A Gap-{ETH}-Tight Approximation Scheme for Euclidean {TSP}

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 \emph{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

Nearly ETH-tight algorithms for planar Steiner Tree with terminals on few faces

\u3cp\u3eThe Steiner Tree problem is one of the most fundamental NP-complete problems as it models many network design problems. Recall that an instance of this problem consists of a graph with edge weights, and a subset of vertices (often called terminals); the goal is to find a subtree of the graph of minimum total weight that connects all terminals. A seminal paper by Erickson et al. [Math. Oper. Res., 1987] considers instances where the underlying graph is planar and all terminals can be covered by the boundary of k faces. Erickson et al. show that the problem can be solved by an algorithm using n\u3csup\u3eO\u3c/sup\u3e(k\u3csup\u3e)\u3c/sup\u3e time and n\u3csup\u3eO\u3c/sup\u3e(k\u3csup\u3e)\u3c/sup\u3e space, where n denotes the number of vertices of the input graph. In the past 30 years there has been no significant improvement of this algorithm, despite several efforts. In this work, we give an algorithm for Planar Steiner Tree with running time 2\u3csup\u3eO\u3c/sup\u3e(k\u3csup\u3e)\u3c/sup\u3en\u3csup\u3eO\u3c/sup\u3e(k\u3csup\u3e)\u3c/sup\u3e using only polynomial space. Furthermore, we show the running time of our algo-rithm is almost tight: we prove that there is no f(k)n\u3csup\u3eo\u3c/sup\u3e(k) algorithm for Planar Steiner Tree for any computable function f, unless the Exponential Time Hypothesis fails.\u3c/p\u3

European Model Company Act (EMCA)

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