On the Existence of Hamiltonian Paths for History Based Pivot Rules on Acyclic Unique Sink Orientations of Hypercubes

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.

💡 Research Summary

The paper investigates whether history‑based pivot rules—most notably Zadeh’s “least entered” rule and several other variants—can be forced to follow a Hamiltonian path on acyclic unique‑sink orientations (USOs) of d‑dimensional hypercubes. A USO is a direction assignment to the edges of a hypercube such that the resulting digraph is acyclic and every face has a unique sink and a unique source. This abstract model captures the combinatorial essence of the simplex method: each vertex corresponds to a basis, each directed edge corresponds to a pivot, and the global sink corresponds to the optimal solution.

The authors first formalize the hypercube labeling: vertices are numbered 0,…,2^d‑1, and each edge is identified by a signed direction ±t (t∈{1,…,d}) indicating which bit flips and whether the flip is 0→1 (+) or 1→0 (–). A pivot rule can therefore be described as a function that, given the current vertex and a history array h, selects an improving direction. The history array records either the total number of times each signed direction has been used (Zadeh), the number of times each unsigned direction has been used (Least‑Used Direction, LUD), or more subtle recency information (Least‑Recently Considered, LRC; Least‑Recently Basic, LRB; Least‑Recently Entered, LRE; Least‑Iterations‑in‑Basis, LIB).

The central question is: for a given rule, does there exist an acyclic USO in which the rule’s deterministic choices generate a Hamiltonian path (a path that visits every vertex exactly once)? A Hamiltonian path is the worst‑case scenario for a pivot rule because it forces 2^d‑1 pivots. To answer this, the authors develop a backtracking enumeration algorithm that builds Hamiltonian paths vertex by vertex. Crucially, once a partial path is known, the orientation of all edges that lie between already visited vertices is forced (edges must point from earlier to later vertices). Using the Williamson‑Hoke theorem, which characterizes USOs by the degree sequence (exactly C(d,k) vertices must have indegree k for each k), the algorithm can prune any partial path that already violates the degree constraints. This “on‑the‑fly” pruning reduces the search space dramatically and makes exhaustive enumeration feasible up to dimension 6.

Experimental results obtained with this algorithm are striking. Zadeh’s rule admits Hamiltonian‑path USOs for dimensions 1 through 9; the authors explicitly construct examples up to d=9 and have not found a counterexample, suggesting that Zadeh’s rule may allow Hamiltonian paths in arbitrarily high dimensions. In contrast, the other five history‑based rules (LUD, LRC, LRB, LRE, LIB) fail to produce a Hamiltonian path already at dimension 6, and the authors prove that for dimensions greater than 5 no USO can support a Hamiltonian path under any of these rules. The proofs rely on structural properties of the rules: for instance, LUD always picks the unsigned direction with smallest usage count, which forces the first occurrence of each direction to create a vertex of indegree 1. Since there are only d such vertices, the rule quickly runs out of “new” directions and inevitably creates a vertex with indegree 0 or 2, violating the Williamson‑Hoke condition. Similar arguments apply to the recency‑based rules, which concentrate pivots on a small subset of directions, leading to degree violations.

A further theoretical contribution concerns Zadeh’s rule. The authors prove that if a Hamiltonian path follows Zadeh’s rule on a USO, then the least‑used signed direction must be used at least 2^{d‑2} − 2^{d‑3} times. This lower bound shows that Zadeh’s rule forces each variable to be pivoted roughly the same exponential number of times, contrasting sharply with the Klee‑Minty construction where some variables pivot only once. Consequently, Zadeh’s rule exhibits a balanced but still exponential pivot count, which aligns with Friedmann’s recent lower‑bound result that Zadeh’s rule requires at least 2^{Ω(√d)} steps in the worst case.

The paper concludes that among the examined history‑based pivot rules, Zadeh’s least‑entered rule is uniquely capable of supporting Hamiltonian paths on USOs, at least up to dimension 9, while the other rules are structurally incapable of doing so beyond very low dimensions. This insight highlights the profound impact that the design of the history component has on the combinatorial behavior of pivot algorithms. It suggests that rules which evenly distribute pivot usage (as Zadeh’s does) may be more robust against pathological exponential behavior, whereas rules that rely on recentness or minimal usage of unsigned directions can be forced into configurations that violate the USO degree constraints. The work thus provides both a methodological framework for exploring worst‑case scenarios in abstract pivot models and concrete evidence about the relative strengths and weaknesses of several prominent history‑based pivot strategies.