On the existence of Hamiltonian paths for history based pivot rules on acyclic unique sink orientations of hypercubes

Abstract

An acyclic USO on a hypercube is formed by directing its edges in such as way that the digraph is acyclic and each face of the hypercube has a unique sink and a unique source. A path to the global sink of an acyclic USO can be modeled as pivoting in a unit hypercube of the same dimension with an abstract objective function, and vice versa. In such a way, Zadeh's 'least entered rule' and other history based pivot rules can be applied to the problem of finding the global sink of an acyclic USO. In this paper we present some theoretical and empirical results on the existence of acyclic USOs for which the various history based pivot rules can be made to follow a Hamiltonian path. In particular, we develop an algorithm that can enumerate all such paths up to dimension 6 using efficient pruning techniques. We show that Zadeh's original rule admits Hamiltonian paths up to dimension 9 at least, and prove that most of the other rules do not for all dimensions greater than 5. √ d) [7]. A non-trivial upper bound on Zadeh's rule is still unknown.

An exponential lower bound for Cunningham’s rule

“…There is a very natural analogy between path following algorithms and pivoting in linear programming. Pivot rules for LPs therefore have natural analogues for AUSOs and a full discussion of this is contained in [1]. In particular the least recently considered rule can be adapted to give a path following algorithm to find the unique sink of an n-cube AUSO starting at any given vertex as follows.…”

The Niceness of Unique Sink Orientations

“…Again, for the coordinate we choose we have two choices (edges oriented forward or backward). Hence, we have 2 . Note that the construction we suggest, i.e.…”

“…First described by Stickney and Watson already in 1978 as abstract models for P-matrix linear complementarity problems (PLCPs) [29], USO were revived by Szabó and Welzl in 2001 [30]. Subsequently, their structural and algorithmic properties were studied extensively ( [27], [28], [23], [14], [7], [2], [17], [15], [20], [18]). In a nutshell, a USO is an orientation of the n-dimensional hypercube graph, with the property that there is a unique sink in every subgraph induced by a nonempty face.…”

Exponential lower bounds for history-based simplex pivot rules on abstract cubes

“…It has an ordered list L that contains all 2n directions; let L[k] indicate the kth direction on the list. There is a marker µ of which direction was used last: if direction L[k] was used at the last step then µ = k. At the next step the algorithm will start checking the directions on the list from L[µ + 1] in a cyclic order (so if it reaches L[2n] it continues from L [1]) and it chooses the first available one. Initially, µ = 2n so that the first direction that the algorithm checks is L [1].…”

“…Finally, Avis and Friedmann write [2]: "More generally it is of interest to determine whether all of the history based rules mentioned in [1] have exponential behaviour on AUSO".…”