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# Bi-Criteria Approximation Algorithms for Bounded-Degree Subset TSP

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## Cite As

Zachary Friggstad and Ramin Mousavi. Bi-Criteria Approximation Algorithms for Bounded-Degree Subset TSP. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 8:1-8:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ISAAC.2022.8

## Abstract

We initiate the study of the Bounded-Degree Subset Traveling Salesman problem (BDSTSP) in which we are given a graph G = (V,E) with edge cost c_e ≥ 0 on each edge e, degree bounds b_v ≥ 0 on each vertex v ∈ V and a subset of terminals X ⊆ V. The goal is to find a minimum-cost closed walk that spans all terminals and visits each vertex v ∈ V at most b_v/2 times. In this paper, we study bi-criteria approximations that find tours whose cost is within a constant-factor of the optimum tour length while violating the bounds b_v at each vertex by additive quantities. If X = V, an adaptation of the Christofides-Serdyukov algorithm yields a (3/2, +4)-approximation, that is the tour passes through each vertex at most b_v/2+2 times (the degree of v in the multiset of edges on the tour being at most b_v + 4). This is enabled through known results in bounded-degree Steiner trees and integrality of the bounded-degree Y-join polytope. The general case X ≠ V is more challenging since we cannot pass to the metric completion on X. However, it is at least simple to get a (5/3, +4)-bicriteria approximation by using ideas similar to Hoogeveen’s TSP-Path algorithm. Our main result is an improved approximation with marginally worse violations of the vertex bounds: a (13/8, +6)-approximation. We obtain this primarily through adapting the bounded-degree Steiner tree approximation to ensure certain "dangerous" nodes always have even degree in the resulting tree which allows us to bound the cost of the resulting degree-bounded Y-join. We also recover a (3/2, +8)-approximation for this general case.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Approximation algorithms
##### Keywords
• Linear programming
• approximation algorithms
• combinatorial optimization

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## References

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