In many wireless networks, there is no fixed physical backbone nor centralized network management. The nodes of such a network have to self-organize in order to maintain a virtual backbone used to route messages. Moreover, any node of the network can be a priori at the origin of a malicious attack. Thus, in one hand the backbone must be fault-tolerant and in other hand it can be useful to monit...

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of n disks in the plane, a TSP tour whose length is at most O(1) times the optimal can be computed in time that is polynomial in n. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and...

The first part of this report describes the following result that, logarithmic approximation factor for hard capacitated set cover can be achieved from Wolsey's work [9], using a simpler and more intuitive analysis. We further show in our work, that O(log n) approximation factor can be achieved for the same problem by applying analysis of general set cover to analyze Wolsey's algorithm [5]. Thi...

Given an instance of a geometric set cover problem on a set of points X and a set of objects R, the dual is a geometric hitting set problem on a set of points P and a set of objects Q, where there exists a one-to-one mapping from each xj ∈ X to a dual object Qj ∈ Q and for each Ri ∈ R to a dual point in pi ∈ P , so that a dual point pi is contained in a dual object Qj if and only if the corresp...

We describe a polynomial time O(k log log OPTk(P ))approximation algorithm for the k-guarding problem of finding a minimum number, OPTk(P ), of vertex guards of an n-vertex simple polygon P so that for every point p ∈ P , the number of guards that see p is at least the minimum of k and the number of vertices that see p. Our approach finds O ( k ε log log 1 ε ) size (k, ε)-nets for instances of ...

This paper uses a realistic problem taken from the telecommunication world as the basis for comparing di.erent combinatorial optimization algorithms. The problem recalls the minimum hitting set problem, and is solved with greedy-like, Darwinism and genetic algorithms. These three paradigms are described and analyzed with emphasis on the Darwinism approach, which is based on the computation of -...

In this report, we will introduce an optimization problem called Maximum Density Set Cover Problem. We firstly prove it is a NP-hard problem and then we analyse its approximation algorithms. The performance guarantee of one of the approximation algorithms is θ( √ n). Experiment is conducted to verify the algorithms’ performance.

A common subproblem of DNF approximate counting and derandomizing RL is the discrepancy problem for combinatorial rectangles. We explicitly construct a poly(n)-size sample space that approximates the volume of any combinatorial rectangle in [n] to within o(1) error (improving on the constructions of [EGLNV92]). The construction extends the techniques of [LLSZ95] for the analogous hitting set pr...

For a polygonal region P with n vertices, a guard cover G is a set of points in P , such that any point in P can be seen from a point in G. In a colored guard cover, every element in a guard cover is assigned a color, such that no two guards with the same color have overlapping visibility regions. The Chromatic Art Gallery Problem (CAGP) asks for the minimum number of colors for which a colored...

We investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems, namely k-VERTEX COVER and k-EDGE COVER. Specifically, we impose the additional requirement that each connected component of a solution have at least t vertices (resp. edges from the solution), and call the problem t-TOTAL VERTEX COVER (resp. t-TOTAL E...