Death Of a Traveling Salesman Problem

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Death Of a Traveling Salesman Problem

• Charles Huber
• Tuesday, March 14

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Traveling Salesman Problem

• Given an undirected, complete graph G and cost
  matrix C, find an optimal Hamiltonian cycle
• Simple enough, but the devil is in the details
• Clipart

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TSP Variants

• Asymmetric
• Edge weights vary by direction
• Non-complete graph
• Hamiltonian may not exist!
• Triangle inequality
• Christofides approx at most 1.5x optimal
• Euclidian
• Subset of triangle inequality
• Makes heuristics happy!

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TSP Solutions

• NP-Hard
• Enumerate all paths, O(n!)
• Not going anywhere for a while? Grab a
  Snickers!
• Approximation algorithms
• Heuristics
• Genetic Algorithms
• Local Search Algorithms
• Fast but sadly non-optimal

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Dynamic Programming

• ltstrikegtLike eXtreme Programming but allocated
  from the heap!lt/strikegt
• Trades memory for time
• Lots of applications
• Edit distance (diff, anyone?)
• Bellman-Ford SSSP
• Knapsack problem

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Dynamic Programming, cont.

• Dynamic programming requirements
• Optimal sub-structure
• Lots of memory
• Cant really be parallelized like divide and
  conquer algorithms
• Too much shared state (ie, results table)

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TSP Solutions, cont.

• Can dynamic programming be applied to the TSP?
• Yes!
• What do we gain?
• Significant reduction in run-time complexity
• O(n22n) vs O(n!)
• Retains optimality of brute-force method

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Algorithm
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Example
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