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The Simultaneous Semi-random Model for TSP

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 13265)

Abstract

Worst-case analysis is a performance measure that is often too pessimistic to indicate which algorithms we should use in practice. A classical example is in the context of the Euclidean Traveling Salesman Problem (TSP) in the plane, where local search performs extremely well in practice even though it only achieves an \(\varOmega (\frac{\log n}{\log \log n})\) worst-case approximation ratio. In such cases, a natural alternative approach to worst-case analysis is to analyze the performance of algorithms in semi-random models.

In this paper, we propose and investigate a novel semi-random model for the Euclidean TSP. In this model, called the simultaneous semi-random model, an instance over n points consists of the union of an adversarial instance over \((1-\alpha )n\) points and a random instance over \(\alpha n\) points, for some \(\alpha \in [0, 1]\). As with smoothed analysis, the semi-random model interpolates between distributional (random) analysis when \(\alpha = 1\) and worst-case analysis when \(\alpha = 0\). In contrast to smoothed analysis, this model trades off allowing some completely random points in order to have other points that exhibit a fully arbitrary structure.

We show that with only an \(\alpha = \frac{1}{\log n}\) fraction of the points being random, local search achieves an \(\mathcal {O}(\log \log n)\) approximation in the simultaneous semi-random model for Euclidean TSP in fixed dimensions. On the other hand, we show that at least a polynomial number of random points are required to obtain an asymptotic improvement in the approximation ratio of local search compared to its worst-case approximation, even in two dimensions.

Keywords

  • Traveling Salesman Problem
  • Semi-random Models
  • Local Search

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Correspondence to Eric Balkanski .

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Balkanski, E., Faenza, Y., Kubik, M. (2022). The Simultaneous Semi-random Model for TSP. In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_4

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  • DOI: https://doi.org/10.1007/978-3-031-06901-7_4

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