Efficient PTAS for the Maximum Traveling Salesman Problem in a Metric Space of Fixed Doubling Dimension

关于线邻域旅行商问题的可近似性

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

研究线邻域旅行商问题的可近似性，证明其APX-难性并给出更优近似算法。

A PTAS for subset TSP in minor-free graphs

We give the first PTAS for the subset Traveling Salesperson Problem (TSP) in $H$-minor-free graphs. This resolves a long standing open problem in a long line of work on designing PTASes for TSP in minor-closed families initiated by Grigni, Koutsoupias and Papadimitriou in FOCS'95. The main technical ingredient in our PTAS is a construction of a nearly light subset $(1+ε)$-spanner for any given edge-weighted $H$-minor-free graph. This construction is based on a necessary and sufficient condition given by \emph{sparse spanner oracles}: light subset spanners exist if and only if sparse spanner or…

Approximation Schemes for Capacitated Clustering in Doubling Metrics

Motivated by applications in redistricting, we consider the uniform capacitated k-median and uniform capacitated k-means problems in bounded doubling metrics. We provide the first QPTAS for both problems and the first PTAS for the uniform capacitated k-median problem for points in R^2 . This is the first improvement over the bicriteria QPTAS for capacitated k-median in low-dimensional Euclidean space of Arora, Raghavan, Rao [STOC 1998] (1 + ε-approximation, 1 + ε-capacity violation) and arguably the first polynomial-time approximation algorithm for a non-trivial metric.

Asymptotic Optimality of the Greedy Patching Heuristic for Max TSP in Doubling Metrics

The maximum traveling salesman problem (Max~TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. We prove that, in the case when the edge weights are induced by a metric space of bounded doubling dimension, asymptotically optimal solutions of the problem can be found by the simple greedy patching heuristic. Taking as a start point a maximum-weight cycle cover, this heuristic iteratively patches pairs of its cycles into one minimizing the weight loss at each step.

Maximum Scatter TSP in Doubling Metrics

We study the problem of finding a tour of $n$ points in which every edge is long. More precisely, we wish to find a tour that visits every point exactly once, maximizing the length of the shortest edge in the tour. The problem is known as Maximum Scatter TSP, and was introduced by Arkin et al. (SODA 1997), motivated by applications in manufacturing and medical imaging. Arkin et al. gave a $0.5$-approximation for the metric version of the problem and showed that this is the best possible ratio achievable in polynomial time (assuming $P \neq NP$). Arkin et al. raised the question of whether a be…

拓扑码最小权重解码的多项式时间近似方案

A polynomial-time approximation scheme for minimum-weight decoding of topological codes

针对二维拓扑平移不变稳定子码的最小权重解码问题，提出一个多项式时间近似方案（PTAS），可在多项式时间内找到权重不超过最优解1+ε倍的恢复算子。

A $(1 + {\varepsilon})$-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs

Graphs with bounded highway dimension were introduced by Abraham et al. [SODA 2010] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V, E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V, E') that distorts shortest path distances of G by at most a 1 + ${\varepsilon}$ factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi…

Approximation Algorithms for Generalized MST and TSP in Grid Clusters

We consider a special case of the generalized minimum spanning tree problem (GMST) and the generalized travelling salesman problem (GTSP) where we are given a set of points inside the integer grid (in Euclidean plane) where each grid cell is $1 \times 1$. In the MST version of the problem, the goal is to find a minimum tree that contains exactly one point from each non-empty grid cell (cluster). Similarly, in the TSP version of the problem, the goal is to find a minimum weight cycle containing one point from each non-empty grid cell. We give a $(1+4\sqrt{2}+ε)$ and $(1.5+8\sqrt{2}+ε)$-approxim…

Estimating the Average of a Lipschitz-Continuous Function from One Sample

We study the problem of estimating the average of a Lipschitz continuous function $f$ defined over a metric space, by querying $f$ at only a single point. More specifically, we explore the role of randomness in drawing this sample. Our goal is to find a distribution minimizing the expected estimation error against an adversarially chosen Lipschitz continuous function. Our work falls into the broad class of estimating aggregate statistics of a function from a small number of carefully chosen samples. The general problem has a wide range of practical applications in areas as diverse as sensor ne…

高速公路维度：一种度量视角

Highway Dimension: a Metric View

本文提出改进的高速公路维度定义，适用于更广泛的度量空间，并基于此构建了TSP的PTAS，同时开发了适用于小高速公路维度空间的度量工具。

欧氏k-Means稳定实例的精确算法与下界

Exact Algorithms and Lower Bounds for Stable Instances of Euclidean k-Means

该研究证明加倍度量下(1+ε)-稳定k-Means/Median可通过多交换局部搜索在多项式时间内求解，同时基于新PCP假设证明维度作为输入时稳定k-Means不存在PTAS，给出了稳定聚类问题的算法与复杂度下界。

Approximation Schemes for Clustering with Outliers

Clustering problems are well-studied in a variety of fields such as data science, operations research, and computer science. Such problems include variants of centre location problems, $k$-median, and $k$-means to name a few. In some cases, not all data points need to be clustered; some may be discarded for various reasons. We study clustering problems with outliers. More specifically, we look at Uncapacitated Facility Location (UFL), $k$-Median, and $k$-Means. In UFL with outliers, we have to open some centres, discard up to $z$ points of $\cal X$ and assign every other point to the nearest o…