Quasi Polynomial Diameter Formulas

• Quasi polynomial diameter formulas are functions that provide explicit or asymptotic bounds on the maximum distance within structures, characterized by growth rates between polynomial and exponential.
• They are derived through methods involving piecewise polynomial functions and recursive techniques, with applications in discrete geometry, combinatorics, and optimization.
• These formulas have practical implications in analyzing algorithm complexity, bounding polyhedral diameters, and understanding structural properties in groups, graphs, and algebraic systems.

Quasi polynomial diameter formulas refer to explicit or asymptotic bounds on the diameter of mathematical objects—graphs, groups, polyhedra, algebraic varieties, or combinatorial structures—that are given by functions exhibiting quasi-polynomial growth. Such formulas frequently appear in discrete geometry, combinatorics, optimization, algebraic settings, and the analysis of algorithms, and often reflect a complexity that is intermediate between strictly polynomial and exponential. Quasi-polynomial behavior is characterized by eventual periodicity in polynomial coefficients (e.g., $f(n)$ expresses as a finite set of polynomials depending on the residue of $n$ modulo some $m$) or by forms such as $n^{O(\log n)}$. This article reviews the definition, structural properties, computational aspects, major results, and representative applications of quasi polynomial diameter formulas, referencing prominent developments across multiple areas.

1. Definitions and General Properties

Quasi-polynomial diameter formulas arise when the maximum metric distance between elements (vertices, points, group elements, etc.) inside a finite or bounded structure is described by a function $g: \mathbb{N} \to \mathbb{Z}$ such that there exists some $m \in \mathbb{N}$ and polynomials $f_0, ..., f_{m-1} \in \mathbb{Q}[t]$ with $g(t) = f_i(t)$ for $t \equiv i \pmod m$, for large $t$ (Bogart et al., 2016). In geometric or combinatorial contexts, these diameter bounds may emerge from recursive or threshold procedures and behave piecewise polynomially across parameter regimes. For example, in parametric Presburger arithmetic, diameter formulas for families of sets parameterized by $t$ (such as sets of lattice points in parametric polytopes) are proven to be eventually quasi-polynomial in $t$ (Bogart et al., 2016).

In network and algebraic structures, diameters can occasionally scale quasi-polynomially in the number and configuration of nodes, constraints, or generators. In certain topological or convex geometric settings, the diameter function is influenced by symmetry, regularity, or the underlying combinatorics of the objects, as in polyhedral graphs, regular graphs with Q-polynomial structure, or partition problems.

2. Diameter in Graphs, Networks, and Groups

Quasi polynomial diameter formulas naturally occur in families of graphs and groups where the growth of metric diameter is closely tied to the combinatorial parameters. For example, the diameter of Knödel graphs $W_{\Delta,n}$ for $n \geq (2\Delta-5)(2^\Delta-2)+4$ is given exactly by

$$\operatorname{diam}(W_{\Delta, n}) = 1 + \left\lceil \frac{n-2}{2^\Delta - 2} \right\rceil,$$

which exhibits quasi-polynomial scaling for fixed $\Delta$ and large $n$, as the dominating term $2^\Delta$ controls the decay in diameter as degree increases (Musawi et al., 2020). Such formulas frequently capture a transition from linear to sublinear or sublogarithmic behavior in $n$ as structural parameters increase.

In the analysis of finite groups, diameter lower bounds grow polynomially in group size when a large abelian or nilpotent section is present. The paper (Sabatini, 2021) proves that if a generating set $S$ generates $G$ and $\operatorname{diam}(G, S) \geq c|G|^\varepsilon$, then $G$ admits a subgroup with an abelian quotient of size exponentially large in $|G|$, and such groups inherently have polynomial or quasi-polynomial diameter.

3. Polyhedral Diameter Bounds

The combinatorial diameter of polyhedra—the maximum length of shortest paths along the graph of the polyhedron—has been the subject of sustained paper, notably in connection with the Hirsch conjecture and pivot complexity in linear programming. Early work by Kalai–Kleitman yielded a quasi-polynomial upper bound for $d$-dimensional polyhedra with $n$ facets: $A_u(d, n) \leq \exp(C\sqrt{\log n}),$ where $C$ depends on the dimension (Gallagher et al., 2016). Todd and Sukegawa–Kitahara, and later (Gallagher et al., 2016), provided tail–quasipolynomial diameter bounds for polyhedra and polytopes for large $n$; e.g.,

$A_u(d, n) \leq (n-d) \log_2(d-1).$

Iterated recursion leads to nearly linear bounds for large $n$, e.g., $A_u(d, n) \leq n^{1+\epsilon}$ for any fixed $\epsilon > 0$.

For polytopes given by $\{x \in \mathbb{R}^n: Ax \leq b \}$ with $A \in \mathbb{Z}^{m \times n}$, the diameter is bounded by (Bonifas et al., 2011)

$O(\Delta^2 n^4 \log n\Delta)$

where $\Delta$ is the maximum absolute value of the sub-determinants of $A$. When $A$ is totally unimodular $(\Delta=1)$, the bound becomes $O(n^4 \log n)$. These results mark a substantial improvement over previous bounds and illustrate quasi-polynomial dependence on geometric and arithmetic parameters.

4. Algorithmic and Complexity Aspects

Computing objects defined by quasi-polynomial diameter formulas, such as lattice diameter segments or extremal paths, is subject to strong complexity considerations. In the plane, (Brose et al., 27 Aug 2025) describes a polynomial-time algorithm for finding a lattice diameter segment in lattice polygons, based on geometric decomposition and normalized segment volumes. In higher dimensions ($d \geq 3$), the problem of computing lattice diameter segments is NP-hard even for bounded semi-algebraic sets.

Quasi-polynomial formulas also appear in counting extremal metric features. For lattice polygons $P$, the function $LD_P(k)$, the number of lattice diameter segments of the dilated polytope $kP$, is eventually a quasi-polynomial of degree one in $k$, with explicit periodic formulas derived by analyzing the normalized segment lengths and congruential classes.

In optimization contexts, the circuit diameter of a polyhedron is shown to satisfy a quasi-polynomial bound: $O(m \min\{m, n-m\} \log(m + \kappa_A) + n \log n)$ where $\kappa_A$ is a circuit imbalance measure of the constraint matrix $A$. This facilitates polynomial augmentation algorithms for linear programming when $\kappa_A$ is polynomially bounded (Dadush et al., 2021).

5. Algebraic and Combinatorial Structures

Quasi-polynomial diameter formulas are encountered in algebraic contexts, such as Q-polynomial distance-regular graphs, affine Hecke algebras, and Macdonald polynomial theory. When a distance-regular graph with valency at least three admits two Q-polynomial structures, the algebraic constraints sharply restrict the possible graph types (e.g., hypercubes, folded cubes, half cubes, dual polar graphs) and yield diameter formulas controlled by explicit algebraic parameters (Ma et al., 2014). The underlying behavior—such as vanishing dual intersection numbers and symmetry constraints—enforces quasi-polynomial scaling in diameter as a function of combinatorial invariants.

In the context of affine Hecke algebras, (Venkateswaran, 2023) establishes precise decomposition formulas for symmetric and antisymmetric quasi-polynomials: $$Y_w(\pi^+_{t,\mathrm{IP}}(f)) = t(w_0) w_0(T^{-1}(T_{w_0 w})^{-1} f),$$ expressing symmetry properties and expansion in terms that deliver support (or "diameter") profiles governed by quasi-polynomial combinatorics, i.e., length functions and partial symmetries.

6. Parametric, Logical, and Geometric Settings

Parametric Presburger arithmetic provides a logical-combinatorial framework for families of sets and functions with quasi-polynomial behavior (Bogart et al., 2016). The Woods conjecture, confirmed in (Bogart et al., 2016), asserts that for sets $S_t$ defined by parametric Presburger formulas,

• The counting function $|S_t|$,
• The coordinates of extremal elements,
• The generating functions are eventually quasi-polynomial in $t$. This has direct implications for combinatorial diameter formulas in parametric families—e.g., counting diameter segments in dilated polytopes or evaluating metric invariants in integer programming.

In real or complex geometry, quasi-polynomial diameter formulas manifest via Chebyshev constants and transfinite diameter invariants. For instance, for compact subsets $K$ and generalized polynomial degree via convex body $C$, (Ma`u, 2019) gives

$$\delta^w_C(K)^{A_n} = \exp\left( \frac{1}{\mathrm{vol}_n(C)} \int_{\mathrm{int}(C)} \log T^w_C(K,\theta) d\theta \right),$$

where $A_n$ is a normalization constant, and $T^w_C(K,\theta)$ are directional Chebyshev constants. These objects encode interpolation, capacity, and approximation phenomena via quasi-polynomial scaling in degree and direction.

7. Applications in Discrete Partition Problems

Quasi-polynomial diameter formulas are instrumental in combinatorial partition problems. For discrete analogues of the Borsuk partition problem, (Brose et al., 27 Aug 2025) proves that any bounded set $S \subset \mathbb{Z}^d$ can be partitioned into at most $2^d$ subsets, each of strictly smaller lattice diameter: $\beta^{\mathbb{Z}}(S) \leq 2^d,$ and this bound is sharp for $S = \{0,1\}^d$. The proof exploits extremal properties of the lattice Borsuk graph, leveraging combinatorial dimension and coloring tools, and illustrates the concrete manifestation of quasi-polynomial support in discrete spaces.

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Quasi polynomial diameter formulas thus represent a broad, structurally significant class of diameter bounds and metric invariants with far-reaching implications across discrete geometry, combinatorics, optimization, algebraic theory, and computational mathematics. They provide quantitative control at the threshold between polynomial and exponential complexity and define a rich territory for algorithmic analysis, logical characterization, and combinatorial synthesis.