Parameterized Approximation Algorithms for TSP
We study the Traveling Salesman problem (TSP), where given a complete undirected graph G = (V,E) with n vertices and an edge cost function c:E↦R_{⩾0}, the goal is to find a minimum-cost cycle visiting every vertex exactly once. It is well-known that unless P = NP, TSP cannot be approximated in polynomial time within a factor of ρ(n) for any computable function ρ, while the metric case of TSP, that the edge cost function satisfies the △-inequality, admits a polynomial-time 1.5-approximation. We investigate TSP on general graphs from the perspective of parameterized approximability. A parameterized ρ-approximation algorithm returns a ρ-approximation solution in f(k)⋅|I|^O(1) time, where f is a computable function and k is a parameter of the input I. We introduce two parameters, which measure the distance of a given TSP-instance from the metric case, and achieve the following two results:
- A 3-approximation algorithm for TSP in O((3k₁)! 8^k₁⋅ n²+n³) time, where k₁ is the number of triangles in which the edge costs violate the △-inequality.
- A 3-approximation algorithm for TSP in O(n^O(k₂)) time and a (6k₂+9)-approximation algorithm for TSP in O(k₂^O(k₂)⋅n³) time, where k₂ is the minimum number of vertices, whose removal results in a metric graph.
To our best knowledge, the above algorithms are the first non-trivial parameterized approximation algorithms for TSP on general graphs.
FPT-approximation algorithms
the Traveling Salesman problem
the triangle inequality
fixed-parameter tractability
metric graphs
Theory of computation~Graph algorithms analysis
Theory of computation~Approximation algorithms analysis
50:1-50:16
Regular Paper
The work has been supported by the National Natural Science Foundation of China (No. 61772314, 61761136017, and 62072275).
The authors thank the reviewers for their valuable comments and constructive suggestions.
Jianqi
Zhou
Jianqi Zhou
School of Computer Science and Technology, Shandong University, Qingdao, China
Peihua
Li
Peihua Li
School of Computer Science and Technology, Shandong University, Qingdao, China
Jiong
Guo
Jiong Guo
School of Computer Science and Technology, Shandong University, Qingdao, China
10.4230/LIPIcs.ISAAC.2022.50
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Jianqi Zhou, Peihua Li, and Jiong Guo
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