{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

L16-TSP-MST

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

This preview shows pages 1–7. Sign up to view the full content.

Approximation Algorithms: Combinatorial Approaches Lecture 16: March 7

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 37

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

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online