# Randomly select a starting node 2 add to the last

• Test Prep
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Randomly select a starting node 2. Add to the last node the closest node until no more node is available 3. Connect the last node with the first node. Running time O( n 2 ) Worst-case guarantee: L/L* 0.5( log 2 (n) +1) Nearest insertion heuristic 1. Select a node and its closest node and build a tour of two nodes. 2. At any stage when we have a tour containing node set C , select the closest node k V \ C , such that { c ( i , k ), i C , k V \ C } is minimized . 3. Insert k into the tour C in the best way (find ( i, j ), the edge in the tour that minimizes c ( i, k )+ c ( k, j )– c ( i, j ), replace ( i, j ) with ( i, k ) and ( k, j ).) 4. Repeat 2-3 until V \ C is empty. Running time O( n 3 ) Worst-Case Guarantee: L/L* 2 Farthest insertion heuristic 1. Select a farthest node pair and build a tour of two nodes. 2. At any stage when we have a tour containing node set C , select the closest node k V \C , such that { c ( i , k ), i C , k V \C } is maximized . 3. Insert k into the tour C in the best way (find ( i, j ), the edge in the tour that minimizes c ( i, k )+ c ( k, j )– c ( i, j ), replace ( i, j ) with ( i, k ) and ( k, j ).) 4. Repeat 2-3 until V \ C is empty. Running time O( n 3 ) , Practically quick efficient Worst-Case Guarantee: L / L* log2(n) Note: The worst-case guarantee of these heuristic algorithms was available in D. J. Rosenkrantz, R. E. Stearns and P. M. Lewis, (1977) An Analysis of Several Heuristics for the Traveling Salesman Problem, SIAM Journal on Computing , 6(3): 563-581. i j k C V\C

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Logistics Systems Analysis Construction Heuristics for TSP Cheapest insertion heuristic 1. Select a node and its closest node and build a tour of two nodes. 2. At any stage when we have a tour containing node set C , select the closest node k V \C , such that { c ( i, k )+ c ( k, j )– c ( i, j ), i,j C, k V \C } is minimized. 3. Insert k into the tour C in the best way (replace ( i,j ) with ( i,k ) and ( k,j ).). 4. Repeat 2-3 until V \ C is empty. Running time O( n 3 ) Worst-Case Guarantee: L / L* 2 Minimum Spanning Tree (MST) -based heuristic 1. Find the minimum spanning tree T . O( n 2 ) 2. Duplicate every edge on the minimum spanning tree to form a multi-graph O( n ) 3. Eliminate repetitions in the graph by taking short cuts. O( n ) ( Shortcut traversal: whenever the depth first traversal would take back to an already-visited vertex, skip ahead in the traversal and go directly to the next unvisited vertex; if all vertices have been visited come back to the start) Running time O( n 2 ) Worst-Case Guarantee: L/ L* 2 Logistics Systems Analysis Construction Heuristics for TSP Christofide’s heuristic 1. Find the minimum spanning tree T . O( n 2 ) 2. Find the set of nodes in T with odd degree, N 1 , and find the shortest complete matching M in the complete graph consisting of the nodes in N 1 only. Let G be the multigraph containing all nodes and edges in T and M . O( n 4 ) 3. Find an Euler tour S on G (each node appears at least once and each edge exactly once); 4. Eliminate repetitions in the Euler tour S by taking short cuts. O( n ) Running time O( n 4 ) Worst-Case Guarantee: L/ L* 3/2 Proposition: MST-based heuristic yields a tour length L such that L/L * 3/2.
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