L15 - 6.889 Lecture 15: Traveling Salesman (TSP) Christian...

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Unformatted text preview: 6.889 Lecture 15: Traveling Salesman (TSP) Christian Sommer csom@mit.edu (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 Polynomial-time 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: Linear-Time 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 tree-width 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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L15 - 6.889 Lecture 15: Traveling Salesman (TSP) Christian...

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