Covering edges in networks

Abstract

In this paper we consider the covering problem on a network G = (V, E) with edge demands. The task is to cover a subset J ⊆ E of the edges with a minimum number of facilities within a predefined coverage radius. We focus on both the nodal and the absolute version of this problem. In the latter, facilities may be placed everywhere in the network. While there already exist polynomial time algorithms to solve the problem on trees, we establish a finite dominating set (i.e., a finite subset of points provably containing an optimal solution) for the absolute version in general graphs. Complexity and approximability results are given and a greedy strategy is proved to be a (1 + ln(|J|))‐approximate algorithm. Finally, the different approaches are compared in a computational study.

A Systematic Review of Hyper-Heuristics on Combinatorial Optimization Problems

“…Traveling Salesman [92] Traveling Salesman [93] Multiple Traveling Salesman [94] Bottleneck Traveling Salesman [95] Cutting Stock [96] Cutting Stock [97] 2D Cutting [98] Packing [99] Packing [100] 2D Packing [101] Bin Packing [102] Knapsack [103] Knapsack [104] Subset Sum [105] Unbounded Knapsack [105] Bounded Knapsack [106] Multiple Knapsack [107] Quadratic Knapsack [108] Scheduling [109] Scheduling [110] Production Scheduling [111] Workforce Scheduling [112] Job-Shop Scheduling [113] Precedence Constrained Scheduling [114] Educational Timetabling [115] Educational Timetabling [116] Facility Location [117] Assignment [118] Quadratic Assignment [119] Spanning Tree [120] Maximum Leaf Spanning Tree [121] Degree Constrained Spanning Tree [122] Minimum Spanning Tree [123] Boolean Satisfiability [124] Boolean Satisfiability [125] Covering [126] Minimum Vertex Cover [127] Set Cover [128] Exact Cover [129] Minimum Edge Cover [130] Vehicle Routing [131] Vehicle Routing…”

Continuous Covering on Networks: Improved Mixed Integer Programming Formulations

“…However, this assumption corresponds to ideal but usually unrealistic scenarios (the reader is referred to the examples of real applications of the above paragraph). Some works addressing network covering with continuous sets of both candidate locations and demand points are [12,13,14], for maximal covering, and [15,16,17], for set-covering. We focus on the latter variant, which we call the continuous set-covering problem.…”

“…This algorithm is polynomial time for the class of networks satisfying that every non-separable component is either an edge, a simple cycle, or a simple cycle with one chord, that is, for "almost tree" networks. More recently, Fröhlich et al [16] also studied the same version of the continuous set-covering with natural numbers. The authors presented three different approaches to solve the problem, including a Mixed Integer Linear Programming (MILP) formulation.…”

“…Secondly, Gurevich et al [15] studied the continuous set-covering problem when the covering radius and the edge's lengths are natural numbers. They presented a FDS for the case of uniform unit length of the edges, which can be easily extended to the case of general edges' lengths (see [16]),…”

“…Note that F DS 2 depends on the edge's length. Lastly, Fröhlich et al [16] Figure 1 also depicts two locations in C (marked with symbols 'x'), which correspond to two possible feasible locations for the remaining facility needed to cover C (note that the one at the bottom only yields a covering of the cycle if p 2 is located, while the other one together with either p 1 or p 2 can completely cover C). The authors of [16] gave the following recursive definition of an FDS for the problem with natural numbers:…”