The simultaneous semi-random model for TSP

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 very well in practice even though it only achieves an $\mathrm{\Omega }(\frac{logn}{loglogn})$ 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 $\alpha =\frac{1}{logn}$ fraction of the points being random, local search achieves an $\mathcal{O}(loglogn)$ 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.