L16-TSP-MST - Approximation Algorithms: Combinatorial...

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Approximation Algorithms: Combinatorial Approaches Lecture 16: March 7
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The Hamiltonian Path Problem Hamiltonian Path Problem: Given an undirected graph, find a cycle visiting every vertex exactly once. The Hamiltonian Path problem is NP-complete. Eulerian Path Problem: Given an undirected graph, find a walk visiting every edge exactly once. Notice that in a walk some vertices may have been visited more than once. The Eulerian Path problem is polynomial time solvable. A graph has an Eulerian path if and only if every vertex has an even degree.
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The Hamiltonian Path Problem Application: Seating assignment problem: Given n persons and a big round table of n seats, each pair of persons may like each other or hate each other. Can we find a seating assignment so that everyone likes his/her neighbors (only two neighbors)? Construct a graph as follows. For each person, create a vertex. For each pair of vertices, add an edge if and only if they like each other. Then the problem of finding a seating assignment is reduced to the Hamiltonian path problem.
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The Traveling Salesman Problem Traveling Salesman Problem (TSP): Given a complete graph with nonnegative edge costs, Find a minimum cost cycle visiting every vertex exactly once. Given a number of cities and the costs of traveling from any city to any other city, what is the cheapest round-trip route that visits each city exactly once and then returns to the starting city? One of the most well-studied problem in combinatorial optimization.
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NP-completeness of Traveling Salesman Problem Hamiltonian Path Problem: Given an undirected graph, find a cycle visiting every vertex exactly once. For each edge, we add an edge of cost 1. For each non-edge, we add an edge of cost 2. Traveling Salesman Problem (TSP): Given a complete graph with nonnegative edge costs, Find a minimum cost cycle visiting every vertex exactly once. The Hamiltonian path problem is a special case of the Traveling Salesman Problem. If there is a Hamiltonian path, then there is a cycle of cost n. If there is no Hamiltonian path, then every cycle has cost at least n+1.
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Theorem : There is no constant factor approximation algorithm for TSP, unless P=NP. Idea: Use the Hamiltonian path problem. For each edge, we add an edge of cost 1. For each non-edge, we add an edge of cost 2 nk. If there is a Hamiltonian path, then there is a cycle of cost n. If there is no Hamiltonian path, then every cycle has cost greater than nk . So, if you have a k-approximation algorithm for TSP, one just needs to check if the returned solution is at most nk . If yes, then the original graph has a Hamiltonian path.
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This note was uploaded on 10/13/2009 for the course CS 5150 taught by Professor Xulei during the Spring '09 term at University of Central Arkansas.

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L16-TSP-MST - Approximation Algorithms: Combinatorial...

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