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The Traveling Salesperson Problem

The Traveling Salesperson Problem. Algorithms and Networks. Instance : n vertices (cities), distance between every pair of vertices Question : Find shortest (simple) cycle that visits every city. 4. 1. 2. 3. 2. 5. 2. 3. 4. 2. Problem. 4. 4. 1. 2. 1. 2. 3. 3. 2. 2. 5. 5.

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The Traveling Salesperson Problem

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  1. The Traveling Salesperson Problem Algorithms and Networks

  2. Instance: n vertices (cities), distance between every pair of vertices Question: Find shortest (simple) cycle that visits every city 4 1 2 3 2 5 2 3 4 2 Problem 4 4 1 2 1 2 3 3 2 2 5 5 2 2 3 4 2 3 4 2 11 A&N: TSP

  3. Assumptions • Lengths are non-negative (or positive) • Symmetric: w(u,v) = w(v,u) • Not always: painting machine application • Triangle inequality: for all x, y, z: w(x,y) + w(y,z) ³ w(x,z) • Always valid? A&N: TSP

  4. Construction heuristic:Nearest neighbor • Start at some vertex s; v=s; • While not all vertices visited • Select closest unvisited neighbor w of v • Go from v to w; • v=w • Go from v to s. Can have performance ratio O(log n) A&N: TSP

  5. Closest insertion heuristic • Build tour by starting with one vertex, and inserting vertices one by one. • Always insert vertex that is closest to a vertex already in tour. A&N: TSP

  6. A dynamic programming algorithm

  7. Held-Karp algorithm for TSP • O(n22n) algorithm for TSP • Uses Dynamic programming • Take some starting vertex s • For set of vertices R (s Î R), vertex w Î R, let • B(R,w) = minimum length of a path, that • Starts in s • Visits all vertices in R (and no other vertices) • Ends in w A&N: TSP

  8. TSP: Recursive formulation • B({s},s) = 0 • If |S| > 1, then • B(S,w) = minv Î S – {x}B(S-{x}, v}) + w(v,x) • If we have all B(V,v) then we can solve TSP. • Gives requested algorithm using DP-techniques. A&N: TSP

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