Related papers: Exponential lower bounds for history-based simplex…
We study the problem of covering a given set of $n$ points in a high, $d$-dimensional space by the minimum enclosing polytope of a given arbitrary shape. We present algorithms that work for a large family of shapes, provided either only…
In this paper, a double-pivot simplex method is proposed. Two upper bounds of iteration numbers are derived. Applying one of the bounds to some special linear programming (LP) problems, such as LP with a totally unimodular matrix and Markov…
Pivoting methods are of vital importance for linear programming, the simplex method being the by far most well-known. In this paper, a primal-dual pair of linear programs in canonical form is considered. We show that there exists a sequence…
Motivated by the problem of bounding the number of iterations of the Simplex algorithm we investigate the possible lengths of monotone paths followed by the Simplex method inside the oriented graphs of polyhedra (oriented by the objective…
We propose to classify the power of algorithms by the complexity of the problems that they can be used to solve. Instead of restricting to the problem a particular algorithm was designed to solve explicitly, however, we include problems…
Policy iteration is a family of algorithms that are used to find an optimal policy for a given Markov Decision Problem (MDP). Simple Policy iteration (SPI) is a type of policy iteration where the strategy is to change the policy at exactly…
We show that deciding simulation equivalence and simulation preorder have quadratic lower bounds assuming that the Strong Exponential Time Hypothesis holds. This is in line with the best know quadratic upper bounds of simulation…
The simplex algorithm is one of the most popular algorithms to solve linear programs (LPs). Starting at an extreme point solution of an LP, it performs a sequence of basis exchanges (called pivots) that allows one to move to a better…
We show that the size of a minimal simplicial cover of a polytope $P$ is a lower bound for the size of a minimal triangulation of $P$, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and…
We present a specialized network simplex algorithm for the budget-constrained minimum cost flow problem, which is an extension of the traditional minimum cost flow problem by a second kind of costs associated with each edge, whose total…
Cubature rules on the triangle have been extensively studied, as they are of great practical interest in numerical analysis. In most cases, the process by which new rules are obtained does not preclude the existence of similar rules with…
Algebraic Branching Programs(ABPs) are standard models for computing polynomials. Syntactic multilinear ABPs (smABPs) are restrictions of ABPs where every variable is allowed to occur at most once in every path from the start to the…
We present a new positive lower bound for the minimum value taken by a polynomial P with integer coefficients in k variables over the standard simplex of R^k, assuming that P is positive on the simplex. This bound depends only on the number…
We study symmetric arithmetic circuits and improve on lower bounds given by Dawar and Wilsenach (ArXiv 2020). Their result showed an exponential lower bound of the permanent computed by symmetric circuits. We extend this result to show a…
This paper studies the polynomial basis that generates the smallest $n$-simplex enclosing a given $n^{\text{th}}$-degree polynomial curve in $\mathbb{R}^n$. Although the Bernstein and B-Spline polynomial bases provide feasible solutions to…
We describe an approach for finding upper bounds on an ODE dynamical system's maximal Lyapunov exponent among all trajectories in a specified set. A minimization problem is formulated whose infimum is equal to the maximal Lyapunov exponent,…
In this paper we study an extension of the Polynomial Calculus proof system where we can introduce new variables and take a square root. We prove that an instance of the subset-sum principle, the binary value principle, requires refutations…
We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over $\mathbb C$. Applying recent…
We improve known upper bounds for the minimal dispersion of a point set in the unit cube and its inverse in both the periodic and non-periodic settings. Some of our bounds are sharp up to logarithmic factors.
Although simplices are trivial from a linear optimization standpoint, the simplex algorithm can exhibit quite complex behavior. In this paper we study the behavior of max-slope pivot rules on (products of) simplices and describe the…