We revisit the classic task of finding the shortest tour of n points in d-dimensional Euclidean space, for any fixed constant d \geq 2. We determine the optimal dependence on \varepsilon in the running time of an algorithm that computes a (1+\varepsilon)-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in 2^{ \mathcal{O (1/\varepsilon^{d-1})} n\log n time. This improves the previously smallest dependence on \varepsilon in the running time (1/\varepsilon)^{ \mathcal{O (1/\varepsilon^{d-1})}n \log n of the algorithm by Rao and Smith (STOC 1998). We also show that a 2^{o(1/\varepsilon^{d-1})} \text{poly (n) algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a simple new idea that we call sparsity-sensitive patching . On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. Our approach is (arguably) simpler than the one by Rao and Smith since it can work without geometric spanners. We demonstrate the technique extends easily to other problems, by showing as an example that it also yields a Gap-ETH-tight approximation scheme for Rectilinear Steiner Tree.
We revisit the classic task of finding the shortest tour of n points in d-dimensional Euclidean space, for any fixed constant d \geq 2. We determine the optimal dependence on \varepsilon in the running time of an algorithm that computes a (1+\varepsilon)-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in 2^{ O (1/\varepsilon^{d-1})} n\log n time. This improves the previously smallest dependence on \varepsilon in the running time (1/\varepsilon)^{ O (1/\varepsilon^{d-1})}n \log n of the algorithm by Rao and Smith (STOC 1998). We also show that a 2^{o(1/\varepsilon^{d-1})} \mathrm{poly (n) algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a new idea that we call sparsity-sensitive patching . On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. We demonstrate that our technique extends to other problems, by showing that for Steiner Tree and Rectilinear Steiner Tree it yields the same running time. We complement our results with a matching Gap-ETH lower bound for Rectilinear Steiner Tree.
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\mathcal{O
O
R
\mathcal{O
O
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\text{poly
\mathrm{poly
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simple
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sparsity-sensitive patching
sparsity-sensitive patching
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Our approach is (arguably) simpler than the one by Rao and Smith since it can work without geometric spanners. We demonstrate the technique extends easily
We demonstrate that our technique extends
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as an example that it also yields a Gap-ETH-tight approximation scheme
that for Steiner Tree and Rectilinear Steiner Tree it yields the same running time. We complement our results with a matching Gap-ETH lower bound