Related papers: The Simplex Algorithm is NP-mighty
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…
We show that there are simplex pivoting rules for which it is PSPACE-complete to tell if a particular basis will appear on the algorithm's path. Such rules cannot be the basis of a strongly polynomial algorithm, unless P = PSPACE. We…
The simplex method is a well-studied and widely-used pivoting method for solving linear programs. When Dantzig originally formulated the simplex method, he gave a natural pivot rule that pivots into the basis a variable with the most…
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 prove that computing a shortest monotone path to the optimum of a linear program over a simple polytope is NP-hard, thus resolving a 2022 open question of De Loera, Kafer, and Sanit\`a. As a consequence, finding a shortest sequence of…
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…
It is well known that the most challenging question in optimization and discrete geometry is whether there is a strongly polynomial time simplex algorithm for linear programs (LPs). This paper gives a positive answer to this question by…
Until now, Computer Scientists have concerned themselves with identifying efficient algorithms for solving the general case of some problem -- that is finding one which performs well when the size of the input tends to infinity. In this…
This paper discusses a data-driven, empirically-based framework to make algorithmic decisions or recommendations without expert knowledge. We improve the performance of two algorithmic case studies: the selection of a pivot rule for the…
We describe an efficient implementation of a recent simplex-type algorithm for the exact solution of separated continuous linear programs, and compare it with linear programming approximation of these problems obtained via discretization of…
In this article we consider a certain sub class of Integer Equal Flow problem, which are known NP hard [8]. Currently there exist no direct solutions for the same. It is a common problem in various inventory management systems. Here we…
A double pivot algorithm that combines features of two recently published papers by these authors is proposed. The proposed algorithm is implemented in MATLAB. The MATLAB code is tested, along with a MATLAB implementation of Dantzig's…
Linear programming has been practically solved mainly by simplex and interior point methods. Compared with the weakly polynomial complexity obtained by the interior point methods, the existence of strongly polynomial bounds for the length…
We construct a family of Markov decision processes for which the policy iteration algorithm needs an exponential number of improving switches with Dantzig's rule, with Bland's rule, and with the Largest Increase pivot rule. This immediately…
The purpose of this article is to examine and limit the conditions in which the P complexity class could be equivalent to the NP complexity class. Proof is provided by demonstrating that as the number of clauses in a NP-complete problem…
The goal of this paper is to design a simplex algorithm for linear programs on lattice polytopes that traces `short' simplex paths from any given vertex to an optimal one. We consider a lattice polytope $P$ contained in $[0,k]^n$ and…
We examine possibility to design an efficient solving algorithm for problems of the class \np. It is introduced a classification of \np problems by the property that a partial solution of size $k$ can be extended into a partial solution of…
We consider the general problem of blocking all solutions of some given combinatorial problem with only few elements. For example, the problem of destroying all maximum cliques of a given graph by forbidding only few vertices. Problems of…
The support of a flow $x$ in a network is the subdigraph induced by the arcs $uv$ for which $x(uv)>0$. We discuss a number of results on flows in networks where we put certain restrictions on structure of the support of the flow. Many of…
The existence of a polynomial pivot rule for the simplex method for linear programming, policy iteration for Markov decision processes, and strategy improvement for parity games each are prominent open problems in their respective fields.…