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Unformatted text preview: 6.889 — Lecture 15: Traveling Salesman (TSP) Christian Sommer [email protected] (figures by Philip Klein) November 2, 2011 Traveling Salesman Problem (TSP) given G = ( V,E ) find a tour visiting each 1 node v ∈ V . NP –hard optimization problem, hard even for planar graphs Polynomialtime approximation for general graphs: Christofides’ algorithm achieves 3 / 2 approximation Assumption (all of Lecture 15) undirected planar G , ‘ : E → R + 2 –approximation simple algorithm, bound approximation ratio in terms of minimum spanning tree • compute minimum spanning tree T . let ‘ ( T ) := ∑ e ∈ T ‘ ( e ) • duplicate all edges Eulerian graph • find Eulerian cycle, length at most 2 ‘ ( T ) • (if G is the complete graph K n , Eulerian cycle can be converted into Hamiltonian cycle by skipping already visited nodes) any tour needs to visit all nodes, total length at least ‘ ( T ) , hence 2 –approximation Recall: LinearTime Approximation Schemes for Planar Graphs (L. 8) Example min VERTEXCOVER Algorithm given G and approximation parameter ∈ (0 , 1) , let k = 1 / 1. BFS in G 2. G ij ← graph induced by k + 1 BFS levels jk + i to ( j + 1) k + i ( shift i , 6 i < k , and slice j ) 3. S ij ← min VERTEXCOVER of G ij (dynamic programming on graph with treewidth O ( k ) ) 4. S i ← S j S ij 5. RETURN best S i (best shift i , 6 i < k , smallest  S i  ) Running Time dynamic program runs in time 2 O ( k )  V ( G ij )  , overall 2 O ( k ) n Correctness and Approximation Ratio two properties used 3. part of OPT in G ij is a feasible solution for G ij . consequence:  OPT ∩ V ( G ij )  >  S ij  optimum solution OPT induces solution on subgraph G ij for at least one shift i overlap  OPT i  is small 2 ( 6  OPT  /k =  OPT  ) 4. solutions in G ij together form a feasible solution for G , S j S ij is a solution for G (for any i ) 1 visiting only a subset U ⊆ V to be discussed in Lectures 16 and 17 2 define OPT i = OPT ∩ { all nodes on BFS level i mod k } 1 Figure 1: Consecutive slices (subgraphs) overlap by one single level (boundary)....
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 Fall '11
 ErikDemaine
 Algorithms, Dynamic Programming, Graph Theory, Hamiltonian path, Travelling salesman problem, Seven Bridges of Königsberg, total length

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