Related papers: Exponential lower bounds for history-based simplex…
In this paper we give an exponential lower bound for Cunningham's least recently considered (round-robin) rule as applied to parity games, Markhov decision processes and linear programs. This improves a recent subexponential bound of…
The question whether the Simplex Algorithm admits an efficient pivot rule remains one of the most important open questions in discrete optimization. While many natural, deterministic pivot rules are known to yield exponential running times,…
The existence of a polynomial-time pivot rule for the simplex method is a fundamental open question in optimization. While many super-polynomial lower bounds exist for individual or very restricted classes of pivot rules, there currently is…
The existence of a pivot rule for the simplex method that guarantees a strongly polynomial run-time is a longstanding, fundamental open problem in the theory of linear programming. The leading pivot rule in theory is the shadow pivot rule,…
An acyclic USO on a hypercube is formed by directing its edges in such as way that the digraph is acyclic and each face of the hypercube has a unique sink and a unique source. A path to the global sink of an acyclic USO can be modeled as…
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.…
We investigate the worst-case behavior of the simplex algorithm on linear programs with three variables, that is, on 3-dimensional simple polytopes. Among the pivot rules that we consider, the ``random edge'' rule yields the best asymptotic…
We prove that the active-set method needs an exponential number of iterations in the worst-case to maximize a convex quadratic function subject to linear constraints, regardless of the pivot rule used. This substantially improves over the…
Unique Sink Orientations (USOs) are an appealing abstraction of several major optimization problems of applied mathematics such as for instance Linear Programming (LP), Markov Decision Processes (MDPs) or 2-player Turn Based Stochastic…
In 2011, Friedmann [F 7] claimed to have proved that pathological linear programs existed for which the Simplex method using Zadeh's least-entered rule [Z 14] would take an exponential number of pivots. In 2019, Disser and Hopp [DH 5]…
We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/sqrt(d) pivot steps on any simple d-polytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a…
The simplex algorithm using the random edge pivot-rule on any realization of a dual cyclic 4-polytope with n facets does not take more than O(n) pivot-steps. This even holds for general abstract objective functions (AOF) / acyclic unique…
We present new pivot rules for the Simplex method for LPs over 0/1 polytopes. We show that the number of non-degenerate steps taken using these rules is strongly polynomial and even linear in the dimension or in the number of variables. Our…
The complexity of Philip Wolfe's method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. The method is important because it is used as a subroutine for one of the…
We provide a monotone non increasing sequence of upper bounds $f^H_k$ ($k\ge 1$) converging to the global minimum of a polynomial $f$ on simple sets like the unit hypercube. The novelty with respect to the converging sequence of upper…
In this paper, we analyze the simplex method with the largest distance rule and derive upper bounds on the number of different basic feasible solutions generated. The pivoting rule was proposed by Pan [10], and in some cases, it was…
Random Edge is the most natural randomized pivot rule for the simplex algorithm. Considerable progress has been made recently towards fully understanding its behavior. Back in 2001, Welzl introduced the concepts of \emph{reachmaps} and…
Circuit-augmentation algorithms are generalizations of the Simplex method, where in each step one is allowed to move along a fixed set of directions, called circuits, that is a superset of the edges of a polytope. We show that in the…
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…
Combining a modular approach to the $abc$ conjecture developed by the second author with the classical method of linear forms in logarithms, we obtain improved unconditional bounds for two classical problems. First, for Szpiro's conjecture…