Finding the Sink Takes Some Time: An Almost Quadratic Lower Bound for Findingthe Sink of Unique Sink Oriented Cubes

We give a worst-case Ω(n 2/log n) lower bound on the number of vertex evaluations a deterministic algorithm needs to perform in order to find the (unique) sink of a unique sink oriented n-dimensional cube. We consider the problem in the vertex-oracle model, introduced in [18]. In this model one can access the orientation implicitly, in each vertex evaluation an oracle discloses the orientation of the edges incident to the queried vertex. An important feature of the model is that the access is indeed arbitrary, the algorithm does not have to proceed on a directed path in a simplex-like fashion, but could "jump around”. Our result is the first superlinear lower bound on the problem. The strategy we describe works even for acyclic orientations. We also give improved lower bounds for small values of n and fast algorithms in a couple of important special classes of orientations to demonstrate the difficulty of the lower bound proble

Unique Sink Orientations of Grids

We introduce unique sink orientations of grids as digraph models for many well-studied problems, including linear programming over products of simplices, generalized linear complementarity problems over P-matrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the Holt-Klee condition known to hold for polytope digraphs, and we give the first expected linear-time algorithms for solving PGLCP with a fixed number of block

A complexity analysis of Policy Iteration through combinatorial matrices arising from Unique Sink Orientations

Unique Sink Orientations (USOs) are an appealing abstraction of several major optimization problems of applied mathematics such as Linear Programming (LP), Markov Decision Processes (MDPs) or 2-player Turn Based Stochastic Games (2TBSGs). A polynomial time algorithm to find the sink of a USO would translate into a strongly polynomial time algorithm to solve the aforementioned problems—a major quest for all three cases. In the case of an acyclic USO of a cube, a situation that captures both MDPs and 2TBSGs, one can apply the well-known Policy Iteration (PI) algorithm. The study of its complexity is the object of this work. Despite its exponential worst case complexity, the principle of PI is a powerful source of inspiration for other methods. In 2012, Hansen and Zwick introduced a new combinatorial relaxation of the complexity problem for PI resulting in what we call Order-Regular (OR) matrices. They conjectured that the maximum number of rows of such matrices—an upper bound on the number of steps of PI—should follow the Fibonacci sequence. As our first contribution, we disprove the lower bound part of Hansen and Zwick's conjecture. Then, for our second contribution, we (exponentially) improve the Ω(1.4142n) lower bound on the number of steps of PI from Schurr and Szabó in the case of OR matrices and obtain an Ω(1.4269n) bound. © 2017 Elsevier B.V