Approximation Algorithms

• An approximation algorithm tries to find the solution to an optimization problem.
• Sometimes an approximation can be close, and sometimes not so much.
• The hope is to find an algorithm which runs in polynomial time, which gives a very good answer.
• The vertex cover problem
  • Given a graph G=(V,E), then V' ⊆ V is a set of vertexes such that for all e = (u,v) ∈ E, either u∈ V' or v ∈ V' or both.
  • 
• An Approximation Algorithm
  • VERTEX-COVER is to find the minimal set of vertexes which cover G
  • VERTEX-COVER ∈ NP-Complete.

  • APPROX_VERTEX_COVER(G)
    
     C ← ∅
     E' ← E[G]
      while E' ≠ ∅ do
         let (u,v) be an arbitrary edge in E'
         C ← C ∪ {u,v}
         Remove from E' every edge incident on either u or v.
     return C
    
        

  • Normal questions.
  • Can you find the minimal cover for the graph?
  • Can you find a bad cover for the graph ?
  • Does this algorithm terminate? work?
  • How much extra will it cost?
    • Assume C is the set of vertexes produced by the algorithm.
    • Assume C^* is the optimal cover.
    • Assume A is the set of edges selected by the algorithm.
    • For every edge in A, at least one of the endpoints is in C^*
    • Therefore |C^* ≥ |A|
    • Also, |A| = 2*|C| (by line 5)
    • So 2|C| ≤ |C^*|
    • So APPROX_VERTEX_COVER is at most twice as bad as an optimal solution.
• Traveling Salesman Problem.
  • TSP ∈ NP-Complete
  • We will assume complete, undirected graph G,
  • With a cost function c(u,v) - the cost of traversing the edge from u to v.
  • Let c(X) be the cost of traversing any path in the graph.
  • c(u,v) ≥ 0, for all (u,v) ∈ E
  • They assume integer cost.
  • The triangle inequality states c(u,w) ≤ c(u,v) + c(v,w) for all v.
  • This does not need to hold for an arbitrary graph, but it makes sense in most "real" world applications.
  • Requiring the triangle inequality does not change this from being in NPC.
  • If we require the triangle inequality, then the following approximation algorithm will work.

    APPROX_TSP_TOUR(G,c)  // c is the cost function for the graph
    
     select a vertex r ∈ V to be the "root" (or starting point)
     compute a mst T for G from root r using MST_PRIM(G,c,r)
     Let L be the list of vertexes visited in a preorder tree walk of T.
     Return L
    
    	 

  • 
  • The MST
  • 
  • The Traversal
  • 
  • The Approximate Solution
  • 
• Let H^* be the optimal solution to the problem for a graph.
• Let T be any MST.
• A Spanning tree T' can be produced from H^* by removing any edge, so c(T) ≤ c(H^*)
• A full walk W, of the T (a,b,c,b,a,d,e,f,e,g,e,d,a) will traverse each edge twice, so
• c(W) = 2c(T)
• But this is not a tour, since it visits every non-leaf vertex at least twice.
• Since the triangle inequality exists, (c,b,a,d) can be replaced with (cd)
• So a new tour H = (a,b,c,d,e,f,g,a) can be constructed such that c(H) ≤ c(W)
• so c(H) ≤ c(W) ≤ 2c(H^*)
• If we remove the triangle inequality there can not be an approximation algorithm that runs in polynomial time.
  • let ρ = cost(C/C^*) (as above)
  • ρ > 1
  • Let G be an instance of the HAM-CYCLE problem.
  • Transform G to G' by:
  • G'= (V,E')
  • E' = {(u,v) | u,v ∈ V and u ≠ v}
    • c(u,v) = 1 if (u,v) ∈ E
    • c(u,v) = ρ|V|+1 otherwise.
  • This transformation can be done in polynomial time on |V| and |E|
  • But, we can run the approximation algorithm for TSP(G',c).
  • If G has a hamiltonin cycle, then it will be the solution to the TSP problem, and the cost will be |V|.
  • If G does not have a hamiltonian cycle, the cost of the solution to TSP will be at least ρ|V|+1 + (|V|-1) = ρ|V|+|V| > ρ|V|
  • But we know, by assumption, that the cost of the approximate TSP problem is no more than ρ|V|,
  • So there can be no "good" value for ρ
• The book has a number of other approximation methods for TSP.
• An a nice little study comparing some of these methods.
• They also have a solution to the Kanpsack Problem.