[Paper Review] Constructing subset partition graphs with strong adjacency and end-point count properties

Nicolai Hähnle|arXiv (Cornell University)|Mar 7, 2012

Advanced Combinatorial Mathematics5 references5 citations

TL;DR

This paper presents a randomized construction using Lovász’s Local Lemma to transform any subset partition graph (SPG) satisfying the singleton property into a higher-dimensional SPG that also satisfies strong adjacency and end-point count properties. The key result is a construction of abstract spindles with exponential length that meet these combinatorial properties, offering a conceptual simplification of prior constructions and advancing the search for high-diameter polytopal graphs.

ABSTRACT

Kim defined a very general combinatorial abstraction of the diameter of polytopes called subset partition graphs to study how certain combinatorial properties of such graphs may be achieved in lower bound constructions. Using Lovász' Local Lemma, we give a general randomized construction for subset partition graphs satisfying strong adjacency and end-point count properties. This can be used as a building block to conceptually simplify the constructions given in [Kim11]. We also use our method to construct abstract spindles, an analogy to the spindles used by Santos to disprove the Hirsch conjecture, of exponential length which satisfy the adjacency and end-point count properties.

Motivation & Objective

• To develop a general method for constructing subset partition graphs (SPGs) that satisfy strong adjacency and end-point count properties, which are critical for modeling polytopal diameter behavior.
• To simplify existing combinatorial constructions for high-diameter SPGs by transforming lower-dimensional SPGs with the singleton property into higher-dimensional ones with enhanced structural properties.
• To construct abstract spindles of exponential length that satisfy strong adjacency and end-point count, providing new candidates for disproving the Hirsch conjecture.
• To explore the interaction between the strong adjacency and end-point count properties and the dimension reduction property, which is essential for modeling actual polytopes.

Proposed method

• A randomized subdivision process is applied to an input SPG with the singleton property, where each edge is subdivided into r segments using a symbol set S' = S × [r].
• The construction uses a random assignment of labels to the subdivision vertices, ensuring that adjacent vertices in the new graph correspond to sets differing in at least two elements, thus avoiding unwanted high intersections.
• Lovász’s Local Lemma is applied to show that with positive probability, no pair of vertices on different edges will have intersection size d-1, which would violate the end-point count property.
• The method bounds the number of conflicting events (bad events) and uses the condition (k+1)pe < 1 to guarantee a positive probability of success, leading to a sufficient value of r = ⌈16eΔ⌉.
• The resulting SPG preserves the graph structure as a subdivision of the original and inherits the singleton property, with each original vertex A mapping to A × [r].
• The construction is applied to a path-based SPG with 2d symbols to generate an abstract spindle of exponential length in d, satisfying strong adjacency and end-point count.

Experimental results

Research questions

• RQ1 Can a general randomized method be used to transform any SPG with the singleton property into one satisfying strong adjacency and end-point count?
• RQ2 What is the minimal dimension increase required to achieve strong adjacency and end-point count via random labeling?
• RQ3 Can exponential-length abstract spindles satisfying strong adjacency and end-point count be constructed using this method?
• RQ4 How do the strong adjacency and end-point count properties interact with the dimension reduction property in the constructed SPGs?

Key findings

• A randomized construction using Lovász’s Local Lemma ensures the existence of an r-dimensional SPG with r ≥ ⌈16eΔ⌉ that satisfies strong adjacency, end-point count, and the singleton property.
• The construction transforms any d-dimensional SPG with the singleton property into an rd-dimensional SPG that is isomorphic to a subdivision of the original graph.
• The method produces abstract spindles of exponential length in d that satisfy strong adjacency and end-point count, improving upon the polynomial-length construction in Kim (2011).
• The resulting SPG preserves the singleton property, with each original vertex A mapping to the set A × [r] in the new graph.
• The construction maintains strong adjacency by ensuring that adjacent vertices in the new graph correspond to sets with intersection size d−1 in the original labeling.
• The method does not preserve dimension reduction when the original graph contains cycles, highlighting a key limitation for modeling polytopal graphs.