Multiway Karmarkar–Karp differencing
Algorithms AIMech:MultiwayKarmarkarKarpDifferencing DRAFT
A multiway number-partitioning heuristic that represents partial partitions by normalized bin-load vectors and repeatedly combines two selected subpartitions by pairing their largest and smallest bin loads, retaining one composition until a full partition remains.
Identity Notes & Concept Scope
Compared all 442 pinned AIO and 590 native IDs, names, aliases and scoped synonyms; no normalized identity match. Ignored- and hidden-inclusive fixed-string searches of the recovery, older discovery and original Documents checkouts returned 50, 0, 0 lines. All distinct match contents were inspected. Semantic matching covered 67 AIO and 97 native definitions, with every matched definition read. 6 full annotated AIO node/edge/generated contexts and 4 full native records were inspected. This record scopes the k>=3 vector-subpartition generalization of KK, with opposing-rank bin composition and normalization. The two-way scalar identity remains in KarmarkarKarpDifferencing. Complete CKK explores alternate compositions, while CGA assigns original items directly. Existing hits are source references and a pending combined multiway lead; no suitable AIO class denotes the procedure.
Claims & Grounded Evidence
6 Claims
Mechanism
#partial-bin-representation
For a fixed k>=3 and a finite nonempty multiset of nonnegative weights, initialize one k-bin subpartition per original item: its weight occupies one bin and the other bins are empty. Each subpartition represents disjoint original items already assigned to relative bins. Subtracting its minimum load from every coordinate preserves all within-subpartition load differences. Original memberships require bookkeeping beyond the normalized numbers.
Mechanism
#opposing-rank-composition
Select two subpartitions with the largest normalized maximum loads. Order the bins of one by decreasing load and the other by increasing load, merge corresponding bin memberships and add their loads, then subtract the new minimum and reinsert the result. This pairs large with small components within the two chosen subpartitions; it is not global placement of each next raw item in the lightest existing bin. Tie policy must be specified to reproduce a particular trace or membership output.
Scope
#normalization-and-reconstruction
A normalized zero denotes a minimum relative load, which may belong to a nonempty subset. Merge the stored original-index groups according to each selected bin permutation; their final sums recover the actual loads, differing from the normalized vector by a common offset. In the source example 8,7,6,5,4 with three bins, successive combined normalized vectors are (8,7,0), (2,1,0), (5,4,0), then (3,3,0). The final actual loads are 11,11,8, so the normalized zero is not an empty bin.
Limitation
#heuristic-not-complete
Only one opposing-rank composition is retained at each merge. This yields a feasible partition but is not guaranteed to minimize range. In a constructed run with declared deterministic tie handling on weights 2,2,2,3,3,5, the heuristic gives loads (7,5,5), range 2; exhaustive assignment admits (6,6,5), range 1. The source’s complete multiway CKK instead explores alternate relative bin permutations. This counterexample is a curator calculation, not a reported author experiment.
Limitation
#range-objective-boundary
The normalized maximum equals the final load range, not the absolute largest bin load. Range minimization and makespan minimization can select different solutions for k>=3. On weights 4,4,4,6,6,9, the constructed heuristic returns loads (13,10,10) with globally minimum range 3, while a maximum load of 12 is possible with (12,12,9). The source’s opening claim that minimizing range also minimizes the largest load is therefore not adopted as a guarantee for the returned partition.
Scope
#variant-and-evaluation-scope
This vector-subpartition record covers k>=3; the two-way scalar heuristic has its own maintained identity. The one-composition rule differs from complete multiway CKK and from randomized cyclic-differencing and compaction schemes in the original report. Korf’s three-way search experiments concern complete algorithms, so their node counts and runtime comparisons are not standalone validation of this heuristic across distributions, bin counts or precisions.
⚠️ Curated Limitations & Negative Results
- The scope is unconstrained range partitioning into k>=3 interchangeable bins. Equal cardinality, nonempty-bin requirements and bin-dependent restrictions are not imposed.
- Tie choices can affect the constructed trace or memberships. The curator calculation declares its policy and does not assert identity with an uninspected author implementation.
- Normalization preserves differences but discards an absolute common offset unless separate membership or offset data are retained. A zero component need not denote an unused bin.
- A heuristic result is not an optimality certificate, and even a range optimum need not be a minimum-makespan partition.
- All 31 pages of the Korf manuscript and all 30 pages of the original report were read; constructed exact-integer examples do not establish typical performance or replace complete search.
Artificial Intelligence Ontology (AIO) Grounding
Explicit AIO Snapshot Assessment: No Match
Rationale: No inspected AIO class denotes normalized arithmetic subpartition composition. A shared graph, heuristic or partitioning word does not identify a learned model, learning task or function entity.
Search Scope: Compared all 442 pinned AIO and 590 native IDs, names, aliases and scoped synonyms; no normalized identity match. Ignored- and hidden-inclusive fixed-string searches of the recovery, older discovery and original Documents checkouts returned 50, 0, 0 lines. All distinct match contents were inspected. Semantic matching covered 67 AIO and 97 native definitions, with every matched definition read. 6 full annotated AIO node/edge/generated contexts and 4 full native records were inspected. This record scopes the k>=3 vector-subpartition generalization of KK, with opposing-rank bin composition and normalization. The two-way scalar identity remains in KarmarkarKarpDifferencing. Complete CKK explores alternate compositions, while CGA assigns original items directly. Existing hits are source references and a pending combined multiway lead; no suitable AIO class denotes the procedure.
| Examined AIO Candidate | Rejection Reason |
|---|---|
AIO:GraphLanguageModel
|
Graph-structured language modeling is distinct from arithmetic subpartition manipulation. |
AIO:ReasoningLLM
|
A language model with explicit reasoning is not a number-partitioning algorithm. |
AIO:NeuroSymbolicLLM
|
A neural-symbolic language-model architecture is not this nonlearning search or heuristic. |
AIO:MachineLearningTask
|
Balancing a given multiset does not learn a method from data. |
AIO:MathematicalFunction
|
An algorithm evaluating differences and choosing a partition is distinct from the mathematical functions it uses. |
AIO:ReinforcementLearning
|
There is no reward-based learned policy or exploration/exploitation learning in the curated deterministic procedure. |
Evidence Sources & Primary Literature
2 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
korf1997
|
Richard E. Korf
|
primary paper
full_text
|
ID: https://doi.org/10.1016/S0004-3702(98)00086-1
sha256:00751dfacfb6…
|
karmarkar-karp-report
|
Narendra Karmarkar, Richard M. Karp
|
primary paper
full_text
|
ID: UCB/CSD-83-113
sha256:f6ca1d4198d4…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-13T08:11:33Z
Updated: 2026-10-13T08:11:33Z
Source File:
data/curated/MultiwayKarmarkarKarpDifferencing.yaml