TSP-0 Traveling Salesperson Problem (TSP) Input: Undirected edge weighted complete graph G = (V, E, W ), where W : e R +. Tour: Find a path that starts at vertex 1, visits every vertex exactly once, and ends at vertex 1. Cost (weight) of a Tour: Sum of the weight of the edges in the tour. TSP: Find a least cost tour. TSP is NP-hard even when the graph is defined by a set of points in 2D. Later on we show that the approximation problem is NP-hard.
TSP-1 Relaxed Version of TSP A relaxed tour is defined as as a path that starts at vertex 1, visits every vertex at least once, and ends at vertex 1. The Relaxed Version of the TSP is to find a least cost relaxed tour.
TSP-2 Euler Tours and Multigraphs A multigraph is a graph that may contain more than one edge between a pair of vertices. A multigraph in which there is a path between every pair vertices (or connected multigraph) and every vertex is of even degree is called an Euler multigraph. An Euler tour of a multigraph is a relaxed tour of a multigraph in which every edge is traversed by the relaxed tour exactly once. There is a linear time (with respect to the number of vertices and edges in the multigraph) algorithm (CS 130A) that given an Eulerean multigraph it constructs an Euler tour.
TSP-3 Approx.: Relaxed TSP (Via Restriction) Construct a tour that uses the least number of different edges. Least number of different edges is n 1. From all sets of n 1 edges, pick the set with least total weight. This is just a minimum cost spanning tree. Make a copy of each edge. The resulting graph is an Eulerean multigraph. Find an Euler Tour The Euler tour is the solution. Claim: ˆf 2f. (Follows from ˆf = 2W (MCST ) and f > W (MCST ), where W (MCST ) is the sum of the weight of the edges in a minimum cost spanning tree for the TSP graph) (We prove this later on).
TSP-4 Metric TSP Graph G is said to satisfy the triangle inequality (or the graph is called a metric graph) if for every pair of vertices u and v, w(u, v) w(p(u, v)), where p(u, v) is any path from vertex u to vertex v and w(p(u, v)) is the sum of the weight of the edges in p(u, v).
TSP-5 Approximation to Metric TSP Approximation algorithm is based on Relaxation - Restriction. Relaxation: May visit vertices more than once. Restriction: Visit least number of different edges. Algorithm MWST (G = (V, E, W )) Construct a min weight (cost) spanning tree S = (V, T ); /* I.e., sum of weight of edges in tree is least possible */; Construct multigraph M = (V, T T ); /* I.e., there are multiple copies of edges between nodes */ Construct Euler circuit EC for M; Apply Transformation R to EC n 1 times; /* Resulting graph is a TOUR */ End of Algorithm MWST
TSP-6 Transformation R Example MWST: (For example use Kruskal s algorithm). Multigraph: M = (V, T T ) Euler Circuit EC for M 12 11 13 10 14 15 16 9 8 7 1 6 2 3 4 5
TSP-7 2W(T) 2W(T) 2W(T) 2W(T) 2W(T) 2W(T) 2W(T) 2W(T) SOL at most 2W(T)
TSP-8 Approximation Bound f*(i) Spanning Tree Deleting the edge with largest weight from the an optimal tour results in a spanning tree. Therefore, f (I)(1 1 n ) W (T ) So, ˆf(I) 2W (T ) 2f (I)(1 1 n ). This is equivalent to ˆf(I) f (I) = 2 2 n
TSP-9 Worst Case Example 1 2 ˆf(I) = 10; f (I) = 6; ˆf(I) f (I) = 5 3 In General, 2 2 n. So the approximation bound is tight.
TSP-10 Christofidas Method Definitions (for graphs) Matching: Subset of edges no two of which are adjacent to the same vertex. Complete Matching: A matching that Covers all the vertices in the graph. Weight of Complete Matching (for edge weighted graphs): Sum of weight of edges in the matching.
TSP-11 MWCM Algorithm MWCM (G = (V, E, W )) Construct a min weight (cost) spanning tree S = (V, T ); Let V be the set of odd degree vertices in S = (V, T ); Let M be the graph induced by V in G; Let C be a minimum weight complete matching in M ; Let M = (V, T C); Construct Euler circuit EC for M; Apply Transformation R to EC until you get a TOUR; /* Resulting graph is a TOUR */ End of Algorithm MWCM
TSP-12 S M M C
TSP-13 ˆf(I) W (T ) + W (C) Since W (T ) (1 1 n )f (I) ˆf(I) (1 1 n )f (I) + W (C) o e o e o e W (e) W (C) and W (o) W (C) f (I) W (Edges e o) = W (e) + W (o) 2W (C) Therefore, ˆf(I) (1 1 n )f (I) + 0.5f (I) ( 3 2 1 n )f (I)
TSP-14
TSP-15 Worst Case Example 1 2 ˆf(I) = 8; f (I) = 6; ˆf(I) f (I) = 4 3 In General, 1 2 1 n. So the approximation bound is tight.
TSP-16 Equivalence of Relaxed and Metric TSP The metric TSP (graph is metric) and the relaxed TSP are equivalent problems. Given an instance of the relaxed TSP for graph G, construct the graph G for the metric TSP as follows. Every vertex in G is a vertex in G. The graph G is complete and the weight of the edge from vertex u to vertex v is defined as w(sp(u, v)) in G, where sp(u, v) is a shortest path from vertex u to vertex v in G, and w(sp(u, v)) is the sum of the weight of the edges in sp(u, v). There is a correspondence between tours in the metric TSP and relaxed tour in the relaxed TSP and vice-versa.