On Friedmann's subexponential lower bound for Zadeh's pivot rule

The question whether the Simplex method admits a polynomial time pivot rule remains one of the most important open questions in discrete optimization. Zadeh's pivot rule had long been a promising candidate, before Friedmann (IPCO, 2011) presented a subexponential instance, based on a close relation to policy iteration algorithms for Markov decision processes (MDPs). We investigate Friedmann's lower bound example and exhibit three flaws in the corresponding MDP: We show that (a) the initial policy for the policy iteration does not produce the required occurrence records and improving switches, (b) the specification of occurrence records is not entirely accurate, and (c) the sequence of improving switches used by Friedmann does not consistently follow Zadeh's pivot rule. In this paper, we resolve each of these issues by adapting Friedmann's construction. While the first two issues require only minor changes to the specifications of the initial policy and the occurrence records, the third issue requires a significantly more sophisticated ordering and associated tie-breaking rule that are in accordance with the Least-Entered pivot rule. Most importantly, our changes do not affect the macroscopic structure of Friedmann's MDP, and thus we are able to retain his original result

On Friedmann's Subexponential Lower Bound for Zadeh's Pivot Rule

The question whether the Simplex method admits a polynomial time pivot rule remains one of the most important open questions in discrete optimization. Zadeh's pivot rule had long been a promising candidate, before Friedmann (IPCO, 2011) presented a subexponential instance, based on a close relation to policy iteration algorithms for Markov decision processes (MDPs). We investigate Friedmann's lower bound example and exhibit three flaws in the corresponding MDP: We show that (a) the initial policy for the policy iteration does not produce the required occurrence records and improving switches, (b) the specification of occurrence records is not entirely accurate, and (c) the sequence of improving switches used by Friedmann does not consistently follow Zadeh's pivot rule. In this paper, we resolve each of these issues by adapting Friedmann's construction. While the first two issues require only minor changes to the specifications of the initial policy and the occurrence records, the third issue requires a significantly more sophisticated ordering and associated tie-breaking rule that are in accordance with the Least-Entered pivot rule. Most importantly, our changes do not affect the macroscopic structure of Friedmann's MDP, and thus we are able to retain his original result

On Friedmann's subexponential lower bound for Zadeh's pivot rule

The question whether the Simplex method admits a polynomial time pivot rule remains one of the most important open questions in discrete optimization. Zadeh's pivot rule had long been a promising candidate, before Friedmann (IPCO, 2011) presented a subexponential instance, based on a close relation to policy iteration algorithms for Markov decision processes (MDPs). We investigate Friedmann's lower bound example and exhibit three flaws in the corresponding MDP: We show that (a) the initial policy for the policy iteration does not produce the required occurrence records and improving switches, (b) the specification of occurrence records is not entirely accurate, and (c) the sequence of improving switches used by Friedmann does not consistently follow Zadeh's pivot rule. In this paper, we resolve each of these issues by adapting Friedmann's construction. While the first two issues require only minor changes to the specifications of the initial policy and the occurrence records, the third issue requires a significantly more sophisticated ordering and associated tie-breaking rule that are in accordance with the Least-Entered pivot rule. Most importantly, our changes do not affect the macroscopic structure of Friedmann's MDP, and thus we are able to retain his original result

Integer Programming and Combinatorial Optimization [electronic resource] : 20th International Conference, IPCO 2019, Ann Arbor, MI, USA, May 22-24, 2019, Proceedings /

This book constitutes the refereed proceedings of the 20th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2019, held in Ann Arbor, MI, USA, in May 2019. The 33 full versions of extended abstracts presented were carefully reviewed and selected from 114 submissions. The conference is a forum for researchers and practitioners working on various aspects of integer programming and combinatorial optimization. The aim is to present recent developments in theory, computation, and applications in these areas.Identically Self-Blocking Clutters -- Min-Max Correlation Clustering via -- Strong Mixed-Integer Programming Formulations for Trained Neural -- Extended Formulations from Communication Protocols in Output-Efficient -- Sub-Symmetry-Breaking Inequalities for ILP with Structured Symmetry -- Intersection Cuts for Polynomial Optimization -- Fixed-Order Scheduling on Parallel Machines -- Online Submodular Maximization: Beating 1/2 Made Simple -- Improving the Integrality Gap for Multiway Cut -- nell 1-sparsity Approximation Bounds for Packing Integer Programs -- A General Framework for Handling Commitment in Online Throughput Maximization -- Lower Bounds and A New Exact Approach for the Bilevel Knapsack with Interdiction Constraints -- On Friedmann's Subexponential Lower Bound for Zadeh's Pivot Rule -- Tight Approximation Ratio for Minimum Maximal Matching -- Integer Programming and Incidence Treedepth -- A Bundle Approach for SDPs with Exact Subgraph Constraints -- Dynamic Flows with Adaptive Route Choice -- The Markovian Price of Information -- On Perturbation Spaces of Minimal Valid Functions: Inverse Semigroup Theory and Equivariant Decomposition Theorem -- On Compact Representations of Voronoi Cells of Lattices -- An Efficient Characterization of Submodular Spanning Tree Games -- The Asymmetric Traveling Salesman Path LP Has Constant Integrality Ratio -- Approximate Multi-Matroid Intersection via Iterative Refinement -- An Exact Algorithm for Robust Influence Maximization -- A New Contraction Technique with Applications to Congruency-Constrained Cuts -- Sparsity of Integer Solutions in the Average Case -- A Generic Exact Solver for Vehicle Routing and Related Problems -- Earliest Arrival Transshipments in Networks With Multiple Sinks -- Intersection Cuts for Factorable MINLP -- Linear Programming Using Limited-Precision Oracles -- Computing the Nucleolus of Weighted Cooperative Matching Games in Polynomial Time -- Breaking Symmetries to Rescue SoS: The Case of Makespan Scheduling -- Random Projections for Quadratic Programs over a Euclidean Ball.This book constitutes the refereed proceedings of the 20th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2019, held in Ann Arbor, MI, USA, in May 2019. The 33 full versions of extended abstracts presented were carefully reviewed and selected from 114 submissions. The conference is a forum for researchers and practitioners working on various aspects of integer programming and combinatorial optimization. The aim is to present recent developments in theory, computation, and applications in these areas