Finding the Exact Integrality Gap for Small Traveling Salesman Problems

Finding the Exact Integrality Gap for Small Traveling Salesman Problems

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Article ID: iaor200954196
Country: United States
Volume: 33
Issue: 4
Start Page Number: 921
End Page Number: 931
Publication Date: Nov 2008
Journal: Mathematics of Operations Research
Authors: ,
Keywords: computational analysis
Abstract:

The symmetric traveling salesman problem (STSP) is to find a minimum weight Hamiltonian cycle in a weighted complete graph on n nodes. One direction which seems promising for finding improved solutions for the STSP is the study of a linear relaxation of this problem called the subtour elimination problem (SEP). A well–known conjecture in combinatorial optimization says that the integrality gap of the SEP is 4/3 in the metric case. Finding the exact value for this integrality gap is challenging even for small values of n as it is difficult to model this problem in a way that can be solved practically. We describe how we are able to overcome such difficulties and obtain the exact integrality gap for all values of n up to 10 and a lower bound for this gap for all values of n from 11 to 14. Our results give rise to a new stronger form of the conjecture which is dependent on n.

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