# 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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### Cite this paper

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