Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, November 11, 2021, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Sándor Kisfaludi-Bak

A Gap-ETH-tight approximation scheme for Euclidean TSP

The Euclidean TSP problem aims to find the shortest round trip of a set of n points in Euclidean space. The celebrated approximation scheme of Arora [JACM'98] can produce a (1+epsilon)-approximate tour in d-dimensional space for any fixed positive epsilon and d in polynomial time. The result (and the independently discovered algorithm of Mitchell) sparked a lot of interest in the problem and in approximation schemes for geometric problems in general. In this talk, we will see an elegant modification to Arora's scheme called "sparsity-sensitive patching", which fine-tunes the granularity with which the tour is simplified. The resulting algorithm has running time 2^O(epsilon^(1-d)) n log n for any fixed d, which is faster than Arora's scheme, and also faster than the approximation scheme of Rao and Smith. In fact, the epsilon-dependence of our algorithm is optimal under the Gap Exponential Time Hypothesis. The talk is based on joint work with Jesper Nederlof and Karol Wegrzycki. A preprint is available here: https://arxiv.org/abs/2011.03778

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