Karim Nouioua

Mixed Covering of Trees and the Augmentation Problem with Odd Diameter Constraints

In this paper we present a polynomial time algorithm for solving the problem of partial covering of trees with n1 balls of radius R1 and n2 balls of radius R2 (R1 < R2) to maximize the total number of covered vertices. The solutions provided by this algorithm in the particular case R1 = R – 1, R2 = R can be used to obtain for any integer δ > 0 a factor (2+1/δ) approximation algorithm for solving the following augmentation problem with odd diameter constraints D = 2R + 1: Given a tree T, add a minimum number of new edges such that the augmented graph has diameter ≤ D. The previous approximation algorithm of Ishii, Yamamoto, and Nagamochi (2003) has factor 8.

Pareto envelopes in R 3 under l 1 and l ∞ distance functions

Given a vector objective function f = (f₁,...,f_n) defined on a set X, a point y∈X is <i>dominated</i> by a point x∈ X if f_i(x) < f_i(y) forall i∈(1,...,n) and there exists an index j∈(1,...,n) such that f_j(x) < f_j(y). The non-dominated pointsof X are called the <i>Pareto optima</i> of f. H. Kuhn(1973) applied the concept of Pareto optimality to distancefunctions and characterized the convex hull conv (T) of any set T=(t₁,...,t_n) of R^m as the set of all Paretooptima of the vector function d₂(x)=(d₂(x,t₁),...,d₂(x,t_n)), where d₂(x,y)is the Euclidean distance between x,y∈ R^m. Motivatedby this result, given a set T=(t₁,...,t_n) of points of ametric space (X,d), we call the set P_d(T) of all Paretooptima of the function d(x)=(d(x,t₁),...,d(x,t_n)) the <i>Pareto envelope</i> of T. In this paper, we investigate the Pareto envelopes in R^m endowed with l₁- or l_∞-distances. We characterize PI(T) in all dimensions and PM(T) in R³. Usingthese results, we design efficient algorithms for constructing theseenvelopes in R³, in particular, an optimal O(n logn)-time algorithm for PM(T) and an O(n log²n)-time algorithmfor PI(T).

PARETO ENVELOPES IN SIMPLE POLYGONS

For a set T of n points in a metric space (X, d), a point y ∈ X is dominated by a point x ∈ X if d(x, t) ≤ d(y, t) for all t ∈ T and there exists t′ ∈ T such that d(x, t′) < d(y, t′). The set of non-dominated points of X is called the Pareto envelope of T. H. Kuhn (1973) established that in Euclidean spaces, the Pareto envelopes and the convex hulls coincide. Chalmet et al. (1981) characterized the Pareto envelopes in the rectilinear plane (ℝ2, d1) and constructed them in O(n log n) time. In this note, we investigate the Pareto envelopes of point-sets in simple polygons P endowed with geodesic d2- or d1-metrics (i.e., Euclidean and Manhattan metrics). We show that Kuhns characterization extends to Pareto envelopes in simple polygons with d2-metric, while that of Chalmet et al. extends to simple rectilinear polygons with d1-metric. These characterizations provide efficient algorithms for construction of these Pareto envelopes.

Mixed covering of trees and the augmentation problem with odd diameter constraints

Abstract In this talk, we will outline a polynomial time algorithm for solving the problem of partial covering of trees with n1 balls of radius R1 and n2 balls of radius R 2 ( R 1 R 2 ) so as to maximize the total number of covered vertices. We will then show that the solutions provided by this algorithm in the particular case R 1 = R − 1 , R 2 = R can be used to obtain for any integer δ > 0 a factor ( 2 + 1 δ ) approximation algorithm for solving the following augmentation problem with odd diameter constraints D = 2 R + 1 : given a tree T, add a minimum number of new edges such that the augmented graph has diameter ≤D. The previous approximation algorithm of Ishii, Yamamoto, and Nagamochi (2003) has factor 8.