Optimality Number: Unifying Invariants

• Optimality number is defined as a minimal certificate size or invariant that guarantees optimal performance across different domains, such as convex minimization and caching.
• It captures exact optimum values in contexts like coded caching, where the exact worst-case load is achieved under specified constraints.
• The concept also encompasses counts of minimal resources in graph pebbling, pegging, and sequential optimization, linking combinatorial structures with optimality.

Searching arXiv for relevant papers using the phrase and nearby formal notions. I’m sorry, but I don’t have access to the arXiv search tool in this environment. I’ll proceed using the arXiv papers provided in the source material and cite them directly. “Optimality number” is not a single standardized invariant across the arXiv literature. Instead, it appears as a family of domain-specific quantities that encode the size of a minimal certificate, the value of an exact optimum, or the least amount of combinatorial resource needed for universal reachability. In convex minimization over nonconvex sets, the relevant quantity is the worst-case size of a strong optimality certificate and is bounded by a Helly number (Basu et al., 2016). In coded caching, the corresponding object is the exact optimal worst-case load under uncoded placement (Wan et al., 2015). In enumerative combinatorics, $C$-sequential optimization numbers $O_C(n,m)$ count configurations according to the size of an optimization set (Hui, 2023). In graph pebbling and pegging, the analogous parameters are the optimal pebbling number $\pi_{\opt}(G)$, the optimal pebbling number $\pi^*(G)$, and the optimal-pegging number $p(G)$, each minimizing resources needed for reachability (Győri et al., 2018, Lewis et al., 2021, Levavi, 2011).

1. Terminological status and unifying schema

Several papers explicitly note that the phrase “optimality number” is not their formal term, even when their results naturally induce one. In the convex-minimization setting, the paper does not use the explicit term “optimality number,” but its results yield a natural quantity: the smallest uniform bound on the size of strong optimality certificates (Basu et al., 2016). In caching, “optimality” is not packaged as a single named parameter like “optimality number,” but the exact optimal load can be interpreted in that role (Wan et al., 2015). In $C$-sequential optimization theory, the phrase is again not formalized in that exact way, although the framework is explicitly built from optimization sets and their weights (Hui, 2023).

A plausible unifying interpretation is that an optimality number measures the minimal complexity needed to certify, attain, or enumerate optimal behavior. The object being minimized varies sharply by field: cardinality of supporting points, transmission load, size of a dominating set, or number of pebbles or pegs.

Domain	Formal quantity	Role of “optimality”
Convex minimization over $S \subseteq \mathbb{R}^n$	size of a strong optimality certificate	minimal certificate size
Uncoded coded-caching	$L^*(M)$	exact optimal worst-case load
$C$-sequential optimization	$O_C(n,m)$	count of instances with optimal weight $O_C(n,m)$0
Graph pebbling	$O_C(n,m)$1, $O_C(n,m)$2	minimal pebbles for solvability
Graph pegging	$O_C(n,m)$3	minimal pegs for full reachability

This diversity is important conceptually. A common misconception is that “optimality number” denotes a single cross-disciplinary notion. The literature instead uses the phrase, or a natural interpretation of it, for structurally different invariants.

2. Convex minimization: certificate size controlled by Helly numbers

For a set $O_C(n,m)$4, not necessarily convex, and a convex function $O_C(n,m)$5, the primal problem is

$O_C(n,m)$6

The paper studies how optimality can be certified by finitely many points and subgradients. Its central object is the strong optimality certificate: points $O_C(n,m)$7 with subgradients $O_C(n,m)$8 such that

$O_C(n,m)$9

is $\pi_{\opt}(G)$0-free and

$\pi_{\opt}(G)$1

This guarantees

$\pi_{\opt}(G)$2

so optimality is reduced to evaluating $\pi_{\opt}(G)$3 on finitely many certificate points (Basu et al., 2016).

The size $\pi_{\opt}(G)$4 of such a certificate is the quantity that naturally functions as an optimality number in this framework. The controlling combinatorial invariant is the Helly number $\pi_{\opt}(G)$5, defined as the largest integer $\pi_{\opt}(G)$6 such that there are convex sets $\pi_{\opt}(G)$7 with

$\pi_{\opt}(G)$8

Under the assumptions that $\pi_{\opt}(G)$9 for all $\pi^*(G)$0, that $\pi^*(G)$1 is finite, and that minima over $\pi^*(G)$2 are attained for every polyhedron $\pi^*(G)$3 with $\pi^*(G)$4, there exists a strong optimality certificate of size at most $\pi^*(G)$5 (Basu et al., 2016).

This yields a direct “optimality number” interpretation. For fixed $\pi^*(G)$6, if $\pi^*(G)$7 denotes the smallest integer such that every admissible problem admits a strong optimality certificate of size at most $\pi^*(G)$8, then

$\pi^*(G)$9

For mixed-integer sets

$p(G)$0

with $p(G)$1 closed and convex, the paper states

$p(G)$2

hence certificate size

$p(G)$3

For the pure integer case $p(G)$4, the bound becomes

$p(G)$5

The associated duality viewpoint is geometric: the certificate corresponds to a maximal $p(G)$6-free gradient polyhedron defined by subgradients at points in $p(G)$7, and the paper shows strong duality for a more restrictive dual family than earlier work had used (Basu et al., 2016).

3. Information theory: exact optimal load as an optimality number

In centralized coded caching with $p(G)$8 files and $p(G)$9 users, the paper considers uncoded cache placement, average per-user memory budget $C$0, worst-case demands, and the regime $C$1. In this setting, the central quantity is the optimal load

$C$2

The paper proves that, under uncoded placement and $C$3, the minimal worst-case load is achieved by making all files equal length, making all caches equal size, and using the Maddah-Ali–Niesen coded caching scheme (Wan et al., 2015).

At the MAN corner points $C$4, with integer $C$5, the achieved load is

$C$6

With memory-sharing between adjacent integer values of $C$7, this gives the exact rate–memory tradeoff under uncoded placement: $C$8 The paper explicitly states that this value is the exact optimal load even when unequal file lengths and unequal cache sizes are allowed subject only to global sum constraints (Wan et al., 2015).

Here “optimality number” is best understood not as a cardinality bound but as an exact optimum value. The proof converts a fixed uncoded placement and demand vector into an index-coding problem, applies an outer bound for every acyclic induced subgraph, and then aggregates the resulting inequalities over demand permutations and user orderings. Introducing

$C$9

the converse derives

$S \subseteq \mathbb{R}^n$0

and then eliminates the $S \subseteq \mathbb{R}^n$1 using the total file-size and cache-size constraints to obtain a lower bound that matches the MAN load (Wan et al., 2015).

A common misconception in this area is that optimality here means only approximate optimality, because the original MAN result was within a factor 12 of the cut-set bound. The paper resolves this in the uncoded-placement regime $S \subseteq \mathbb{R}^n$2: the approximation factor becomes 1. It also states the practical implication sharply: any improvement over MAN in this regime must come from coded placement rather than from a different delivery rule (Wan et al., 2015).

4. Enumerative combinatorics: $S \subseteq \mathbb{R}^n$3-sequential optimization numbers

The paper on $S \subseteq \mathbb{R}^n$4-sequential optimization numbers starts from an optimization-theoretic primitive. For a set $S \subseteq \mathbb{R}^n$5, a relation vector $S \subseteq \mathbb{R}^n$6, and $S \subseteq \mathbb{R}^n$7, a majorization set of $S \subseteq \mathbb{R}^n$8 by $S \subseteq \mathbb{R}^n$9 is a set $L^*(M)$0 such that for every $L^*(M)$1,

$L^*(M)$2

If $L^*(M)$3 is a majorization set of minimal cardinality, then $L^*(M)$4 is an optimization set of $L^*(M)$5 by $L^*(M)$6, denoted

$L^*(M)$7

and its cardinality is the weight $L^*(M)$8 (Hui, 2023).

The higher-dimensional construction fixes a $L^*(M)$9-tuple

$C$0

and forms coordinatewise optimization sets $C$1 from 2D projections $C$2. The final set $C$3 is then determined by the pattern prescribed by $C$4. The resulting set is a $C$5-sequential optimization set, denoted

$C$6

and $C$7 is the sequential optimization weight. The paper’s formal counting invariant is then:

$C$8

In this literature, the “optimality number” is therefore not a single optimum value but an enumerator indexed by the optimal weight $C$9. The paper identifies a classical special case: $O_C(n,m)$0 so the unsigned Stirling numbers of the first kind are exactly the $O_C(n,m)$1-sequential optimization numbers (Hui, 2023).

The structure is governed by explicit recurrences and generating polynomials. With $O_C(n,m)$2, $O_C(n,m)$3, and

$O_C(n,m)$4

the central recurrence is

$O_C(n,m)$5

The total mass identity is

$O_C(n,m)$6

and the symmetry relation is

$O_C(n,m)$7

The generating polynomial factorization is

$O_C(n,m)$8

with zeros at $O_C(n,m)$9 and $O_C(n,m)$00 for $O_C(n,m)$01 (Hui, 2023).

This framework extends beyond exact counting. The paper defines an upper bound $O_C(n,m)$02, proves

$O_C(n,m)$03

and derives exponential tail estimates for the probability mass function

$O_C(n,m)$04

This suggests an “optimality-number” interpretation in which $O_C(n,m)$05 organizes the distribution of optimal solution sizes rather than isolating a single extremal value (Hui, 2023).

5. Graph pebbling: optimal pebbling numbers on grids and binary trees

In graph pebbling, a pebbling move removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A distribution is solvable if every vertex is reachable. The optimal pebbling number

$O_C(n,m)$06

is therefore the minimal number of pebbles for which there exists a distribution making all vertices reachable (Győri et al., 2018). A closely related notation in the complete-binary-tree literature is

$O_C(n,m)$07

which serves the same best-case role (Lewis et al., 2021).

For the square grid $O_C(n,m)$08, the paper develops a lower-bound method based on the effect function, excess, cooperation, and cooperation excess. Its principal numerical result is

$O_C(n,m)$09

The method proceeds through the effect–excess inequality

$O_C(n,m)$10

a unit decomposition of the distribution, and the key cooperation bound

$O_C(n,m)$11

which together yield a strengthened lower bound on $O_C(n,m)$12 for vertex-transitive graphs (Győri et al., 2018).

The same framework gives a new proof of the classical exact values for paths and cycles: $O_C(n,m)$13 For complete binary trees, a different paper provides an exact formula using symmetric even configurations, a reduction operation compatible with the decomposition of a complete binary tree into two copies of the next smaller one, and an arithmetic sequence built from lists of 1s and 5s (Lewis et al., 2021).

Let $O_C(n,m)$14, let $O_C(n,m)$15 be the entries of the infinite list $O_C(n,m)$16, and define

$O_C(n,m)$17

Then

$O_C(n,m)$18

and the exact optimal pebbling number of the complete binary tree $O_C(n,m)$19 is

$O_C(n,m)$20

The same paper proves a precise connection to the Conolly-Fox sequence $O_C(n,m)$21: if $O_C(n,m)$22, then $O_C(n,m)$23 is the largest index satisfying $O_C(n,m)$24, so the optimal pebbling numbers are encoded through a meta-Fibonacci sequence (Lewis et al., 2021).

These two pebbling papers exemplify two distinct optimality-number regimes: density-type lower bounds for grids and exact recursive formulas for highly structured trees.

6. Graph pegging and structural non-monotonicity

The pegging game is a different reachability process on graphs. A pegging move $O_C(n,m)$25 removes pegs from adjacent occupied vertices $O_C(n,m)$26 and $O_C(n,m)$27, then places one peg on an empty neighbor $O_C(n,m)$28 of $O_C(n,m)$29. The optimal-pegging number is

$O_C(n,m)$30

the size of the smallest distribution from which every vertex is reachable (Levavi, 2011).

For complete binary trees $O_C(n,m)$31, the paper shows that the optimal-pegging number is linear in the size of the tree: $O_C(n,m)$32 By contrast, the pegging number $O_C(n,m)$33, which is the worst-case analogue over all distributions of a given size, satisfies

$O_C(n,m)$34

for every $O_C(n,m)$35, and

$O_C(n,m)$36

This sharply separates best-case and worst-case optimality numbers on the same family of graphs (Levavi, 2011).

For complete infinitary trees of height $O_C(n,m)$37, the optimal-pegging number is exactly Fibonacci: $O_C(n,m)$38 For caterpillar graphs $O_C(n,m)$39 of diameter $O_C(n,m)$40,

$O_C(n,m)$41

For lobster graphs $O_C(n,m)$42 of diameter $O_C(n,m)$43, the paper proves the upper bound

$O_C(n,m)$44

The analysis uses a weight function with

$O_C(n,m)$45

together with the reach lemma: if $O_C(n,m)$46, then $O_C(n,m)$47 (Levavi, 2011).

A notable structural point is that optimal-pegging number is not monotone under leaf deletion. The paper deduces that there exist infinitely many trees whose optimal-pegging numbers increase with the removal of a leaf. This directly contradicts an intuition borrowed from many classical static graph invariants and is one of the clearest warnings against treating all optimality numbers as monotone minimization parameters (Levavi, 2011).

7. Comparative interpretation and recurrent themes

Across these literatures, “optimality number” names or suggests three recurrent patterns. First, it can be a certificate-size invariant, as in convex minimization, where Helly numbers bound the size of strong optimality certificates. Second, it can be an exact optimal objective value, as in uncoded coded caching, where the rate–memory tradeoff itself is the quantity of interest. Third, it can be a resource-minimization parameter or an enumerator of optimal weights, as in graph pebbling, graph pegging, and $O_C(n,m)$48-sequential optimization numbers (Basu et al., 2016, Wan et al., 2015, Hui, 2023, Győri et al., 2018, Levavi, 2011, Lewis et al., 2021).

A plausible cross-domain implication is that these quantities are best understood through the structure of the admissible certificates or moves rather than through optimization in the abstract. In the convex setting, the decisive structure is the geometry of maximal $O_C(n,m)$49-free gradient polyhedra and the combinatorics of Helly numbers. In caching, it is the induced index-coding graph under uncoded placement. In $O_C(n,m)$50-sequential optimization, it is the interaction between coordinatewise optimization sets and the pattern vector $O_C(n,m)$51. In pebbling and pegging, it is the graph’s reachability geometry under lossy local moves.

Another common misconception is that exact optimality numbers are invariably algorithmically simple once formally defined. The papers point in the opposite direction. Convex minimization requires delicate geometric duality and local attainability hypotheses; caching exactness hinges on a converse via index coding; grid pebbling lower bounds require cooperation-excess machinery; and complete binary tree pebbling leads to a meta-Fibonacci connection rather than a simple closed recurrence in $O_C(n,m)$52 alone (Basu et al., 2016, Wan et al., 2015, Győri et al., 2018, Lewis et al., 2021).

In this sense, the encyclopedia-level meaning of “optimality number” is necessarily plural. It denotes a class of mathematically rigorous invariants whose shared purpose is to compress optimality into a single parameter, but whose formal definitions, proof techniques, and structural consequences are determined entirely by the ambient theory.