On the randomized simplex algorithm in abstract cubes Jiři Matoušek Charles University Prague Tibor Szabó ETH Zürich

Linear Programming the geometric view Given a convex polytope P in R n with m facets and a linear objective function c, Find the minimum value of c on P. The minimum is taken at a vertex of P. A simplex algorithm moves from vertex to vertex along an edge each time decreasing the objective function value.

Pivot Rules Which improving edge to choose: the pivot rule No deterministic pivot rule is known to yield a polynomial or even subexponential running time. In fact almost all pivot rules are known to have bad instances. Randomized pivot rules are a bit more succesful. There is a subexponential randomized pivot rule and there are no known superpolynomial lower bounds for any decent randomized pivot rule.

LP Algorithms Simplex method [Dantzig 1947] –very fast in practice –very good “average case” –very bad/unknown “worst-case” Ellipsoid method [Khachyian], interior-point methods [Karmakar],… –weakly polynomial but NO (worst-case) bound in terms of n and m alone

Abstract frameworks Abstract objective functions Acyclic unique sink orientations LP-type problems [Sharir, Welzl] Abstract optimization problems [Gärtner]

Abstract Objective Functions P is a polytope, f : V(P) → R is a function f is unimin on P if there is no local minima other than the global minima. f is an abstract objective function on P if it is unimin on any face F of P. Adler and Saigal, Williamson Hoke, Kalai, 1988.

Unimin functions on the cube Any randomized algorithm needs at least queries for some unimin function on the hypercube [Aldous ’84] There is a (simple) randomized algorithm which works in steps Improvement: [Aaronson, ’04] Quantum query complexity

RandomFacet on AOF Kalai (1992): the simplex algorithm RandomFacet finishes in subexponential time on any AOF. ( in cubes.) (also: Matoušek, Sharir and Welzl in a dual setting) Still the best known! Matoušek gave AOFs on which Kalai’s analysis is essentially tight.

RandomEdge RandomEdge is the simplex algorithm which selects an improving edge uniformly at random. Its running time –on the n-dimensional simplex is Liebling –on n-dimensional polytopes with n+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)

RandomEdge on AOFs RandomEdge is quadratic on Matoušek’s orientations (which kill RandomFacet ) Williamson Hoke (1988) conjectured that RandomEdge is quadratic on all AOFs. (cf. Tovey, 1997)

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

RandomEdge is slow Theorem. [Matoušek, Sz., FOCS’04] 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 Klee-Minty cube Blowup construction [Schurr-Sz., ‘02] Hypersink reorientation [Schurr-Sz., ‘02] Randomness

Klee-Minty cube

Blowup Construction

A very special case: the Klee-Minty cube reversed KM m-1 KM m-1 KM m

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 permute (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

Difficulties… 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.

AiAi Rand KM m Hypersink reorientation to ensure that when the walk enters the sink of any of the small blocks it enters a random copy of A i on the first n coordinate A i is an (n+ikm)-cube, A 0 is an arbitrary n-cube constrcut A i+1 from A i recursively

Claim: The first 2 i steps visit vertices with outdegree at least k AiAi Rand KM m When the walk enters the sink of any of the small blocks it enters a random copy of A i on the first n coordinate 1. Phase: first 2 i steps (Note: k≥11m) 2. Phase: in between (still no KM m is in its sink) 3. Phase: one of the KM m is in its sink Proof: induction on i

Conclusion: The first 2 t steps of RandomEge in the 2n-dimensional cube A t visit vertices with outdegree at least k A t is a (n+tkm)-cube, Choose

An upper bound, please! Obtain any reasonable upper bound on the running time of RandomEdge Best known upper bound is, where p(n) is an arbitrary polynomial [Gärtner and Kaibel, ’05] Find an algorithm which gets to the minima of AOFs on the n-cube faster than exp(  n)

BottomTop From v move to the sink in the subcube spanned by the outgoing edges. (Note: BottomTop is NOT an algorithm!) [suggested by Kaibel] Theorem [Schurr, Sz., IPCO’05] There is an AUSO of the n-cube on which BottomTop, starting at a random vertex, takes at least c2 n/2 steps.

Lower bounds Improve on the current modest lower bounds for AUSOs: Deterministic complexity: Ω(n 2 /log n) Randomized complexity: Ω(n)

Realizability Can one modify the construction such that the cube is realizable? (Probably not …) Or at least it satisfies the Holt-Klee condition? Or at least each three-dimensional subcube satisfies the Holt-Klee condition?

Unique Sink Orientations of Cubes The model of unique sink orientations of cubes (possibly with cycles) includes LP on an arbitrary polytope. Find a subexponential algorithm!

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