Related papers: $k$-Transmitter Watchman Routes
In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G=(V, E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k>0; the goal is to find a minimum-cost…
We show that the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem is $\text{polyloglog}(n)$. In other words, there is a polynomial time algorithm that approximates the value of the optimum tour…
The disjoint paths problem is a fundamental problem in algorithmic graph theory and combinatorial optimization. For a given graph $G$ and a set of $k$ pairs of terminals in $G$, it asks for the existence of $k$ vertex-disjoint paths…
In this paper, we study the problem of map matching with travel time constraints. Given a sequence of $k$ spatio-temporal measurements and an embedded path graph with travel time costs, the goal is to snap each measurement to a close-by…
This paper considers the problem of computing the weak visibility polygon (WVP) of any query line segment pq (or WVP(pq)) inside a given simple polygon P. We present an algorithm that preprocesses P and creates a data structure from which…
Shortest paths problems are subject to extensive studies in classic distributed models such as the CONGEST or Congested Clique. These models dictate how nodes may communicate in order to determine shortest paths in a distributed input…
Given a polygon representing a transportation network together with a point p in its interior, we aim to extend the network by inserting a line segment, called a feed-link, which connects p to the boundary of the polygon. Once a feed link…
In this paper, we consider the 1.5-dimensional orthogonal terrain guarding problem. In this problem, we assign an x-monotone chain T because each edge is either horizontal or vertical, and determine the minimal number of vertex guards for…
For a given polygonal region $P$, the Lawn Mowing Problem (LMP) asks for a shortest tour $T$ that gets within Euclidean distance 1 of every point in $P$; this is equivalent to computing a shortest tour for a unit-disk cutter $C$ that covers…
We prove that it is NP-hard to decide whether two points in a polygonal domain with holes can be connected by a wire. This implies that finding any approximation to the shortest path for a long snake amidst polygonal obstacles is NP-hard.…
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…
In the Disjoint Shortest Paths problem one is given a graph $G$ and a set $\mathcal{T}=\{(s_1,t_1),\dots,(s_k,t_k)\}$ of $k$ vertex pairs. The question is whether there exist vertex-disjoint paths $P_1,\dots,P_k$ in $G$ so that each $P_i$…
The art gallery problem enquires about the least number of guards that are sufficient to ensure that an art gallery, represented by a polygon $P$, is fully guarded. In 1998, the problems of finding the minimum number of point guards, vertex…
The Hierarchical Chinese Postman Problem is finding a shortest traversal of all edges of a graph respecting precedence constraints given by a partial order on classes of edges. We show that the special case with connected classes is NP-hard…
A well-studied continuous model of graphs considers each edge as a continuous unit-length interval of points. In the problem $\delta$-Tour defined within this model, the objective to find a shortest tour that comes within a distance of…
The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First…
Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance…
The $k$-cut problem asks, given a connected graph $G$ and a positive integer $k$, to find a minimum-weight set of edges whose removal splits $G$ into $k$ connected components. We give the first polynomial-time algorithm with approximation…
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…
In this paper we study a variant of the shortest path problem in graphs: given a weighted graph G and vertices s and t, and given a set X of forbidden paths in G, find a shortest s-t path P such that no path in X is a subpath of P. Path P…