Closure problem¶
The optimization problem of selecting a maximum-weight vertex set in a directed graph subject to the rule that selecting a vertex also selects every vertex reachable along its outgoing dependencies.
Core Idea¶
A closure of a directed graph contains every successor of each selected vertex; the maximum-closure problem asks for a closure maximizing the sum of vertex weights. Positive and negative vertex weights are encoded as source and sink capacities, dependency arcs receive effectively infinite capacity, and a minimum cut identifies the optimal closed set. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Closure problem belongs to combinatorial optimization and is useful where the analyst can specify the typed combinatorial optimization carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the selected vertex set is successor-closed under every directed edge and has maximum total declared weight among all such sets. The scope is broad within that domain but bounded by the need for the selected vertex set is successor-closed under every directed edge and has maximum total declared weight among all such sets. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the selected vertex set is successor-closed under every directed edge and has maximum total declared weight among all such sets the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Closure problem can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Closure problem. Closure problem compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed combinatorial optimization carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the selected vertex set is successor-closed under every directed edge and has maximum total declared weight among all such sets independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of combinatorial optimization because they reuse the typed combinatorial optimization carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Positive and negative vertex weights are encoded as source and sink capacities, dependency arcs receive effectively infinite capacity, and a minimum cut identifies the optimal closed set., and type the carrier, state every parameter and convention in the definition, test that the selected vertex set is successor-closed under every directed edge and has maximum total declared weight among all such sets, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Closure problem Domain-specific
Parents (1) — more general patterns this builds on
-
Closure problem is a kind of Constraint Prime
The proposed strict upward parent is
prime:constraint.
Hierarchy path (1) — routes to 1 parentless root
- Closure problem → Constraint
Neighborhood in Abstraction Space¶
Closure problem sits in a crowded region of the domain-specific corpus (7th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Combinatorial Optimization & Network Flows (24 abstractions)
Nearest neighbors
- Quasi-bipartite graph — 0.94
- Dissociation number — 0.94
- Set TSP problem — 0.94
- 3-dimensional matching — 0.94
- Gomory–Hu tree — 0.93
Computed from structural-signature embeddings · 2026-09-08