The delivery man problem on a tree network
Annals of Operations Research
Analyzing the Held-Karp TSP bound: a monotonicity property with application
Information Processing Letters
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Paths, Trees, and Minimum Latency Tours
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
On the Integrality Ratio for Asymmetric TSP
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs
Journal of the ACM (JACM)
On The Approximability Of The Traveling Salesman Problem
Combinatorica
Traveling salesman path problems
Mathematical Programming: Series A and B
A Faster, Better Approximation Algorithm for the Minimum Latency Problem
SIAM Journal on Computing
The Directed Minimum Latency Problem
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
A new formulation for the Traveling Deliveryman Problem
Discrete Applied Mathematics
Improved Approximation Ratios for Traveling Salesperson Tours and Paths in Directed Graphs
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Poly-logarithmic Approximation Algorithms for Directed Vehicle Routing Problems
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
An improved integrality gap for asymmetric TSP paths
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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We study integrality gaps and approximability of two closely related problems on directed graphs. Given a set V of n nodes in an underlying asymmetric metric and two specified nodes s and t, both problems ask to find an s-t path visiting all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the objective is to minimize the total cost of this path. In the directed latency problem, the objective is to minimize the sum of distances on this path from s to each node. Both of these problems are NP-hard. The best known approximation algorithms for ATSPP had ratio O(log n) [7, 9] until the very recent result that improves it to O(log n/ log log n) [3, 9]. However, only a bound of O(√n) for the integrality gap of its linear programming relaxation has been known. For directed latency, the best previously known approximation algorithm has a guarantee of O(n1/2+ε), for any constant ε 0 [23]. We present a new algorithm for the ATSPP problem that has approximation ratio of O(log n), but whose analysis also bounds the integrality gap of the standard LP relaxation of ATSPP by the same factor. This solves an open problem posed in [7]. We then pursue a deeper study of this LP and its variations and their use in approximating directed latency. Our second major result is an O(log n)-approximation to the directed latency problem. This also places an O(log n) bound on the integrality gap of a new LP relaxation of the latency problem that we introduce.