Maximum matching¶
In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts.
Core Idea¶
Maximum matching is treated here as the recurring mathematics_logic_statistics identity summarized by this source-grounded definition: In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts.
In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts. Given a graph , a matching is a subgraph where no two edges share a vertex. The cardinality of the matching is the number of edges in the subgraph, and the maximum cardinality is the largest number of edges a matching can contain.
graph is a maximum-cardinality matching if its cardinality is this maximum cardinality. If we think of each edge as "covering" the vertices it connects exactly once,. then a maximal matching is also the largest non-overlapping cover of the graph.
For Maximum matching, the abstraction is narrower than the article's general subject matter: a positive case must preserve In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.
Structural Signature¶
Sig role-phrases:
- Defining carrier — The Ford–Fulkerson algorithm proceeds by repeatedly finding an augmenting path from some to some and updating the matching by taking the symmetric difference of that path with (assuming such a path exists).
- Constitutive relation — As each path can be found in time, the running time is , and the maximum matching consists of the edges of that carry flow from to .
- Operating condition — An improvement to this algorithm is given by the more elaborate Hopcroft–Karp algorithm, which searches for multiple augmenting paths simultaneously.
- Recognition evidence — The algorithm of Chandran and Hochbaum for bipartite graphs runs in time that depends on the size of the maximum matching , which for is.
- Admissible variation — For planar bipartite graphs, the problem can be solved in time where is the number of vertices, by reducing the problem to maximum flow with multiple sources and sinks.
- Characteristic consequence — The same bound was achieved by an algorithm by and an algorithm by Gabow and Tarjan.
- Failure boundary — Other algorithms for the task are reviewed by Duan and Pettie (see Table I).
What It Is Not¶
- Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts.
- Not an over-broad reading. For finite graphs, a maximum-cardinality matching always exists, but is not usually unique.
- Not an over-broad reading. The simplest way to compute a maximum-cardinality matching is to follow the Ford–Fulkerson algorithm.
- Not an over-broad reading. This algorithm solves the more general problem of computing the maximum flow.
- Not automatically Matching (graph theory). Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.
Scope of Application¶
Maximum matching applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:
- Algorithms for arbitrary graphs. This is better in theory for sufficiently dense graphs, but in practice the algorithm is slower.
- Algorithms for bipartite graphsFlow-based algorithm. The simplest way to compute a maximum-cardinality matching is to follow the Ford–Fulkerson algorithm.
- Algorithms for bipartite graphsFlow-based algorithm. This algorithm solves the more general problem of computing the maximum flow.
- Algorithms for bipartite graphsFlow-based algorithm. A bipartite graph can be converted to a flow network as follows.
- Algorithms for bipartite graphsFlow-based algorithm. Add a source vertex ; add an edge from to each vertex in .
- Algorithms for bipartite graphsFlow-based algorithm. Add a sink vertex ; add an edge from each vertex in to .
Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.
Clarity¶
A clear use of Maximum matching names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts. The strongest recognition evidence in the frozen account is: The algorithm of Chandran and Hochbaum for bipartite graphs runs in time that depends on the size of the maximum matching , which for is. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification For finite graphs, a maximum-cardinality matching always exists, but is not usually unique. so that a reader can reproduce the classification rather than infer it from topical resemblance.
Manages Complexity¶
Maximum matching compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—as each path can be found in time, the running time is , and the maximum matching consists of the edges of that carry flow from to .—and the practical consequence—the same bound was achieved by an algorithm by and an algorithm by Gabow and Tarjan. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.
Abstract Reasoning¶
- Type the carrier. Identify the mathematics_logic_statistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts.
- Check operation and conditions. An improvement to this algorithm is given by the more elaborate Hopcroft–Karp algorithm, which searches for multiple augmenting paths simultaneously.
- Demand recognition evidence. The algorithm of Chandran and Hochbaum for bipartite graphs runs in time that depends on the size of the maximum matching , which for is.
- Test variation. Change an implementation or setting while preserving for planar bipartite graphs, the problem can be solved in time where is the number of vertices, by reducing the problem to maximum flow with multiple sources and sinks.
- Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
- Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.
Knowledge Transfer¶
Within the home domain. Knowledge about Maximum matching transfers literally when a new case preserves the same carrier type, relation, and recognition test. This is better in theory for sufficiently dense graphs, but in practice the algorithm is slower. The simplest way to compute a maximum-cardinality matching is to follow the Ford–Fulkerson algorithm.
Beyond the home domain. No canonical parent is asserted for Maximum matching. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.
Examples¶
Canonical¶
In this case, the problem can be efficiently solved with simpler algorithms than in the general case. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.
Mapped back: carrier → the entities in the documented case; operation → In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts; recognition evidence → The algorithm of Chandran and Hochbaum for bipartite graphs runs in time that depends on the size of the maximum matching , which for is
Applied / In Practice¶
An important special case of the maximum-cardinality matching problem is when is a bipartite graph representing a binary relation, whose vertices are partitioned between left vertices in and right vertices in , and edges in always connect a left vertex to a right vertex. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.
Mapped back: changed setting → the applied context; invariant → In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts; boundary → the case exits the class when for finite graphs, a maximum-cardinality matching always exists, but is not usually unique
Structural Tensions¶
T1 — Stable identity versus admissible variation. For finite graphs, a maximum-cardinality matching always exists, but is not usually unique. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.
Diagnostic: Which changes preserve the defining relation, and which replace it?
T2 — Recognition versus proxy. The simplest way to compute a maximum-cardinality matching is to follow the Ford–Fulkerson algorithm. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.
Diagnostic: Does the cited evidence establish the identity or only a correlated sign?
T3 — Definition versus implementation. This algorithm solves the more general problem of computing the maximum flow. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.
Diagnostic: Is the observed implementation constitutive, optional, or merely common?
T4 — Scope versus overextension. A bipartite graph can be converted to a flow network as follows. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.
Diagnostic: Can every claimed application fill the same typed roles without metaphor?
T5 — Transfer versus domain accent. The Ford–Fulkerson algorithm proceeds by repeatedly finding an augmenting path from some to some and updating the matching by taking the symmetric difference of that path with (assuming such a path exists). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.
Diagnostic: Does the receiving case instantiate Maximum matching literally, co-instantiate Pattern, or only resemble it?
T6 — Autonomy versus reduction. As each path can be found in time, the running time is , and the maximum matching consists of the edges of that carry flow from to . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.
Diagnostic: What does Maximum matching distinguish that the broader parent Pattern leaves together?
Structural–Framed Character¶
Maximum matching is structural-leaning. Its structural side is the repeatable organization summarized by In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.
Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: An improvement to this algorithm is given by the more elaborate Hopcroft–Karp algorithm, which searches for multiple augmenting paths simultaneously. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.
Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.
Structural Core vs. Domain Accent¶
What is skeletal. In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The Ford–Fulkerson algorithm proceeds by repeatedly finding an augmenting path from some to some and updating the matching by taking the symmetric difference of that path with (assuming such a path exists). As each path can be found in time, the running time is , and the maximum matching consists of the edges of that carry flow from to . It further constrains recognition and variation through: An improvement to this algorithm is given by the more elaborate Hopcroft–Karp algorithm, which searches for multiple augmenting paths simultaneously. The algorithm of Chandran and Hochbaum for bipartite graphs runs in time that depends on the size of the maximum matching , which for is.
What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Maximum matching literal. Its documented scope includes the condition that This is better in theory for sufficiently dense graphs, but in practice the algorithm is slower. Another bounded application condition is that The simplest way to compute a maximum-cardinality matching is to follow the Ford–Fulkerson algorithm. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.
Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—For planar bipartite graphs, the problem can be solved in time where is the number of vertices, by reducing the problem to maximum flow with multiple sources and sinks.—and future graph densification may discover a defensible relation only if it preserves that boundary.
Instantiates / Related Primes¶
- Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Maximum matching. The reviewed identity is: In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
- Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.
Neighborhood in Abstraction Space¶
Maximum matching sits in a sparse region of the domain-specific corpus (77th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Convex Optimization & Iterative Methods (8 abstractions)
Nearest neighbors
- Convex bipartite graph — 0.89
- Matching — 0.83
- Factor-critical graph — 0.82
- A-star algorithm — 0.82
- Treewidth — 0.81
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
- Pattern. The parent omits the specialist differentia. Tell: Can the case establish In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts?
- Matching (graph theory). A set of graph edges with no shared endpoint. Tell: Which entry's carrier, operation, and failure condition are satisfied?
- 3-dimensional matching. A matching in a tripartite 3-uniform hypergraph: a set of triples no two of which share any coordinate, with optimization and perfect-decision variants. Tell: Which entry's carrier, operation, and failure condition are satisfied?
- Matching. Cast a pairing problem as a largest (or minimum-cost, or perfect) set of pairwise vertex-disjoint edges on an explicit graph, then split on bipartiteness to select the theorems and polynomial-time algorithm that solve it. Tell: Which entry's carrier, operation, and failure condition are satisfied?
- A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Maximum matching remain present if the detector or downstream effect changed?
- A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?
References¶
- Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Maximum-cardinality_matching (revision 1365861338).
- Preserved source candidate: https://web.eecs.umich.edu/~pettie/matching/Blum-matching-ICALP90.pdf
- Preserved source candidate: https://web.eecs.umich.edu/~pettie/matching/Gabow-Tarjan-scaling-general-graph-matching.pdf
- Preserved source candidate: https://www.mimuw.edu.pl/~mucha/pub/mucha_sankowski_focs04.pdf
- Preserved source candidate: https://web.eecs.umich.edu/~pettie/papers/ApproxMWM-JACM.pdf
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.