RandomEdge can be mildly exponential on abstract cubes Jiri Matousek Charles University Prague Tibor Szabó ETH Zürich

Linear Programming Given a convex polyhedron P in R n with at most m facets and a linear objective function c, one would like to determine the minimum value of c on P. The minimum is taken at a vertex of P. The simplex algorithm moves from vertex to vertex along an edge each time decreasing the objective function value. The way to select the next vertex is the pivot rule

RandomEdge RandomEdge is the simplex algorithm which selects an improving edge uniformly at random. Its running time –on the d-dimensional simplex is Liebling –on d-dimensional polytopes with d+2 facets is Gärtner et al. (2001) –on the n-dimensional Klee-Minty cube is Williamson Hoke (1988) Gärtner, Henk, Ziegler (1995) Balogh, Pemantle (2004)

Abstract Objective Functions P is a polytope f : V(P) → R is an abstract objective function if a local minimum of any face F is also the unique global minimum of F. Adler and Saigal, Williamson Hoke, Kalai, 1988.

RandomFacet on AOF Kalai (1992): the simplex algorithm RandomFacet finishes in subexponential time on any AOF. (also: Matousek, Sharir and Welzl in a dual setting) Matousek gave AUSOs on which Kalai’s analysis is essentially tight.

RandomEdge is quadratic on Matousek’s orientations Williamson Hoke (1988) conjectured that RandomEdge is quadratic on all AOFs.

Acyclic Unique Sink Orientations Let P be a polytope. An orientation of its graph is called an acyclic unique sink orientation or AUSO if every face has a unique sink (that is a vertex with only incoming edges) and no directed cycle. AUSOs and AOFs are the same

Killing RandomEdge Theorem. There exists an AUSO of the n-dimensional cube, such that RandomEdge started at a random vertex, with probability at least, makes at least moves before reaching the sink.

Ingredients of the good pasta The flour: The water: The eggs: The mixing: Ingredients of a slow cube Klee-Minty cube Blowup construction Hypersink reorientation Randomness

Klee-Minty cube reversed KM m-1 KM m-1 KM m

Blowup Construction

Hypersink reorientation

A simpler construction Let A be an n-dimensional cube, on which RandomEdge is slow. Let. Take the blowup of A with random KM m whose sink is in the same copy of A Reorient the hypersink by placing a random copy of A.

A A A A rand A A simpler construction

A typical RandomEdge move Move in frame: –RandomEdge move in KM m –Stay put in A Move within a hypervertex: –RandomEdge move in A –Move to a random vertex of KM m on the same level A rand A A A v Random walk with reshuffles on KM m RandomEdge on A

Walk with reshuffles on KM m Start at a random v (0) of KM m v (i) is chosen as follows: –With probability p i,step we make a step of RandomEdge from v (i-1). –With probability p i,resh we reshuffle the coordinates of v (i-1) to obtain v (i). –With probability 1- p i,step - p i,resh, v (i) = v (i-1).

Walk with reshuffles on KM m is slow Proposition. Suppose that Then with probability at least The random walk with reshuffles makes at least steps. (α and β are constants)

Reaching the hypersink Either we reach the sink by reaching the sink of a copy of A and then perform RandomEdge on KM m. This takes at least T(n) time. Or we reach the hypersink without entering the sink of any copy of A. That is the random walk with reshuffles reaches the sink of KM m. This takes at least time.

The recursion RandomEdge arrives to the hypersink at a random vertex. Then it needs T(n) more steps. So passing from dimension n to n+  n the expected running time of RandomEdge doubles. Iterating  n - times gives In order to guarantee that reshuffles are frequent enough we need a more complicated construction and that is why we are only able to prove a running time of.

Open questions Obtain any reasonable upper bound on the running time of RandomEdge Can one modify the construction such that the cube is realizable? (I don’t think so …) Or at least it satisfies the Holt-Klee condition? Or at least each three-dimensional subcube satisfies the Holt-Klee condition?

More open questions The model of unique sink orientations of cubes (possibly with cycles) include LP on an arbitrary polytope. Find a subexponential algorithm.

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