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 better approximation ratio can be obtained in the Euclidean plane. We answer this question in the affirmative in a more general setting, by giving a $(1-\epsilon)$-approximation algorithm for $d$-dimensional doubling metrics, with running time $\tilde{O}\big(n^3 + 2^{O(K \log K)}\big)$, where $K \leq \left( \frac{13}{\epsilon} \right)^d$. As a corollary we obtain (i) an efficient polynomial-time approximation scheme (EPTAS) for all constant dimensions $d$, (ii) a polynomial-time approximation scheme (PTAS) for dimension $d = \log\log{n}/c$, for a sufficiently large constant $c$, and (iii) a PTAS for constant $d$ and $\epsilon = \Omega(1/\log\log{n})$. Furthermore, we show the dependence on $d$ in our approximation scheme to be essentially optimal, unless Satisfiability can be solved in subexponential time

A time- and space-optimal algorithm for the many-visits TSP

The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of $n$ cities that visits each city $c$ a prescribed number $k_c$ of times. Travel costs may be asymmetric, and visiting a city twice in a row may incur a non-zero cost. The MV-TSP problem finds applications in scheduling, geometric approximation, and Hamiltonicity of certain graph families. The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou (SICOMP, 1984). It runs in time $n^{O(n)} + O(n^3 \log \sum_c k_c )$ and requires $n^{\Theta(n)}$ space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the total length $\sum_c k_c$ of the tour, allowing the algorithm to handle instances with very long tours. The \emph{superexponential} dependence on the number of cities in both the time and space complexity, however, renders the algorithm impractical for all but the narrowest range of this parameter. In this paper we improve upon the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time $2^{O(n)}$, i.e.\ \emph{single-exponential} in the number of cities, using \emph{polynomial} space. Our algorithm is deterministic, and arguably both simpler and easier to analyse than the original approach of Cosmadakis and Papadimitriou. It involves an optimization over directed spanning trees and a recursive, centroid-based decomposition of trees.Comment: Small fixes, journal versio

A 3/2-Approximation for the Metric Many-visits Path TSP

In the Many-visits Path TSP, we are given a set of $n$ cities along with their pairwise distances (or cost) $c(uv)$, and moreover each city $v$ comes with an associated positive integer request $r(v)$. The goal is to find a minimum-cost path, starting at city $s$ and ending at city $t$, that visits each city $v$ exactly $r(v)$ times. We present a $\frac32$-approximation algorithm for the metric Many-visits Path TSP, that runs in time polynomial in $n$ and poly-logarithmic in the requests $r(v)$. Our algorithm can be seen as a far-reaching generalization of the $\frac32$-approximation algorithm for Path TSP by Zenklusen (SODA 2019), which answered a long-standing open problem by providing an efficient algorithm which matches the approximation guarantee of Christofides' algorithm from 1976 for metric TSP. One of the key components of our approach is a polynomial-time algorithm to compute a connected, degree bounded multigraph of minimum cost. We tackle this problem by generalizing a fundamental result of Kir\'aly, Lau and Singh (Combinatorica, 2012) on the Minimum Bounded Degree Matroid Basis problem, and devise such an algorithm for general polymatroids, even allowing element multiplicities. Our result directly yields a $\frac32$-approximation to the metric Many-visits TSP, as well as a $\frac32$-approximation for the problem of scheduling classes of jobs with sequence-dependent setup times on a single machine so as to minimize the makespan.Comment: arXiv admin note: text overlap with arXiv:1911.0989