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We consider the NP-hard 2-period balanced travelling salesman problem. In this problem the salesman needs to visit a set of customers in two time periods. A given subset of the customers has to be visited in both periods while the rest of…
We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a…
The purpose of this note is to give a simple proof for a necessary and sufficient condition for visibility paths in simple polygons. A visibility path is a curve such that every point inside a simple polygon is visible from at least one…
Given a simple polygon $\mathcal{P}$ on $n$ vertices, two points $x,y$ in $\mathcal{P}$ are said to be visible to each other if the line segment between $x$ and $y$ is contained in $\mathcal{P}$. The Point Guard Art Gallery problem asks for…
Let $P$ be a set of $n$ points in the plane. We consider the problem of partitioning $P$ into two subsets $P_1$ and $P_2$ such that the sum of the perimeters of $\text{CH}(P_1)$ and $\text{CH}(P_2)$ is minimized, where $\text{CH}(P_i)$…
Given a graph $G$, and terminal vertices $s$ and $t$, the TRACKING PATHS problem asks to compute a minimum number of vertices to be marked as trackers, such that the sequence of trackers encountered in each s-t path is unique. TRACKING…
We consider the problem of designing sublinear time algorithms for estimating the cost of a minimum metric traveling salesman (TSP) tour. Specifically, given access to a $n \times n$ distance matrix $D$ that specifies pairwise distances…
In the pairwise weighted spanner problem, the input consists of an $n$-vertex-directed graph, where each edge is assigned a cost and a length. Given $k$ vertex pairs and a distance constraint for each pair, the goal is to find a…
The input to the unrooted traveling repairman problem is an undirected metric graph and a subset of nodes, each of which has a time window of unit length. Given that a repairman can start at any location, the goal is to plan a route that…
The problem of finding multiple simple shortest paths in a weighted directed graph $G=(V,E)$ has many applications, and is considerably more difficult than the corresponding problem when cycles are allowed in the paths. Even for a single…
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…
We provide exact and approximation methods for solving a geometric relaxation of the Traveling Salesman Problem (TSP) that occurs in curve reconstruction: for a given set of vertices in the plane, the problem Minimum Perimeter Polygon (MPP)…
We consider approximating the minmax value of a multi-player game in strategic form. Tightening recent bounds by Borgs et al., we observe that approximating the value with a precision of epsilon log n digits (for any constant epsilon>0 is…
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…
The Traveling Tournament Problem (TTP-$k$) is a well-known benchmark problem in tournament timetabling, which asks us to design a double round-robin schedule such that the total traveling distance of all $n$ teams is minimized under the…
This paper deals with the problem of autonomous navigation of a mobile robot in an unknown 2D environment to fully explore the environment as efficiently as possible. We assume a terrestrial mobile robot equipped with a ranging sensor with…
Prize-Collecting TSP is a variant of the traveling salesperson problem where one may drop vertices from the tour at the cost of vertex-dependent penalties. The quality of a solution is then measured by adding the length of the tour and the…
We are given a set of points $p_1,\ldots , p_n$ and a symmetric distance matrix $(d_{ij})$ giving the distance between $p_i$ and $p_j$. We wish to construct a tour that minimizes $\sum_{i=1}^n \ell(i)$, where $\ell(i)$ is the {\em latency}…
Optimal transport (OT) provides powerful tools for comparing probability measures in various types. The Wasserstein distance which arises naturally from the idea of OT is widely used in many machine learning applications. Unfortunately,…
Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least $k$ vertices is considered long. When $k \le 3$, the…