Gale’s theorem on flows with convex costs
E612747
Gale’s theorem on flows with convex costs is a fundamental result in mathematical optimization and network flow theory that characterizes optimal flows in networks when the cost functions on edges are convex rather than linear.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Gale’s theorem on flows with convex costs canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6710765 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Gale’s theorem on flows with convex costs Context triple: [David Gale, notableWork, Gale’s theorem on flows with convex costs]
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A.
Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
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B.
Lipton–Tarjan separator theorem
The Lipton–Tarjan separator theorem is a fundamental result in graph theory that shows any planar graph can be efficiently divided into roughly equal parts by removing only a relatively small set of vertices, enabling faster algorithms for many computational problems.
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C.
Steiner tree problem
The Steiner tree problem is a classic optimization problem in combinatorial mathematics and computer science that seeks the shortest network of line segments connecting a given set of points, potentially adding extra intermediate points to minimize total length.
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D.
Monge problem in optimal transport
The Monge problem in optimal transport is a foundational mathematical formulation that seeks the most efficient way to move mass from one distribution to another, minimizing a given transportation cost.
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E.
Graph Algorithms (book)
"Graph Algorithms" is a foundational textbook by Shimon Even that systematically presents the theory, design, and analysis of algorithms for solving fundamental problems on graphs.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Gale’s theorem on flows with convex costs Target entity description: Gale’s theorem on flows with convex costs is a fundamental result in mathematical optimization and network flow theory that characterizes optimal flows in networks when the cost functions on edges are convex rather than linear.
-
A.
Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
-
B.
Lipton–Tarjan separator theorem
The Lipton–Tarjan separator theorem is a fundamental result in graph theory that shows any planar graph can be efficiently divided into roughly equal parts by removing only a relatively small set of vertices, enabling faster algorithms for many computational problems.
-
C.
Steiner tree problem
The Steiner tree problem is a classic optimization problem in combinatorial mathematics and computer science that seeks the shortest network of line segments connecting a given set of points, potentially adding extra intermediate points to minimize total length.
-
D.
Monge problem in optimal transport
The Monge problem in optimal transport is a foundational mathematical formulation that seeks the most efficient way to move mass from one distribution to another, minimizing a given transportation cost.
-
E.
Graph Algorithms (book)
"Graph Algorithms" is a foundational textbook by Shimon Even that systematically presents the theory, design, and analysis of algorithms for solving fundamental problems on graphs.
- F. None of above. chosen
Statements (36)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in mathematical optimization ⓘ result in network flow theory ⓘ |
| appliesTo | flows in networks ⓘ |
| assumes |
convex cost functions defined on edge flows
ⓘ
convex cost functions on edges ⓘ feasible flow satisfying flow conservation constraints ⓘ |
| characterizes | optimal flows in networks with convex costs ⓘ |
| clarifies | relationship between primal flow variables and dual node potentials ⓘ |
| concerns |
existence of optimal flows
ⓘ
structure of optimal flows ⓘ |
| context |
cost minimization subject to flow constraints
ⓘ
finite directed networks ⓘ |
| ensures | existence of optimal solutions under convexity and feasibility assumptions ⓘ |
| field |
mathematical optimization
ⓘ
network flow theory ⓘ operations research ⓘ |
| generalizes | minimum-cost flow theory with linear costs ⓘ |
| implies |
existence of dual variables associated with nodes or edges
ⓘ
optimality conditions for convex-cost flows ⓘ |
| influenced |
development of polynomial-time algorithms for convex-cost flows
ⓘ
subsequent work on convex network optimization ⓘ |
| namedAfter | David Gale ⓘ |
| provides | necessary and sufficient conditions for optimality of a flow under convex costs ⓘ |
| relatedTo |
Kuhn–Tucker optimality conditions
ⓘ
linked to:
Karush–Kuhn–Tucker conditions
Lagrangian duality ⓘ convex network flow problem ⓘ minimum-cost flow problem ⓘ potential function methods in networks ⓘ |
| usedIn |
design of algorithms for convex-cost network flows
ⓘ
economic equilibrium models on networks ⓘ telecommunication network design with congestion costs ⓘ transportation and logistics optimization with nonlinear costs ⓘ |
| uses |
convex analysis
ⓘ
duality theory ⓘ potential functions on nodes ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Gale’s theorem on flows with convex costs Description of subject: Gale’s theorem on flows with convex costs is a fundamental result in mathematical optimization and network flow theory that characterizes optimal flows in networks when the cost functions on edges are convex rather than linear.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.