The multi-vehicle covering tour problem is de"ned on a graph G"(<X=, E), where= is a set of vertices that must collectively be covered by up to m vehicles. The problem consists of determining a set of total minimum length vehicle routes on a subset of <, subject to side constraints, such that every vertex of= is within a prespeci"ed distance from a route. Three heuristics are developed for this...

In the Covering Salesman Problem (CSP), a distribution of nodes is provided, and the objective is to identify the shortest-length tour of a subset of all given nodes such that each node is not on the tour which is within a radius r of any node on the tour. In this paper, we define a new covering problem called the CSP with Nodes and Segments (CSPNS). The main difference between the CSP and the ...

Let G = (V ∪W,E) be an undirected graph, where V ∪W = {1, ..., n} is the vertex set and E = {(i, j) | i, j ∈ V ∪W, i < j} is the edge set. Vertex s = 1 is the source vertex, V is a set of vertices that might be visited, T ⊆ V is a set of vertices that must be visited (s ∈ T ), and W is a set of vertices that must be covered. A symmetric distance matrix C = (cij), defined on E, uses the Euclidea...

Given a set P of n points in the plane, Covering Points by Lines is the problem of finding a minimum-cardinality set L of lines such that every point p ∈ P is incident to some line l ∈ L. As a geometric variant of Set Cover, Covering Points by Lines is still NP-hard. Moreover, it has been proved to be APX-hard, and hence does not admit any polynomial-time approximation scheme unless P = NP. In ...

We consider a generalized version of the well known Traveling Salesman Problem called Covering Salesman problem. In this problem, we are given a set of vertices while each vertex i can cover a subset of vertices within its predetermined covering distance ri. The goal is to construct a minimum length Hamiltonian cycle over a subset of vertices in which those vertices not visited on the tour has ...

Wepropose a generalization of themulti-depot capacitated vehicle routing problemwhere the assumption of visiting each customer does not hold. In this problem, called the Multi-Depot Covering Tour Vehicle Routing Problem (MDCTVRP), the demand of each customer could be satisfied in two different ways: either by visiting the customer along the tour or by “covering” it. When a customer is visited, ...

We introduce a geometric version of the Covering Salesman Problem: Each of the n salesman's clients speciies a neighborhood in which they are willing to meet the salesman. Identifying a tour of minimum length that visits all neighborhoods is an NP-hard problem, since it is a generalization of the Traveling Salesman Problem. We present simple heuristic procedures for constructing tours, for a va...

abstract one of the basic assumptions in hub covering problems is considering the covering radius as an exogenous parameter which cannot be controlled by the decision maker. practically and in many real world cases with a negligible increase in costs, to increase the covering radii, it is possible to save the costs of establishing additional hub nodes. change in problem parameters during the pl...

The subpath planning problem (SPP) is a branch of path planning problem, which has widespread applications in automated manufacturing process as well as vehicle and robot navigation. This problem aims to find the shortest path or tour subject to covering a set of given subpaths. By casting SPP to a graph routing problem, we propose a deterministic 2-approximation algorithm finding near optimal ...

Given an undirected graph G = (V ∪W,E), where V ∪W is the vertex set and E is the edge set, the COVERING TOUR PROBLEM (CTP) consists of determining a minimum length cycle over a subset of V which contains all vertices of T ⊆ V , and every vertex of W is covered by the tour. This work presents a new integer linear programming formulation, a new reduction rule and new heuristic algorithms for the...