Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number

Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number

Glover and Punnen (1997) asked whether there exists a polynomial time algorithm that always produces a tour which is not worse than at least n!=p(n) tours for some polynomial p(n) for every TSP instance on n cities. They conjectured that, unless P=NP, the answer to this question is negative. We prove that the answer to this question is, in fact, positive. A generalization of the TSP, the quadra...

Polynomial Algorithms for the Tsp and the Qap with a Factorial Domination Number

Glover and Punnen (1997) asked whether there exists a polynomial time algorithm that always produces a tour which is not worse than at least n!=p(n) tours for some polynomial p(n) for every TSP instance on n cities. They conjectured that, unless P=NP, the answer to this question is negative. We prove that the answer to this question is, in fact, positive. A generalization of the TSP, the quadra...

Tsp Heuristics with Large Domination Number

New approximation algorithms for the TSP

The traveling salesman problem (TSP) is probably the best known combinatorial optimization problem. Although studied intensively for sixty years, the TSP continues to pose grand challenges. Cook’s [2012] recent book gives an excellent introduction. Since the TSP is NP-hard (Karp [1972]), it is natural to ask for approximation algorithms. How good solutions can we guarantee to find in polynomial...

Dominance guarantees for above-average solutions

Gutin et al. [G. Gutin, A. Yeo, Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number, Discrete Applied Mathematics 119 (1–2) (2002) 107–116] proved that, in the ATSP problem, a tour of weight not exceeding the weight of an average tour is of dominance ratio at least 1/(n − 1) for all n 6= 6. (Tours with this property can be easily obtained.) In [N. Alon...