OBDD-Based Representation of Interval Graphs

A graph $G = (V,E)$ can be described by the characteristic function of the edge set $\chi_E$ which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using \emph{Ordered Binary Decision Diagrams} (OBDDs) to store $\chi_E$ can lead to a (hopefully) compact representation. Given the OBDD as an input, symbolic/implicit OBDD-based graph algorithms can solve optimization problems by mainly using functional operations, e.g. quantification or binary synthesis. While the OBDD representation size can not be small in general, it can be provable small for special graph classes and then also lead to fast algorithms. In this paper, we show that the OBDD size of unit interval graphs is $O(\ | V \ | /\log \ | V \ |)$ and the OBDD size of interval graphs is $O(\ | V \ | \log \ | V \ |)$which both improve a known result from Nunkesser and Woelfel (2009). Furthermore, we can show that using our variable order and node labeling for interval graphs the worst-case OBDD size is$\Omega(\ | V \ | \log \ | V \ |)$. We use the structure of the adjacency matrices to prove these bounds. This method may be of independent interest and can be applied to other graph classes. We also develop a maximum matching algorithm on unit interval graphs using$O(\log \ | V \ |)$operations and a coloring algorithm for unit and general intervals graphs using$O(\log^2 \ | V \ |)$ operations and evaluate the algorithms empirically.Comment: 29 pages, accepted for 39th International Workshop on Graph-Theoretic Concepts 201

Quality of the Solution Sets of Parameter-Dependent Interval Linear Systems

Introduction By R , R denote the set of real vectors with n components, resp. the set of real n matrices. The set of all real compact intervals [a] := [a, a] := {a # a} is denoted by IR. We assume the reader is familiar with interval arithmetic [1]. By IR , IR denote the sets of interval n-vectors, resp. n matrices. Consider the linear system x = b(p), (1) where A(p) and b(p) depend a#ne-linearly on a parameter vector p . When p varies within a range [p] , the set of solutions to all A(p) x = b(p), p [p], called parametric solution set, is := # (A(p), b(p), [p]) := x = b(p) for some p [p]} . Denote by [A] = [A, A] := A([p]) , resp. [b] = [b, b] := b([p]) the interval matrix, resp. the interval vector, obtained from A(p), resp. b(p), substituting p by [p] and performing the corresponding interval operations. Hence the interval linear system x = [b], that is A([p]) x = b([p]) is a general non-parametric interval l

Construct Linear Quasi-Interpolants on Infinite Intervals

In solving the data interpolation problem, which is fundamental in data analysis, we typically deal with the data samples spread in a finite interval [a, b], which results in the operations involving finite-dimensional matrices. There are many interesting results developed under this framework. However, when the data samples are given from an infinite interval [a, ∞) (for certain special types of real-world applications), many existing results would not work anymore due to the special properties of the infinite data samples. A new framework should be established to support the infinite data samples. In this dissertation, we develop a special tool called local linear quasi-interpolant for an infinite interval with the following properties: 1) Each linear functional of the quasi-interpolant is determined by at most three data samples, so that the spline coefficients can be calculated in real-time; 2) The quasi-interpolant preserves all the linear polynomials; 3) Our framework does not impose any restriction on the relationship between the sample locations and the spline knots, which provides us the necessary flexibility in the real-world applications. Our construction is based on a matrix factorization method with respect to infinite-dimensional matrices. In order to ensure that the infinite version of the Shoenberg-Whitney matrices are invertible, we take the constructive approach that results in both the left-inverses and the right-inverses. Furthermore, since the associative law of the matrix multiplication does not work for the infinite matrices, we verify all the formulas derived from the infinite matrix operations. Finally, our local method allows us to calculate the spline interpolating coefficients in real-time on the fly for the infinite data samples

Formally Verified Conditions for Regularity of Interval Matrices

The final publication is available at www.springerlink.comInternational audienceWe propose a formal study of interval analysis that concentrates on theoretical aspects rather than on computational ones. In particular we are interested in conditions for regularity of interval matrices. An interval matrix is called regular if all scalar matrices included in the interval matrix have non-null determinant and it is called singular otherwise. Regularity plays a central role in solving systems of linear interval equations. Several tests for regularity are available and widely used, but sometimes rely on rather involved results, hence the interest in formally verifying such conditions of regularity. In this paper we set the basis for this work: we define intervals, interval matrices and operations on them in the proof assistant Coq, and verify criteria for regularity and singularity of interval matrices