path TSP
Chfistofides last section on the wiki page says that the Blossom christoffides is only a subroutine if the goal is to find a min-weight or max-weight maximal matching on a weighted graph, and that a combinatorial algorithm needs to encapsulate the blossom algorithm. The Kolmogorov paper references an overview paper W. Computing minimum-weight perfect matchings.
In that paper the weighted version is also attributed to Edmonds: That sounds promising, I’ll have to study that algorithm, thanks for the reference. Combinatorial means that it operates in a discrete way. There are several polytime algorithms for minimum matching.
I’m not sure what this adds over the existing answer. Can I encourage you to take a look at some of our unanswered questions and see if you can contribute a useful answer to them? After reading the existing answer, it wasn’t clear to me why the blossom algorithm was useful in this case, so I thought I’d elaborate.
Computer Science > Data Structures and Algorithms
Feel free to delete this answer – I just thought the extra comments would be useful for the next dummy like me that is struggling with the same problem. Sign up or log in Sign up using Google. Sign up using Facebook.