Completely unimodal numberings of a simple polytope
Discrete Applied Mathematics
A Subexponential Algorithm for Abstract Optimization Problems
SIAM Journal on Computing
Linear programming, the simplex algorithm and simple polytopes
Mathematical Programming: Series A and B - Special issue: papers from ismp97, the 16th international symposium on mathematical programming, Lausanne EPFL
Explicit and implicit enforcing: randomized optimization
Computational Discrete Mathematics
Linear Programming - Randomization and Abstract Frameworks
STACS '96 Proceedings of the 13th Annual Symposium on Theoretical Aspects of Computer Science
Combinatorial Linear Programming: Geometry Can Help
RANDOM '98 Proceedings of the Second International Workshop on Randomization and Approximation Techniques in Computer Science
Unique Sink Orientations of Cubes
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Jumping doesn't help in abstract cubes
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
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We give a worst-case 驴(n2/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 [17]. 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 super-linear 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 problem.