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.
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.
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. Inclusion test: Require an exhaustive disjoint partition of a common item set, matroid-independence feasibility for each assigned subset, and an explicitly nested weight objective. Exclusion test: Exclude ordinary number partitioning with no matroid restrictions, matroid partition feasibility with no numeric objective, overlapping independent-set selection, and arbitrary bin constraints not representable as matroids. Nearest boundary: 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. Exit condition: The problem changes identity when items may be fractionally assigned, subsets overlap, feasibility is non-matroidal, or no weight-based partition objective remains. Common misclassifications: 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. Nearest named distinctions: Multiway number partitioning: Usually has no matroid feasibility constraints. Matroid partition: Focuses on covering by independent sets and may lack a weight-balancing objective. Matroid intersection: Selects a set independent in multiple matroids rather than assigning all items among subsets. Generalized assignment: Uses eligibility and capacities that need not form matroids.
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.
Relationships to Other Abstractions¶
Current abstraction Matroid-Constrained Number Partitioning Domain-specific
Parents (1) — more general patterns this builds on
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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 (27th 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
- Partition problem — 0.93
- Maximum subarray problem — 0.90
- Steiner system — 0.89
- ÉLECTRE — 0.89
- Quadratic knapsack problem — 0.88
Computed from structural-signature embeddings · 2026-10-08