Bottleneck Poset Metrics

• Bottleneck poset metrics are order-sensitive distance functions defined by poset structures and ideals that capture dominant error, stability, and complexity features in coding theory and metric geometry.
• They enable code decompositions that isolate P-irreducible components, reducing decoding complexity by focusing on maximal displacement and bottleneck phenomena.
• Extensions of these metrics include path, tree, and interleaving distances that unify approaches in representation theory, persistent homology, and combinatorial metric analysis.

Bottleneck poset metrics arise in several adjacent literatures rather than from a single universally fixed definition. In coding theory, the term appears explicitly for poset metrics built from bottleneck posets and for evaluation codes that are MDS with respect to those metrics; in representation-theoretic stability, it appears in bottleneck distances on minimal projective resolutions over finite metric posets; and in order-theoretic metric geometry, related work develops “bottleneck-style” distances on posets through shortest paths, fences, and interleavings on thin categories (Can et al., 22 Sep 2025, Asashiba et al., 17 Feb 2026, Olave, 4 Jun 2026). Taken together, these constructions treat a partial order not merely as a combinatorial background but as the mechanism that determines which coordinates, relations, or homological generators dominate distance, complexity, or stability.

1. Poset metrics, ideals, and decoding bottlenecks

The basic coding-theoretic poset metric starts with a poset $P=([n],\preceq)$ on the coordinate set of $\mathbb{F}_q^n$. For $x\in\mathbb{F}_q^n$, one defines $\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$, the ideal generated by a subset $X\subseteq[n]$ as $\langle X\rangle$, and the $P$-weight

$$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$

The induced metric is

$d_P(x,y)=\omega_P(x-y).$

The anti-chain recovers the Hamming metric, while the chain yields the Niederreiter–Rosenbloom–Tsfasman metric; hierarchical posets interpolate between these extremes through a level decomposition in which all lower levels lie below all higher ones (Firer et al., 2014).

Within this framework, bottlenecks appear through code decomposition. A $P$-decomposition allows one first to replace a code $\mathbb{F}_q^n$0 by a $\mathbb{F}_q^n$1-isometric image and then split it into components supported on disjoint coordinate sets. A code is $\mathbb{F}_q^n$2-irreducible if no non-trivial $\mathbb{F}_q^n$3-decomposition exists, and a maximal $\mathbb{F}_q^n$4-decomposition is one whose components are all $\mathbb{F}_q^n$5-irreducible. The complexity of syndrome decoding for a decomposition with profile $\mathbb{F}_q^n$6 is

$\mathbb{F}_q^n$7

and a primary $\mathbb{F}_q^n$8-decomposition minimizes this quantity over all $\mathbb{F}_q^n$9-decompositions of the code. The resulting invariant $x\in\mathbb{F}_q^n$0 is the minimal decoding complexity achievable under the poset metric, and the hard parts of decoding are precisely the $x\in\mathbb{F}_q^n$1-irreducible components with large $x\in\mathbb{F}_q^n$2 (Firer et al., 2014).

The same paper makes the dependence on order structure explicit. If $x\in\mathbb{F}_q^n$3 in the poset lattice, then $x\in\mathbb{F}_q^n$4 for every code $x\in\mathbb{F}_q^n$5; hierarchical upper and lower neighbors $x\in\mathbb{F}_q^n$6 and $x\in\mathbb{F}_q^n$7 satisfy

$x\in\mathbb{F}_q^n$8

This means that less hierarchical structure tends to preserve larger, more entangled components, whereas chain-like structure can collapse them. The binary code generated by $x\in\mathbb{F}_q^n$9 is $\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$0-irreducible for the anti-chain, but under the chain order there is a $\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$1-isometry sending it to $\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$2, so the nontrivial part of the code becomes a one-coordinate component. This is a canonical decoding bottleneck phenomenon: the poset can either force global coupling or localize the error structure (Firer et al., 2014).

2. Shortest-path, fence, and maximum-common-structure metrics on posets

A different line of work develops bottleneck-style distances directly on posets. For a path-connected poset, a path is a finite sequence of cover moves, and the 1-climber distance $\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$3 is defined through iterated up-down operators on subsets of the poset and then shown to admit a path realization

$\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$4

where $\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$5 is a shortest-path length corrected by the number of alternations in direction. More generally, the $\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$6-climber metrics $\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$7 replace chain lengths by their ceilings after division by $\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$8, and in discrete fence-connected posets one has the limit

$\mathrm{supp}(x)=\{i\in[n]\mid x_i\neq 0\}$9

where $X\subseteq[n]$0 is the shortest-fence metric. These metrics are not induced by valuations, they exhibit Chebyshev behavior on products,

$X\subseteq[n]$1

and they define interleaving distances when the posets are viewed as thin categories. The paper does not call them bottleneck metrics, but it explicitly describes them as “bottleneck-style” because they minimize over global paths, fences, and iterated expansions rather than summing local costs (Olave, 4 Jun 2026).

These distances are also structurally rigid. For most discrete path-connected posets, the pair $X\subseteq[n]$2 characterizes the poset up to isomorphism, and up to duality in the modular case. The exceptional behavior of modular posets is encoded by the equalities $X\subseteq[n]$3 for all $X\subseteq[n]$4, so orientation becomes invisible. A plausible implication is that, in this setting, bottleneck behavior is governed as much by the alternation structure of comparable chains and fences as by raw path length (Olave, 4 Jun 2026).

Another comparison metric comes from the directed maximum common edge subgraph problem. Represent a finite poset by its transitively closed directed acyclic graph $X\subseteq[n]$5, and for two labeled posets define

$X\subseteq[n]$6

Here $X\subseteq[n]$7 is the maximum number of directed edges preserved by a label-respecting injection. This yields a genuine metric on weakly connected labeled posets, allows the underlying sets to differ, and can be computed by reduction to maximum clique through an extended line digraph, with specialized alternatives for transitively closed digraphs (Nerem et al., 2019). In this construction the bottleneck is the maximal common order structure itself.

3. Metric posets, Galois transport, and bottleneck stability of resolutions

For a finite metric poset $X\subseteq[n]$8, one may study not only points of $X\subseteq[n]$9 but also $\langle X\rangle$0-modules $\langle X\rangle$1. A Galois coupling of two such modules $\langle X\rangle$2 and $\langle X\rangle$3 consists of a finite apex poset $\langle X\rangle$4, two Galois insertions $\langle X\rangle$5, and a $\langle X\rangle$6-module $\langle X\rangle$7 whose pullbacks recover $\langle X\rangle$8 and $\langle X\rangle$9. The cost of the coupling is the maximal displacement in the metric poset,

$P$0

and the Galois transport distance is obtained by infimizing this cost over all couplings. This distance is an extended metric on isomorphism classes of $P$1-modules and generalizes the interleaving distance in the one-parameter and multiparameter settings (Asashiba et al., 17 Feb 2026).

The same paper defines a bottleneck distance on minimal projective resolutions. Indecomposable projective $P$2-modules are representables $P$3, and a minimal projective resolution $P$4 of $P$5 is compared to $P$6 by matching indecomposable projective summands degreewise. Contractible cones play the role of diagonal terms, exactly as unmatched bars are sent to the diagonal in ordinary bottleneck distance. The ground cost is the metric on $P$7: $P$8 After allowing padding by contractible cones, one obtains an extended pseudometric on minimal projective resolutions (Asashiba et al., 17 Feb 2026).

The main stability theorem states that the resolution-level bottleneck distance is controlled by module-level transport: $P$9 Passing to the interval poset $$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$0 and a kernel functor $$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$1, persistence diagrams are interpreted as minimal projective resolutions of kernel modules, yielding

$$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$2

In the one-parameter case this recovers classical bottleneck stability, while in the multiparameter case it produces stable signed diagrams arising from minimal projective resolutions. Via the relationship between minimal resolutions and Möbius inversion, the same inequality is also a stability theorem for Möbius homology (Asashiba et al., 17 Feb 2026).

4. Bottleneck posets in coding theory

The term “bottleneck metric” appears explicitly in coding theory through the posets $$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$3. Start from the NRT poset $$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$4, the disjoint union of $$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$5 chains of length $$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$6, and collapse all vertices at one fixed rank into a single vertex. The resulting bottleneck poset $$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$7 has

$$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$8

vertices and a Hasse diagram in which $$\omega_P(x)=\bigl|\langle \mathrm{supp}(x)\rangle\bigr|.$$9 chains merge into a single neck and then branch again. The corresponding metric lives on the matrix subspace

$d_P(x,y)=\omega_P(x-y).$0

and its weight is the poset weight induced by $d_P(x,y)=\omega_P(x-y).$1. Below the bottleneck row the metric agrees with the NRT metric; above it, the bottleneck contributes a fixed cost that counts the entire collapsed layer and everything below it (Can et al., 22 Sep 2025).

Within this setting, analogues of Reed–Solomon codes are constructed by hyperderivative evaluation with a constraint forcing the relevant hyperderivative row to be constant across evaluation points. For parameters satisfying $d_P(x,y)=\omega_P(x-y).$2, the resulting code

$d_P(x,y)=\omega_P(x-y).$3

has dimension $d_P(x,y)=\omega_P(x-y).$4 and minimum bottleneck distance $d_P(x,y)=\omega_P(x-y).$5. Since the ambient bottleneck length is $d_P(x,y)=\omega_P(x-y).$6, this meets the Singleton-type bound and yields MDS codes in the bottleneck metric. The construction extends to algebraic-geometry codes over global function fields, with MDS or near-MDS conclusions under the genus and divisor-count hypotheses stated in the paper (Can et al., 22 Sep 2025).

A broader ambient generalization is the weighted poset block metric. Here $d_P(x,y)=\omega_P(x-y).$7, each block $d_P(x,y)=\omega_P(x-y).$8 has size $d_P(x,y)=\omega_P(x-y).$9, and a base weight $P$0 on the alphabet defines local block weights

$P$1

The weighted poset block weight is

$P$2

which unifies the Hamming, Lee, poset, pomset, poset block, and pomset block metrics. The group of linear isometries is described as a semi-direct product, a Singleton-type bound is established, and when $P$3 is a chain, MDS codes are equivalent to perfect codes (Ma et al., 2023). This suggests that bottleneck posets sit naturally inside a wider hierarchy of weighted block-poset geometries.

5. Persistence diagrams, bottleneck profiles, and merge trees

Bottleneck geometry also appears for poset-indexed invariants of topological data. For a metric pair $P$4, one forms the space $P$5 of persistence diagrams modulo points on the distinguished closed set $P$6, equipped with bottleneck distance. The resulting space is a pseudometric space in general. It is a genuine metric iff $P$7 is discrete; it is complete iff the quotient $P$8 is complete; it is separable iff every annulus $P$9 is totally bounded; and, for proper $\mathbb{F}_q^n$00, it is geodesic under the criterion stated in the paper for $\mathbb{F}_q^n$01 (Che et al., 2022). These results describe the ambient metric geometry in which many poset-indexed persistence objects live.

A refinement of bottleneck distance is given by the bottleneck profile

$\mathbb{F}_q^n$02

which records, for each threshold $\mathbb{F}_q^n$03, the minimal number of matched points that must exceed $\mathbb{F}_q^n$04. It is monotone decreasing and satisfies

$\mathbb{F}_q^n$05

From this profile one defines discrete Prokhorov metrics

$\mathbb{F}_q^n$06

for admissible superadditive functions $\mathbb{F}_q^n$07. The case $\mathbb{F}_q^n$08 recovers bottleneck distance, the case $\mathbb{F}_q^n$09 gives the discrete Prokhorov metric, and the family $\mathbb{F}_q^n$10 satisfies stability and comparison bounds with respect to Wasserstein distances. A plausible implication is that admissible functions form a partial order of tolerated bottlenecks, with larger $\mathbb{F}_q^n$11 producing coarser metrics (Dłotko et al., 2021).

Merge trees supply a further intrinsic comparison. The barcode map from merge trees to degree-zero persistence diagrams is not injective, so $\mathbb{F}_q^n$12 and the interleaving distance $\mathbb{F}_q^n$13 do not coincide pointwise on merge trees. However, when both are replaced by their intrinsic path metrics, the equality

$\mathbb{F}_q^n$14

holds on merge tree space. The result is established by decomposing the space into finitely many closed regions on which $\mathbb{F}_q^n$15, and then using pruning operations to reduce arbitrary paths to bounded-leaf subspaces. This shows that, in this special case, bottleneck distance can recover interleaving distance after passage to infinitesimal path length, even though the barcode representation itself forgets merge structure (Beers et al., 2 Sep 2025).

6. Combinatorial and order-theoretic extremal metrics

Poset-induced bottleneck behavior also appears in permutation families defined by restriction graphs. A restriction graph $\mathbb{F}_q^n$16 is an oriented acyclic graph on $\mathbb{F}_q^n$17 whose valid permutations satisfy $\mathbb{F}_q^n$18 for every directed edge $\mathbb{F}_q^n$19. Reachability in $\mathbb{F}_q^n$20 induces a poset $\mathbb{F}_q^n$21, and the $\mathbb{F}_q^n$22-diameter of the permutation family is exactly

$\mathbb{F}_q^n$23

where $\mathbb{F}_q^n$24 and $\mathbb{F}_q^n$25 are the reachable sets below and above $\mathbb{F}_q^n$26. For the Kendall–Tau metric, the maximal possible value is bounded by the number of incomparable pairs, and equality holds if and only if the induced poset has dimension at most $\mathbb{F}_q^n$27. In that case, the extremal permutations form a minimal realizer of the poset (Tymoshenko et al., 8 Jul 2025). This is a precise instance in which a poset invariant controls a metric bottleneck.

Perfect-code theory provides another extremal view. For binary poset codes of codimension $\mathbb{F}_q^n$28, an $\mathbb{F}_q^n$29-perfect code exists only when the poset has exactly one ideal of size $\mathbb{F}_q^n$30; $\mathbb{F}_q^n$31-perfect codes exist only under a three-branch configuration over a common $\mathbb{F}_q^n$32-ideal; and $\mathbb{F}_q^n$33-perfect codes occur only for three specific small essential posets. The same paper derives nonexistence conditions for crown posets and unions of disjoint chains (0705.2807). This suggests that highly symmetric or multiply branching posets can themselves be perfect-code bottlenecks.

A third extremal problem is the packing radius. For a poset metric, the packing radius of a vector reduces to a partition problem on maximal elements of the induced support poset. The discordancy of a partition $\mathbb{F}_q^n$34 is

$\mathbb{F}_q^n$35

and the packing radius of a poset is

$\mathbb{F}_q^n$36

Because the classical number partition problem is contained as the special case of disjoint ideals, the general problem is NP-hard (D'Oliveira et al., 2013). In this formulation, overlap of generated ideals is literally the bottleneck term.

7. Canonical forms, tree approximations, and broad structural themes

Canonical forms for codes over poset metrics sharpen the structural picture. A $\mathbb{F}_q^n$37-canonical form is a generalized inverse row-echelon generator matrix on which no further reduction is possible via the poset isometry group. Such a form determines a maximal $\mathbb{F}_q^n$38-decomposition into $\mathbb{F}_q^n$39-irreducible components, and this decomposition is unique up to permutation of components. It yields bounds for the packing radius, identifies hierarchical upper and lower neighbors $\mathbb{F}_q^n$40 and $\mathbb{F}_q^n$41, and reduces syndrome decoding by decomposing the lookup table into smaller tables attached to the components or to hierarchical levels (Pinheiro et al., 2017). This framework supplies a canonical language for locating bottlenecks: they are the components whose supports remain inseparable under the admissible poset isometries.

Filtered posets lead to a different sort of bottleneck metric. For a finite poset $\mathbb{F}_q^n$42 with order-preserving $\mathbb{F}_q^n$43, one defines

$\mathbb{F}_q^n$44

where the minimum ranges over comparable-step paths $\mathbb{F}_q^n$45 from $\mathbb{F}_q^n$46 to $\mathbb{F}_q^n$47. The associated Reeb poset and Reeb tree produce a tree metric $\mathbb{F}_q^n$48, and the deviation between $\mathbb{F}_q^n$49 and $\mathbb{F}_q^n$50 satisfies

$\mathbb{F}_q^n$51

where $\mathbb{F}_q^n$52 is the maximum length of a fence in the poset and $\mathbb{F}_q^n$53 is the hyperbolicity of the Reeb poset. After embedding finite metric spaces into finite metric graphs and then into filtered posets, this improves Gromov’s tree-approximation bound by replacing $\mathbb{F}_q^n$54 with a term depending on the maximum fence length or, in the graph setting, the first Betti number (Mémoli et al., 2018). Here the bottleneck is the $\mathbb{F}_q^n$55-distortion between a metric and its tree approximation.

Across these strands, a common pattern emerges. Bottleneck poset metrics emphasize maximal displacement, maximal common substructure, maximal unmatched mass, or the smallest irreducible component that controls decoding or packing. Some constructions use the term explicitly, others only exhibit the same geometry. The recurring role of ideals, maximal elements, hierarchical levels, interleavings, and $\mathbb{F}_q^n$56-type distortions suggests that the subject is best understood not as a single metric but as a family of order-sensitive bottleneck formalisms connecting coding theory, representation theory, persistence, and combinatorial metric geometry.