Approximation Algorithms for Stochastic k-TSP

This paper studies the stochastic variant of the classical k-TSP problem where rewards at the vertices are independent random variables which are instantiated upon the tour\u27s visit. The objective is to minimize the expected length of a tour that collects reward at least k. The solution is a policy describing the tour which may (adaptive) or may not (non-adaptive) depend on the observed rewards. Our work presents an adaptive O(log k)-approximation algorithm for Stochastic k-TSP, along with a non-adaptive O(log^2 k)-approximation algorithm which also upper bounds the adaptivity gap by O(log^2 k). We also show that the adaptivity gap of Stochastic k-TSP is at least e, even in the special case of stochastic knapsack cover

Algorithms and Adaptivity Gaps for Stochastic k-TSP

Given a metric $(V,d)$ and a $\textsf{root} \in V$, the classic \textsf{k-TSP} problem is to find a tour originating at the $\textsf{root}$ of minimum length that visits at least $k$ nodes in $V$. In this work, motivated by applications where the input to an optimization problem is uncertain, we study two stochastic versions of \textsf{k-TSP}. In Stoch-Reward $k$-TSP, originally defined by Ene-Nagarajan-Saket [ENS17], each vertex $v$ in the given metric $(V,d)$ contains a stochastic reward $R_v$. The goal is to adaptively find a tour of minimum expected length that collects at least reward $k$; here "adaptively" means our next decision may depend on previous outcomes. Ene et al. give an $O(\log k)$-approximation adaptive algorithm for this problem, and left open if there is an $O(1)$-approximation algorithm. We totally resolve their open question and even give an $O(1)$-approximation \emph{non-adaptive} algorithm for this problem. We also introduce and obtain similar results for the Stoch-Cost $k$-TSP problem. In this problem each vertex $v$ has a stochastic cost $C_v$, and the goal is to visit and select at least $k$ vertices to minimize the expected \emph{sum} of tour length and cost of selected vertices. This problem generalizes the Price of Information framework [Singla18] from deterministic probing costs to metric probing costs. Our techniques are based on two crucial ideas: "repetitions" and "critical scaling". We show using Freedman's and Jogdeo-Samuels' inequalities that for our problems, if we truncate the random variables at an ideal threshold and repeat, then their expected values form a good surrogate. Unfortunately, this ideal threshold is adaptive as it depends on how far we are from achieving our target $k$, so we truncate at various different scales and identify a "critical" scale.Comment: ITCS 202

Asymptotic constant-factor approximation algorithm for the Traveling Salesperson Problem for Dubins' vehicle

This article proposes the first known algorithm that achieves a constant-factor approximation of the minimum length tour for a Dubins' vehicle through $n$ points on the plane. By Dubins' vehicle, we mean a vehicle constrained to move at constant speed along paths with bounded curvature without reversing direction. For this version of the classic Traveling Salesperson Problem, our algorithm closes the gap between previously established lower and upper bounds; the achievable performance is of order $n^{2/3}$

The random link approximation for the Euclidean traveling salesman problem

The traveling salesman problem (TSP) consists of finding the length of the shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where the cities are distributed randomly and independently in a d-dimensional unit hypercube. Working with periodic boundary conditions and inspired by a remarkable universality in the kth nearest neighbor distribution, we find for the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive analytical predictions for these quantities using the random link approximation, where the lengths between cities are taken as independent random variables. From the ``cavity'' equations developed by Krauth, Mezard and Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3, numerical results show that the random link approximation is a good one, with a discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we argue that the approximation is exact up to O(1/d^2) and give a conjecture for beta_E(d), in terms of a power series in 1/d, specifying both leading and subleading coefficients.Comment: 29 pages, 6 figures; formatting and typos correcte