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In the Euclidean $k$-traveling salesman problem ($k$-TSP), we are given $n$ points in the $d$-dimensional Euclidean space, for some fixed constant $d\geq 2$, and a positive integer $k$. The goal is to find a shortest tour visiting at least…
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…
The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+eps)-approximation to the optimal tour, for any fixed eps>0,…
We study exact algorithms for Euclidean TSP in $\mathbb{R}^d$. In the early 1990s algorithms with $n^{O(\sqrt{n})}$ running time were presented for the planar case, and some years later an algorithm with $n^{O(n^{1-1/d})}$ running time was…
We revisit the traveling salesman problem with neighborhoods (TSPN) and propose several new approximation algorithms. These constitute either first approximations (for hyperplanes, lines, and balls in $\mathbb{R}^d$, for $d\geq 3$) or…
We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds…
We present a polynomial-time approximation scheme (PTAS) for the min-max multiple TSP problem in Euclidean space, where multiple traveling salesmen are tasked with visiting a set of $n$ points and the objective is to minimize the maximum…
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…
Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We…
We study sublinear time algorithms for the traveling salesman problem (TSP). First, we focus on the closely related {\em maximum path cover} problem, which asks for a collection of vertex disjoint paths that include the maximum number of…
We give a polynomial time, $(1+\epsilon)$-approximation algorithm for the traveling repairman problem (TRP) in the Euclidean plane and on weighted trees. This improves on the known quasi-polynomial time approximation schemes for these…
One of the most studied extensions of the famous Traveling Salesperson Problem (TSP) is the {\sc Multiple TSP}: a set of $m\geq 1$ salespersons collectively traverses a set of $n$ cities by $m$ non-trivial tours, to minimize the total…
We consider the traveling salesman problem when the cities are points in R^d for some fixed d and distances are computed according to geometric distances, determined by some norm. We show that for any polyhedral norm, the problem of finding…
With applications to many disciplines, the traveling salesman problem (TSP) is a classical computer science optimization problem with applications to industrial engineering, theoretical computer science, bioinformatics, and several other…
We consider the Travelling Salesman Problem with Neighbourhoods (TSPN) on the Euclidean plane ($\mathbb{R}^2$) and present a Polynomial-Time Approximation Scheme (PTAS) when the neighbourhoods are parallel line segments with lengths between…
We study the traveling salesman problem in the hyperbolic plane of Gaussian curvature $-1$. Let $\alpha$ denote the minimum distance between any two input points. Using a new separator theorem and a new rerouting argument, we give an…
The dynamic programming solution to the traveling salesman problem due to Bellman, and independently Held and Karp, runs in time $O(2^n n^2)$, with no improvement in the last sixty years. We break this barrier for the first time by…
We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity. Our first set of results is motivated by the Bitonic TSP problem: given a set of $n$ points in the plane, compute a shortest…
Parameterized runtime analysis seeks to understand the influence of problem structure on algorithmic runtime. In this paper, we contribute to the theoretical understanding of evolutionary algorithms and carry out a parameterized analysis of…
The question of whether all problems in NP class are also in P class is generally considered one of the most important open questions in mathematics and theoretical computer science as it has far-reaching consequences to other problems in…