24 research outputs found
P-matrix recognition is co-NP-complete
This is a summary of the proof by G.E. Coxson that P-matrix recognition is
co-NP-complete. The result follows by a reduction from the MAX CUT problem
using results of S. Poljak and J. Rohn.Comment: 9 page
An Overview of Polynomially Computable Characteristics of Special Interval Matrices
It is well known that many problems in interval computation are intractable,
which restricts our attempts to solve large problems in reasonable time. This
does not mean, however, that all problems are computationally hard. Identifying
polynomially solvable classes thus belongs to important current trends. The
purpose of this paper is to review some of such classes. In particular, we
focus on several special interval matrices and investigate their convenient
properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse
M-matrices, inverse nonnegative matrices, nonnegative matrices, totally
positive matrices and some others. We focus in particular on computing the
range of the determinant, eigenvalues, singular values, and selected norms.
Whenever possible, we state also formulae for determining the inverse matrix
and the hull of the solution set of an interval system of linear equations. We
survey not only the known facts, but we present some new views as well
Combinatorial Characterizations of K-matrices
We present a number of combinatorial characterizations of K-matrices. This
extends a theorem of Fiedler and Ptak on linear-algebraic characterizations of
K-matrices to the setting of oriented matroids. Our proof is elementary and
simplifies the original proof substantially by exploiting the duality of
oriented matroids. As an application, we show that a simple principal pivot
method applied to the linear complementarity problems with K-matrices converges
very quickly, by a purely combinatorial argument.Comment: 17 pages; v2, v3: clarified proof of Thm 5.5, minor correction
The Complexity of Recognizing Unique Sink Orientations
Given a Boolean Circuit with n inputs and n outputs, we want to decide if it represents a Unique Sink Orientation (USO). USOs are useful combinatorial objects that serve as abstraction of many relevant optimization problems. We prove that recognizing a USO is coNP-complete. However, the situation appears to be more complicated for recognizing acyclic USOs. Firstly, we give a construction to prove that there exist cyclic USOs where the smallest cycle is of superpolynomial size. This implies that the straightforward representation of a cycle (i.e. by a list of vertices) does not make up for a coNP certificate. Inspired by this fact, we investigate the connection of recognizing an acyclic USO to PSPACE and we prove that the problem is PSPACE-complete
Computational Complexity of the ?-Ham-Sandwich Problem
?_d from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the ?-Ham-Sandwich theorem. They gave an algorithm to find the hyperplane in time O(n (log n)^{d-3}), where n is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]).
Recently, Fearnley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called Unique End-of-Potential Line (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the ?-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the P-matrix linear complementarity problem
An algorithmic characterization of P-matricity
International audienceIt is shown that a matrix M is a P-matrix if and only if, whatever is the vector q, the Newton-min algorithm does not cycle between two points when it is used to solve the linear complementarity problem 0 ≤ x ⊥ (Mx+q) ≥ 0.Nous montrons dans cet article qu'une matrice M est une P-matrice si, et seulement si, quel que soit le vecteur q, l'algorithme de Newton-min ne fait pas de cycle de deux points lorsqu'il est utilisé pour résoudre le problème de compl\émentarité linéaire 0 ≤ x ⊥ (Mx+q) ≥ 0