Random monotone paths on polyhedra

Consider the following simple random process, on a 3-dimensional bounded convex polyhedron, with a given starting vertex: Repeatedly choose a random edge that goes "down" (decreasing the z-coordinate), and follow it to the vertex at the other end -- if there are several choices, pick one of them, with equal probability. Stop once you have reached a vertex with minimal z-coordinate.

If the polyhedron has <i>n</i> faces, <i>in the worst case</i> you could visit <i>2n-4</i> vertices: By Euler's equation, the polyhedron can't have more. But you hope to reach a "minimal" vertex much faster.

So, how about the <i>expected</i> number of vertices to be visited? Is it guaranteed to be substantially smaller? This question is surprisingly difficult to answer: The trivial bounds for the worst-case expected number of steps are <i>n</i> and <i>2n</i> (ignoring constants). Harder work yields a lower bound of 1.3333<i>n</i> and 1.50<i>n</i>, while we can now prove that the answer lies between the following bounds: 1.3473<i>n</i>≤<i>E(n)</i>≤1.4943<i>n</i>.

The upper bound relies on a good measure for "progress" -- we find that via an auxiliary linear program. The lower bound asks for "bad examples" -- these we find via explicit enumeration of small polyhedral (planar, 3-connected) graphs and objective functions on these (in a combinatorial model).

If you ask why we'd consider this problem: It is a wonderful, simple, and already quite challenging test-instance for the performance of randomized pivot rules for the Simplex Algorithm for Linear Programming. In view of the key open problems in this context (such as the Hirsch Conjecture, and the search for a strongly-polynomial Linear Programming algorithm), it makes sense to consider the simple and intuitive 3-dimensional case in detail, and to ask for precise answers. The RANDOM-EDGE rule discussed here is a prime candidate for good results: It is extremely simple to describe, and seems "hard to fool." But it is also remarkably <i>difficult</i> to analyze, as has also been observed in other contexts.

So, we'll also look at other pivot-rules: But most of these show essentially worst-possible behavior; this includes both Kalai's RANDOM-FACET rule, which is known to be subexponential without dimension restriction, as well as Zadeh's deterministic history-dependent rule, for which no non-polynomial instances in general dimensions have been found so far.