Approximation algorithms for the black and white traveling salesman problem

Authors:
Binay Bhattacharya;Yuzhuang Hu;Alexander Kononov
Affiliations:
School of Computing Science, Simon Fraser University, Burnaby, Canada;School of Computing Science, Simon Fraser University, Burnaby, Canada;Laboratory "Mathematical Models of Decision Making", Sobolev Institute of Mathematics, Novosibirsk, Russia
Venue:
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Year:
2007

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Abstract

The black and white traveling salesman problem (BWTSP) is to find the minimum cost hamiltonian tour of an undirected complete graph G, containing black and white vertices, subject to two restrictions: the number of white vertices, and the cost of the subtour between two consecutive black vertices are bounded. This paper focuses on designing approximation algorithms for the BWTSP in a graph satisfying the triangle inequality. We show that approximating the tour which satisfies the length constraint is NP-hard. We then show that the BWTSP can be approximated with tour cost (4 - 3/2Q) times the optimal cost, when at most Q white vertices appear between two consecutive black vertices. When exactly Q white vertices appear between two consecutive black vertices, the approximation bound can be slightly improved to (4- 15/8Q). This approximation bound is further improved to 2.5 when Q = 2.