Approximation Algorithms

• Since we can't find a good way to solve problems in NP-Complete, what can we do?
• TSP: Traveling Salesman Problem
  • Given a list of n interconnected cities and the cost to travel between them, find the least cost cycle that visits each city exactly once returning to the starting point.
  • This is modeled as an undirected complete weighted graph.
• This has been shown to be NP-Complete.
• If you place two reasonable restrictions on this problem, the problem becomes Euclidean
  • w(a,b) = w(b,a), or symmetry.
  • w(a,c) ≤ w(a,b) + w(b,c) , or the triangle inequality
• These assumptions are reasonable for cities in a plane, or even three space.
  • You frequently see such restrictions to make the problem more solvable.
• Given these assumptions you can use the following algorithm

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     Sort the edges by weight, lowest to highest.
     Repeat until all cities have been added
         Select the lowest cost edge which does not
    
             produce a vertex with degree larger than 2
             produce a cycle of length less than n 
             Add this edge to the solution.  
     

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  • It can be shown that
    • Let $c_a$ be the cost of the approximate solution.
    • Let $c_o$ be the cost of the optimal solution.
    • Let $n$ be the number of cities.
    • $\frac{c_a}{c_o} \le \frac{1}{2}(\lceil \log_2 n\rceil + 1)$
• A second solution is

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     form a minimal spanning tree
     Perform a DFS on the tree, recording the vertexes passed.
     Scan this list, remove all repeated occurrences.
     Connect the last vertex to the first vertex.  
     

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  • It can be shown that $c_a \le 2c_o$ when the Euclidean restrictions are applied.
• Currently the Christofides-Serdyukov algorithm is the best we know.
  • Find the MST
  • Add in edges on the odd degree vertexes
  • This can then be modified by several polynomial time algorithms we have not yet studied to produce a solution to the TSP.
  • It can be shown that $c_a \le \frac{3}{2}c_o$.
• But there might be a better solution see this article.