Matroid-Constrained Number Partitioning¶

A multiway number-partitioning problem in which every assigned subset must be independent in a corresponding matroid, while a declared aggregate objective balances or optimizes the subsets' weights.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10618
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Combinatorial Optimization → Mathematics
Aliases: Optimal matroid partitioning, Weighted matroid partitioning

Core Idea¶

This problem combines two structures: every item must belong to exactly one part, and each part must be independent in its associated matroid. The matroids can encode cardinality, category, or richer hereditary restrictions on which items coexist.

Weights then define what makes one feasible partition better than another. Because max, min, and sum can be applied inside each subset and again across subsets, the name covers a family whose scheduling meaning, complexity, and algorithms depend on the exact objective.

Structural Signature¶

Sig role-phrases:

Ground set — Supplies items that must be assigned. It is input set. Counterfactual: Omitted or duplicated items violate partitioning.
Target subsets — Receive a disjoint exhaustive allocation. It is partition slots. Counterfactual: Overlapping selected sets are not a partition.
Matroid family — Defines independence separately for each subset. It is feasibility system. Counterfactual: Arbitrary constraints need not preserve matroid structure.
Weight functions — Assign numerical contributions, possibly slot-specific. It is objective data. Counterfactual: Unweighted feasibility is a different problem.
Aggregation operators — Combine weights within and across subsets. It is objective form. Counterfactual: Changing max to sum changes the optimization problem.
Optimization direction — Selects minimize or maximize among feasible partitions. It is decision goal. Counterfactual: A feasible assignment alone may not be optimal.

What It Is Not¶

A single maximum-weight independent set is not a partition problem.
Ordinary number partitioning is only the free-matroid special case.
Overlapping independent subsets do not satisfy disjoint assignment.
Matroid-constrained feasibility alone does not specify which weighted objective is being optimized.
Closest near-miss. Matroid partition asks whether a ground set can be covered or partitioned by independent sets; matroid-constrained number partitioning adds weighted balancing or optimization across those sets.

Scope of Application¶

Constrained scheduling. Balances jobs subject to independence restrictions.
Resource allocation. Assigns items across eligibility systems.
Fair division. Maximizes minimum bundle value under matroid feasibility.
Algorithm design. Studies approximation and reductions across objective variants.

Clarity¶

State the ground set, number and representation of matroids, whether weights vary by subset, partition coverage, inner and outer aggregation operators, and minimize/maximize direction. Cite results only for that complete variant.

Manages Complexity¶

The abstraction layers a global allocation requirement over several local hereditary systems, then adds a nested numerical objective. Small changes in matroid class or aggregation can move a case from a familiar polynomial problem to a difficult approximation setting.

Abstract Reasoning¶

Define the common ground set and number of subsets.
Specify each matroid by an independence oracle or explicit representation.
Give slot-specific item weights where applicable.
Write the inner and outer aggregation operators and optimization direction.
Verify exhaustive disjoint feasibility before evaluating algorithms, approximations, or complexity.

Knowledge Transfer¶

Results transfer only when both matroid type and nested objective are preserved. A free matroid may recover ordinary scheduling, but conclusions for max-of-sums need not apply to sum-of-max or max-min variants.

Examples¶

Canonical¶

Jobs are assigned to machines; each machine's job set must satisfy its matroid and the maximum total processing weight across machines is minimized.

Mapped back: items → jobs; subsets → machines; constraint → machine-specific independence; objective → minimize maximum sum.

Applied / In Practice¶

Choosing one maximum-weight independent set from a matroid selects rather than partitions the entire ground set and is a different optimization problem.

Mapped back: subsets → one; coverage → partial; partition → absent; verdict → different problem.

Structural Tensions¶

T1 — Feasibility Structure versus Load Balance. Matroid independence controls allowable combinations while the weight objective may favor a different allocation.

Diagnostic: Which constraint blocks the apparently best numerical balance?

T2 — General Objective Family versus Specific Complexity Result. Different nested operators and matroid classes yield different algorithms and hardness.

Diagnostic: Is the theorem attached to the exact objective and special case stated?

Structural–Framed Character¶

Matroid-Constrained Number Partitioning is structural as exhaustive assignment under per-part matroid independence and framed by combinatorial optimization. Its numeric identity is completed only by the declared nested objective.

Structural Core vs. Domain Accent¶

The general core is constrained multi-bin allocation. Matroid theory supplies hereditary feasibility and exchange structure; partitioning supplies exhaustive disjointness; scheduling and fairness interpretations arise from the selected weight aggregations.

This entry is a kind of Partition problem.

Approved unparented root. No reviewed parent entails the conjunction of matroid-feasible parts and weighted multiway partitioning.
Related — matroid partition, intersection, and scheduling. Each captures a slice of feasibility or objective structure, but not the complete family.

Relationships to Other Abstractions¶

Current abstraction Matroid-Constrained Number Partitioning Domain-specific

Parents (1) — more general patterns this builds on

Matroid-Constrained Number Partitioning is a kind of Partition problem Domain-specific

Matroid-Constrained Number Partitioning is a strict kind of Partition problem: it partitions numbers among subsets while adding matroid-independence constraints.

Hierarchy path (1) — routes to 1 parentless root

Matroid-Constrained Number Partitioning → Partition problem

Neighborhood in Abstraction Space¶

Matroid-Constrained Number Partitioning sits in a crowded region of the domain-specific corpus (27^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Combinatorial Optimization & Game Problems (12 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Multiway number partitioning. Tell: Usually has no matroid feasibility constraints.
Matroid partition. Tell: Focuses on covering by independent sets and may lack a weight-balancing objective.
Matroid intersection. Tell: Selects a set independent in multiple matroids rather than assigning all items among subsets.
Generalized assignment. Tell: Uses eligibility and capacities that need not form matroids.

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Matroid-constrained_number_partitioning (revision 1364598679).
Preserved source candidate: https://link-springer-com.mgs.ariel.ac.il/article/10.1007/s00453-021-00797-9
Preserved source candidate: https://doi.org/10.1631/jzus.A071606
Preserved source candidate: https://doi.org/10.1145/2487575.2487636

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.