Karmarkar–Karp differencing
Algorithms AIMech:KarmarkarKarpDifferencing DRAFT
Aliases: Karmarkar–Karp heuristic
A two-way number-partitioning heuristic that repeatedly replaces the two largest remaining nonnegative residuals by their difference, then reconstructs a partition whose absolute sum difference is the final residual.
Identity Notes & Concept Scope
Compared all 442 pinned AIO and 587 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 172, 120, 119 lines. All distinct match contents were inspected. Semantic matching covered 67 AIO and 94 native definitions, with every matched definition read. 6 full annotated AIO node/edge/generated contexts and 3 full native records were inspected. The repeated text matches are a pending combined KK/CKK frontier entry, not existing records. The identity here is the two-way largest-pair differencing heuristic, AlgorithmA in the original report and KK in Korf Section2.6. It is distinct from AlgorithmB, interval-based random differencing, complete Karmarkar–Karp search, and generic DFS or discrepancy ordering. Multiway variants remain explicit separate leads. The inspected AIO graph has no suitable counterpart; no forced related mapping is asserted.
Claims & Grounded Evidence
6 Claims
Mechanism
#largest-pair-reduction
For a finite nonempty multiset of nonnegative weights, repeatedly remove two largest residuals a>=b and insert a-b until one remains. The final value is the achieved absolute difference between the two subset sums. Residuals include combinations formed at earlier steps, so this is not simply assigning the next original weight to the currently lighter bin. Equal-valued input items retain distinct identities.
Mechanism
#relative-orientation-reconstruction
A residual represents a partial signed sum of original weights. Combining a and b by a-b preserves the orientation of one partial partition and reverses the other. Recording the combination tree permits signs to be propagated back to all original item indices; the positive and negative groups reconstruct a partition with the reported residue. A later reversal can flip an entire partial partition, so a differencing step does not irrevocably name each raw item’s final bin.
Scope
#heap-and-arithmetic-cost
A max-priority queue performs n-1 pair reductions with O(n log n) comparison/arithmetic operations. Retaining one binary combination node per reduction uses O(n) items, and a traversal reconstructs the partition in O(n) operations. These are unit-cost operation and item-count statements, not bounds independent of numeric bit length, precision or implementation overhead.
Limitation
#heuristic-not-certificate
The heuristic is not an exact optimizer. On weights8,7,6,5,4, its residual sequence is(6,5,4,1), (4,1,1), (3,1), then2, while subsets{8,7} and{6,5,4} have equal sum15 and residue0. Thus the achieved residue is an upper bound on the minimum residue, not a lower bound or proof of optimality. The procedure does not enforce equal subset cardinalities.
Scope
#distinct-randomized-theorem-algorithm
AlgorithmA is distinct from the original report’s adjacent-pair AlgorithmB and the later interval-based random differencing procedure with compaction. The report’s high-probability asymptotic analysis uses that randomized procedure and assumptions on independent input values with a Lipschitz density. Its opening linear-time and vanishing-residue claims are not adopted as unconditional guarantees for the bare largest-pair heuristic. The k-way cyclic-differencing construction is also outside this two-way record.
Scope
#reported-empirical-scope
Korf’s Figure2 compares average residues from KK and sorted greedy partitioning on sampled uniformly distributed ten-digit integers. The reported average advantage depends on that input distribution and precision; it does not establish that KK beats greedy on every instance. CKK’s later improvements arise from additional search, not an optimality guarantee of this heuristic. No historical runtime or stochastic experiment was replicated here.
⚠️ Curated Limitations & Negative Results
- Scope is two-way unconstrained partitioning of a finite nonempty nonnegative multiset. Multiway, cardinality-balanced, randomized-compaction and complete-search variants have distinct identities.
- The returned partition is feasible but can be suboptimal. Ties can change reconstructed item membership even when the residue is unchanged.
- Unit-cost complexity assumes exact comparisons and arithmetic; floating-point cancellation and very large numeric representations require separate implementation analysis.
- The Berkeley archive metadata and Korf’s1982 report citation differ. The inspected versions are identified explicitly, without inferring publication priority.
- All30 original-report pages and all31 Korf manuscript pages were read. Small constructed calculations were run; author implementations and the final1998 journal full text were not inspected.
Artificial Intelligence Ontology (AIO) Grounding
Explicit AIO Snapshot Assessment: No Match
Rationale: No inspected AIO class denotes this nonlearning arithmetic partitioning heuristic. Graph use, heuristic choice and evaluation of differences do not make it a language model, learned policy, learning task or mathematical-function entity.
Search Scope: Compared all 442 pinned AIO and 587 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 172, 120, 119 lines. All distinct match contents were inspected. Semantic matching covered 67 AIO and 94 native definitions, with every matched definition read. 6 full annotated AIO node/edge/generated contexts and 3 full native records were inspected. The repeated text matches are a pending combined KK/CKK frontier entry, not existing records. The identity here is the two-way largest-pair differencing heuristic, AlgorithmA in the original report and KK in Korf Section2.6. It is distinct from AlgorithmB, interval-based random differencing, complete Karmarkar–Karp search, and generic DFS or discrepancy ordering. Multiway variants remain explicit separate leads. The inspected AIO graph has no suitable counterpart; no forced related mapping is asserted.
| 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 |
|---|---|---|---|
karmarkar-karp-report
|
Narendra Karmarkar, Richard M. Karp
|
primary paper
full_text
|
ID: UCB/CSD-83-113
sha256:f6ca1d4198d4…
|
korf1997
|
Richard E. Korf
|
primary paper
full_text
|
ID: https://doi.org/10.1016/S0004-3702(98)00086-1
sha256:00751dfacfb6…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-13T07:51:19Z
Updated: 2026-10-13T07:51:19Z
Source File:
data/curated/KarmarkarKarpDifferencing.yaml