Linear time algorithms for approximating the facility terminal cover problem

Abstract

In this paper, we consider an interesting generalization of the weighted vertex cover problem, called the Facility Terminal Cover (FTC) problem. In the FTC problem, each vertex is associated with a positive weight, each edge is associated with a positive demand, and the objective is to determine a subset of vertices and a capacity for each selected vertex so that the demand of each edge is covered by the capacity of one of its two endpoints and the total weighted capacity of all selected vertices is minimized. The FTC problem is motivated by several key network optimization problems, such as the power assignment problem in ad hoc networks, and could be used as a subroutine to solve such problems. No quality-guaranteed solution is previously known for the FTC problem. In this paper, we present two linear time approximation algorithms for this problem. Our first algorithm achieves deterministically an approximation ratio of 8 by using an interesting rounding technique and a lower-bounding technique. Based on interesting randomization techniques, our second algorithm further improves the approximation ratio to 2e, where e is the natural logarithmic base. The second algorithm can be easily derandomized in quadratic time. Our algorithms are relatively simple and can be easily implemented for networking applications. Experiments show that the two algorithms behave rather similarly, especially in largesize graphs, indicating that the solutions yielded by one or both algorithms are much closer to the optimum.

Optimizing mix-zone coverage in pervasive wireless networks1

“…In the following sections, we outline two approximation strategies for the MC problem. The first is based on a linear programming formulation of the problem, whereas the other two algorithms employ a "divide and conquer" strategy by utilizing the round and group approach for solving the FTC problem [52].…”

“…The MC problem is very similar to another generalization of the VC problem called the Facility Terminal Cover (FTC) problem [52,31], but there is an important difference between the two problems. Given a graph G ≡ (V, E, w, d), where w : V → R + and d : E → R + (denoted as w v and d e for vertex v and edge e, respectively), the FTC problem is to find a set V F T C ⊆ V and a capacity c(v) for each vertex v ∈ V F T C such that for each edge e ≡ (u, v) ∈ E at least one of the vertices u and v is in V F T C and associated with a capacity c(u) ≥ d e , and the total weighted capacity x∈V F T C c(x)w x is minimized.…”

“…The FTC problem can be considered as a special case of the MC problem, i.e., the MC problem reduces to the FTC problem when both the demand values are equal for all the edges in the graph. The formulation and algorithms of the FTC problem cannot be directly used to solve the much more general MC problem; although we will use one of the solution strategies [52] of the FTC problem for solving the MC problem.…”

“…The problem of optimal mix-zone placement -we refer to it as the Mix Cover problem (MC) -is then to determine a set of intersections (or vertices) for mix-zone placement, such that each road (connecting two intersections) in the network is associated with at least one mix-zone and the network-wide mixing cost due to such a placement of mix-zones is minimized. The mix cover problem is a generalization of a well-known problem in combinatorics, called the Vertex Cover (VC) problem [37]; The Facility Terminal Cover (FTC) problem discussed in [52] is a special case of the mix cover problem. To the best of our knowledge, the mix cover generalization, specifically in the setting of pervasive networks, has never been studied before.…”

“…In this pa-per, we show that the Mix Cover problem is a combinatorially hard problem and propose three bounded-ratio approximation algorithms for the same. The first algorithm is based on a linear programming relaxation of an Integer Program (IP) formulation of the problem, whereas the remaining two algorithms take advantage of the "divide and conquer" strategy which was used in [52] to solve the FTC problem. We analytically study the solution quality and running-time guarantees of these algorithms by deriving their worst-case approximation ratio and running-time, respectively.…”