Approximation algorithms for multi-dimensional assignment problems with decomposable costs
Discrete Applied Mathematics - Special volume: viewpoints on optimization
P-Complete Approximation Problems
Journal of the ACM (JACM)
Confronting hardness using a hybrid approach
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Paired approximation problems and incompatible inapproximabilities
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
On the approximation hardness of some generalizations of TSP
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
Structural properties of hard metric TSP inputs
SOFSEM'11 Proceedings of the 37th international conference on Current trends in theory and practice of computer science
Improved approximations for hard optimization problems via problem instance classification
Rainbow of computer science
Algorithmics – is there hope for a unified theory?
CSR'10 Proceedings of the 5th international conference on Computer Science: theory and Applications
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In this paper, we consider variants of the traveling salesman problem with precedence constraints. We characterize hard input instances for Christofides' algorithm and Hoogeveen's algorithm by relating the two underlying problems, i. e., the traveling salesman problem and the problem of finding a minimum-weight Hamiltonian path between two prespecified vertices. We show that the sets of metric worst-case instances for both algorithms are disjoint in the following sense. There is an algorithm that, for any input instance, either finds a Hamiltonian tour that is significantly better than 1.5-approximative or a set of Hamiltonian paths between all pairs of endpoints, all of which are significantly better than 5/3-approximative. In the second part of the paper, we give improved algorithms for the ordered TSP, i. e., the TSP, where the precedence constraints are such that a given subset of vertices has to be visited in some prescribed linear order. For the metric case, we present an algorithm that guarantees an approximation ratio of 2.5−2/k, where k is the number of ordered vertices. For near-metric input instances satisfying a β-relaxed triangle inequality, we improve the best previously known ratio to $k\beta^{\log_2 (3k-3)}$.