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Is their a variant of Traveling Salesman Problem or other algorithms about the following problem:

Say G is an incomplete undirected weighted graph. V is a subset of vertices of G.

How to find a simple closed circuit along V (and probably some other vertices of G), which has a minimal weight between each two vertices of V.

Thank you

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Is there a name or published document or related research paper for this problem?

Luca
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1 Answers1

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First, because V (which is a subset of vertices of G as you state) is itself a weighted graph - calculate the edges of V first so you can ignore G.

For example this input of G:

G1 (Va) -- 5 meter --> G2 (no V) -- 10 meter --> G3 (Vb)

Can be simplified to this input of V:

Va -- 15 meter --> Vb

The fun starts when you have the input has multiple paths between to V vertices (with no other V vertices in between):

G1 (Va) -- 5 meter --> G2 (no V) -- 10 meter --> G3 (Vb)
G1 (Va) -- 7 meter --> G4 (no V) -- 7 meter --> G3 (Vb)

Then the simplified form is (it takes the second route):

Va -- 14 meter --> Vb

Use the Dijkstra algorithm during that simplification.

Second, apply a good TSP algorithm on V. It's NP-complete so there is no perfect algorithm. There are many available frameworks depending on your programming language (java, C/C++, ...).

Geoffrey De Smet
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  • Thanks! That's pretty awesome! One follow up question: is there any related research paper or any other published document about this problem? – Luca Aug 05 '13 at 10:05
  • Or is there a name for this sort of problems? – Luca Aug 05 '13 at 10:07
  • TSP is one of the most intensively studied research problems. There are plenty of papers and books about it (see Google Scolar). I doubt you 'll find one specifically of your form however, because - from a research perspective - the complexity it adds can be simplified away without any trade-off in the TSP. – Geoffrey De Smet Aug 05 '13 at 10:19
  • I really appreciate your help, and I really want to know is there a name for this particular Dijkstra algorithm + TSP problem, so that I can know the efficiency and compare different solutions? – Luca Aug 05 '13 at 10:23