Hervé Fournier

Computing the Gromov hyperbolicity of a discrete metric space

We give exact and approximation algorithms for computing the Gromov hyperbolicity of an n-point discrete metric space. We observe that computing the Gromov hyperbolicity from a fixed base-point reduces to a (max,min) matrix product. Hence, using the (max,min) matrix product algorithm by Duan and Pettie, the fixed base-point hyperbolicity can be determined in O ( n 2.69 ) time. It follows that the Gromov hyperbolicity can be computed in O ( n 3.69 ) time, and a 2-approximation can be found in O ( n 2.69 ) time. We also give a ( 2 log 2 ? n ) -approximation algorithm that runs in O ( n 2 ) time, based on a tree-metric embedding by Gromov. We also show that hyperbolicity at a fixed base-point cannot be computed in O ( n 2.05 ) time, unless there exists a faster algorithm for (max,min) matrix multiplication than currently known. We consider the problem of computing the Gromov hyperbolicity of a metric space.We present an efficient exact algorithm for this problem.We also present efficient approximation algorithms.We give a hardness result.

Fitting a Step Function to a Point Set

We consider the problem of fitting a step function to a set of points. More precisely, given an integer kand a set Pof npoints in the plane, our goal is to find a step function fwith ksteps that minimizes the maximum vertical distance between fand all the points in P. We first give an optimal i¾?(nlogn) algorithm for the general case. In the special case where the points in Pare given in sorted order according to their x-coordinates, we give an optimal i¾?(n) time algorithm. Then, we show how to solve the weighted version of this problem in time O(nlog4n). Finally, we give an O(nh2logh) algorithm for the case where houtliers are allowed, and the input is sorted. The running time of all our algorithms is independent of k.

Classical and intuitionistic logic are asymptotically identical

This paper considers logical formulas built on the single binary connector of implication and a finite number of variables. When the number of variables becomes large, we prove the following quantitative results: asymptotically, all classical tautologies are simple tautologies. It follows that asymptotically, all classical tautologies are intuitionistic.

Lower Bounds for Comparison Based Evolution Strategies Using VC-dimension and Sign Patterns

We derive lower bounds on the convergence rate of comparison based or selection based algorithms, improving existing results in the continuous setting, and extending them to non-trivial results in the discrete case. This is achieved by considering the VC-dimension of the level sets of the fitness functions; results are then obtained through the use of the shatter function lemma. In the special case of optimization of the sphere function, improved lower bounds are obtained by an argument based on the number of sign patterns.

Lower Bounds for Evolution Strategies Using VC-Dimension

We derive lower bounds for comparison-based or selection-based algorithms, improving existing results in the continuous setting, and extending them to non-trivial results in the discrete case. This is achieved by considering the VC-dimension of the level sets of the fitness functions; results are then obtained through the use of Sauers lemma. In the special case of optimization of the sphere function, improved lower bounds are obtained by bounding the possible number of sign conditions realized by some systems of equations.

Monomials in Arithmetic Circuits: Complete Problems in the Counting Hierarchy

We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting hierarchy which had few or no known natural complete problems before. We also study these questions for circuits computing multilinear polynomials and for univariate multiplicatively disjoint circuits.

A deterministic algorithm for fitting a step function to a weighted point-set

Given a set of n points in the plane, each point having a positive weight, and an integer k>0, we present an optimal O(nlogn)-time deterministic algorithm to compute a step function with k steps that minimizes the maximum weighted vertical distance to the input points. It matches the expected time bound of the best known randomized algorithm for this problem. Our approach relies on Cole@?s improved parametric searching technique. As a direct application, our result yields the first O(nlogn)-time algorithm for computing a k-center of a set of n weighted points on the real line.

Monomials in arithmetic circuits: Complete problems in the counting hierarchy.

We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting hierarchy which had few or no known natural complete problems before. We also study these questions for circuits computing multilinear polynomials.

Tautologies over implication with negative literals

We consider logical expressions built on the single binary connector of implication and a finite number of literals (Boolean variables and their negations). We prove that asymptotically, when the number of variables becomes large, all tautologies have the following simple structure: either a premise equal to the goal, or two premises which are opposite literals (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A Tight Lower Bound for Computing the Diameter of a 3D Convex Polytope

Abstract The diameter of a set P of n points in ℝd is the maximum Euclidean distance between any two points in P. If P is the vertex set of a 3-dimensional convex polytope, and if the combinatorial structure of this polytope is given, we prove that, in the worst case, deciding whether the diameter of P is smaller than 1 requires Ω(nlog n) time in the algebraic computation tree model. It shows that the O(nlog n) time algorithm of Ramos for computing the diameter of a point set in ℝ3 is optimal for computing the diameter of a 3-polytope. We also give a linear time reduction from Hopcroft’s problem of finding an incidence between points and lines in ℝ2 to the diameter problem for a point set in ℝ7.