arXiv subjects

Nils Mosis

Publications and source records attributed to Nils Mosis.

5 recordsLinked to original sources

arXiv2025

Lower bounds for ranking-based pivot rules

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. While numerous natural candidates for efficient rules have been eliminated, all existing lower bound constructions are tailored to individual or small sets of pivot rules. We introduce a unified framework for formalizing classes of rules according to the information about the input that they rely on. Within this framework, we show lower bounds for ranking-based classes of rules that base their decisions on orderings of the improving pivot steps induced by the underlying data. Our first result is a superpolynomial lower bound for strategy improvement, obtained via a family of sink parity games, which applies to memory-based generalizations of Bland's rule that only access the input by comparing the ranks of improving edges in some global order. Our second result is a subexponential lower bound for policy iteration, obtained via a family of Markov decision processes, which applies to memoryless rules that only access the input by comparing improving actions according to their ranks in a global order, their reduced costs, and the associated improvements in objective value. Both results carry over to the simplex method for linear programming.

math.OC↗

arXiv2025

An unconditional lower bound for the active-set method in convex quadratic maximization

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 best previously known lower bound [IPCO 2025], which needs objective functions of polynomial degrees $\omega(\log d)$ in dimension $d$, to a bound using a convex polynomial of degree 2. In particular, our result firmly resolves the open question [IPCO 2025] of whether a constant degree suffices, and it represents significant progress towards linear objectives, where the active-set method coincides with the simplex method and a lower bound for all pivot rules would constitute a major breakthrough. Our result is based on a novel extended formulation, recursively constructed using deformed products. Its key feature is that it projects onto a polygonal approximation of a parabola while preserving all of its exponentially many vertices. We define a quadratic objective that forces the active-set method to follow the parabolic boundary of this projection, without allowing any shortcuts along chords corresponding to edges of its full-dimensional preimage.

cs.DM↗

arXiv2025

An unconditional lower bound for the active-set method on the hypercube

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 little hope for an unconditional lower bound that addresses all pivot rules. We approach this question by considering the active-set method as a natural generalization of the simplex method to non-linear objectives. This generalization allows us to prove the first unconditional lower bound for all pivot rules. More precisely, we construct a multivariate polynomial of degree linear in the number of dimensions such that the active-set method started in the origin visits all vertices of the hypercube. We hope that our framework serves as a starting point for a new angle of approach to understanding the complexity of the simplex method.

cs.DM↗

arXiv2023

A unified worst case for classical simplex and policy iteration pivot rules

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 translates to a family of linear programs for which the simplex algorithm needs an exponential number of pivot steps with the same three pivot rules. Our results yield a unified construction that simultaneously reproduces well-known lower bounds for these classical pivot rules, and we are able to infer that any (deterministic or randomized) combination of them cannot avoid an exponential worst-case behavior. Regarding the policy iteration algorithm, pivot rules typically switch multiple edges simultaneously and our lower bound for Dantzig's rule and the Largest Increase rule, which perform only single switches, seem novel. Regarding the simplex algorithm, the individual lower bounds were previously obtained separately via deformed hypercube constructions. In contrast to previous bounds for the simplex algorithm via Markov decision processes, our rigorous analysis is reasonably concise.

cs.DM↗

arXiv2022

An Improved Algorithm for Open Online Dial-a-Ride

We consider the open online dial-a-ride problem, where transportation requests appear online in a metric space and need to be served by a single server. The objective is to minimize the completion time until all requests have been served. We present a new, parameterized algorithm for this problem and prove that it attains a competitive ratio of $1 + φ\approx 2.618$ for some choice of its parameter, where $φ$ is the golden ratio. This improves the best known bounds for open online dial-a-ride both for general metric spaces as well as for the real line. We also give a lower bound of~$2.457$ for the competitive ratio of our algorithm for any parameter choice.

cs.DS↗