Matching & Coloring Operator Set

Operator Category: Matching & Coloring (graph matching, edge cover, vertex coloring, and chromatic polynomial)

Number of Algorithms: 10

Applicable Stages: One‑to‑one pairing, optimal matching, perfect matching in bipartite graphs, fast matching, edge cover, matching validation, conflict‑free grouping, balanced grouping, graph coloring complexity analysis, combinatorial structure analysis

Product Positioning: Provides a capability base for matching, covering, and coloring operators to answer “how to pair without conflicts / how to pair optimally / how to achieve a complete matching in a bipartite graph / how to cover all vertices / how to group under conflict constraints / how to analyse the number of coloring schemes”.

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I. Operator Set Overview

The Matching & Coloring operator set focuses on pairing optimisation, covering relationships, conflict‑free grouping, and coloring structure analysis on graphs, addressing the following key questions:

1. Matching and Pairing Optimisation

   • How to select a set of non‑conflicting edges from a graph?
   • How to maximise the total weight of a matching?
   • How to minimise the total weight of a matching?
   • How to quickly generate a feasible matching?

2. Bipartite Matching

   • How to achieve a low‑cost perfect matching in a bipartite graph?
   • How to handle pairings between supply and demand, jobs and candidates, tasks and resources?
   • How to find matchings quickly in bipartite graphs?

3. Edge Cover

   • How to cover all vertices with as few edges as possible?
   • How to ensure that every vertex participates in at least one connection?

4. Conflict‑Free Grouping and Vertex Coloring

   • How to assign groups (colours) to vertices so that adjacent vertices receive different colours?
   • How to generate feasible groupings for exam scheduling, frequency assignment, or resource conflicts?
   • How to keep group sizes balanced while satisfying conflict constraints?

5. Coloring Structure Analysis

   • How many proper colourings exist for a given number of colours?
   • How can the complexity of conflicts be expressed as a polynomial?

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II. Operator Capability Classification

Capability Type	Corresponding Operator	Description
Maximum weight matching	max_weight_matching	Maximise total edge weight under one‑to‑one constraints
Minimum weight matching	min_weight_matching	Minimise total edge weight under one‑to‑one constraints
Maximal matching	maximal_matching	Quickly generate a matching that cannot be extended further
Minimum weight full matching	minimum_weight_full_matching	Find a minimum‑weight perfect matching covering one partition of a bipartite graph
Fast bipartite matching	eppstein_matching	Compute a matching in a bipartite graph using Eppstein‑style algorithms
Minimum edge cover	min_edge_cover	Cover all vertices with as few edges as possible
Matching validation	is_matching	Check whether a given edge set satisfies matching constraints
Greedy vertex coloring	greedy_color	Assign colours to vertices such that adjacent vertices have different colours
Equitable vertex coloring	equitable_color	Produce a colouring with given number of colours and balanced colour class sizes
Chromatic polynomial	chromatic_polynomial	Compute the chromatic polynomial of the graph

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III. General Input/Output Conventions

• Input G: NetworkX Graph / DiGraph

  • Matching algorithms typically work on undirected graphs
  • Bipartite matching algorithms usually require a bipartite graph
  • Edge cover algorithms are typically for undirected graphs
  • Coloring algorithms are based on vertex adjacency conflict

• Common Input Parameters

  • weight: Edge weight attribute name
  • maxcardinality: Whether to prioritise maximising the number of matching edges
  • matching: Edge set or mapping to validate
  • top_nodes: Node set specifying one partition of a bipartite graph
  • strategy: Vertex ordering strategy for greedy coloring
  • interchange: Whether to enable colour interchange optimisation
  • num_colors: Number of colours to use for equitable coloring

• Common Output Types

  • Matching: set[(u, v), ...] or dict[node → node]
  • Validation: bool
  • Edge cover: set[(u, v), ...]
  • Coloring: dict[node → color_id]
  • Polynomial: symbolic polynomial expression

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IV. Detailed Operator Descriptions

1. max_weight_matching – Maximum Weight Matching

Description
Finds a matching (set of pairwise non‑adjacent edges) that maximises the total sum of edge weights in an undirected weighted graph.

Matching constraints:

• Each vertex can appear in at most one matching edge
• No two matching edges share a vertex

Product Value

• Supports one‑to‑one optimal pairing
• Maximises value based on profit, preference, similarity, transaction amount, etc.
• Suitable for scenarios requiring “maximise pairing quality”

Typical Scenarios

• Job‑candidate matching
• Supply‑demand pairing
• Order‑capacity matching
• Social / collaboration partner pairing
• High‑value one‑to‑one connections in recommendation systems

Key Parameters

• weight: Edge weight attribute, defaults to weight
• maxcardinality:
  • False: Prioritise maximising total weight
  • True: Prioritise maximising number of matching edges, then maximise weight

Applicability & Characteristics

• Graph type: Undirected
• Output: Set of matching edges
• Complexity: Typically O(V^3)
• Suitable for small‑ to medium‑scale high‑value pairing problems

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2. min_weight_matching – Minimum Weight Matching

Description
Finds a matching that minimises the total sum of edge weights in an undirected weighted graph.

Edge weights often represent:

• Cost
• Distance
• Loss
• Dissimilarity
• Risk penalty

Product Value

• Minimises cost under one‑to‑one constraints
• Suitable for resource scheduling, logistics assignment, task allocation
• Converts “minimum cost pairing” into a graph matching problem

Typical Scenarios

• Logistics resource pairing
• Task‑assignee allocation
• Supply chain cost optimisation
• Pairing similar objects
• Risk or loss minimisation matching

Key Parameters

• weight: Edge weight attribute, defaults to weight
• Missing weights may be treated as default values, affecting interpretation

Applicability & Characteristics

• Graph type: Undirected
• Output: Set of matching edges
• Complexity: Typically O(V^3)
• Suitable for cost‑oriented one‑to‑one pairing

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3. maximal_matching – Maximal Matching

Description
Produces a maximal matching: a matching that cannot be extended by adding any other edge without violating the matching constraints.

Important notes:

• Maximal matching ≠ maximum matching
• Maximal matching only guarantees “no further extension possible”
• It does not guarantee the largest number of edges nor optimal weight

Product Value

• Quickly obtains a conflict‑free pairing
• Suitable when speed is more important than optimality
• Can serve as an initial solution for subsequent optimisation

Typical Scenarios

• Rapid task assignment
• Preliminary online matching results
• Fast feasible solution on large graphs
• Temporary scheduling or pairing
• Preprocessing for matching algorithms

Applicability & Characteristics

• Graph type: Undirected
• Output: Set of matching edges
• Complexity: Typically O(E)
• Features: Fast, but not globally optimal

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4. minimum_weight_full_matching – Minimum Weight Full Matching

Description
In a bipartite graph, finds a minimum‑weight matching that covers all vertices on one specified side (a perfect matching for that partition). This is often used for full assignment problems, e.g.:

• Every task is assigned a resource
• Every job is matched to a candidate
• Every order is assigned to a performer

Product Value

• Supports full low‑cost matching in bipartite graphs
• Suitable for assignment problems that require covering an entire partition
• Minimises cost, distance, or loss for full allocation

Typical Scenarios

• Task‑person assignment
• Order‑driver pairing
• Job‑candidate matching
• Supplier‑buyer allocation
• Minimum‑cost assignment from a resource pool to a task pool

Key Parameters

• G: Bipartite graph
• top_nodes: The set of nodes that must be fully matched (one partition)
• weight: Edge weight attribute (e.g., cost, distance, loss)

Applicability & Characteristics

• Graph type: Bipartite
• Output: Matching mapping or edge set
• Suitable for complete matching that must cover one side
• If no feasible perfect matching exists, may return no valid result

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5. eppstein_matching – Eppstein Bipartite Matching

Description
Computes a matching in a bipartite graph, often used for fast maximum matching algorithms.

It focuses on:

• Feasible pairings between the two partitions
• Building as many matching edges as possible without sharing vertices
• Providing a basic matching capability for covering, assignment, and bipartite structure analysis

Product Value

• Tailored for bipartite matching scenarios
• Performs well on large‑scale bipartite graphs
• Underpins resource allocation, relationship pairing, and edge cover

Typical Scenarios

• User‑item matching
• Worker‑job matching
• Task‑machine assignment
• Supply‑demand pairing
• Bipartite network structure analysis

Applicability & Characteristics

• Graph type: Bipartite
• Output: Matching mapping or edge set
• Suitable for maximum matching problems in bipartite graphs
• Often used as a base algorithm in bipartite matching toolchains

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6. min_edge_cover – Minimum Edge Cover

Description
Finds a set of edges such that every vertex in the graph is incident to at least one edge in the set, while minimising the number of edges used.

Difference from matching:

• Matching requires each vertex appear at most once
• Edge cover requires each vertex appear at least once
• Edge cover allows a vertex to be incident to multiple covering edges

Product Value

• Uses the fewest relationships to cover all objects
• Suitable for constructing a minimal representative connection subset
• Useful when “every vertex must be reached / associated / sampled”

Typical Scenarios

• Cover all accounts with the fewest transaction records
• Cover all users with the fewest relationships
• Build minimal participation structure for jobs/persons
• Minimum covering relationships for data sampling
• Object coverage analysis in bipartite graphs

Key Parameters

• matching_algorithm: Internal maximum matching algorithm can be specified
• Default implementation is usually based on a maximum matching

Applicability & Characteristics

• Graph type: Undirected, often used on bipartite graphs
• Output: Set of covering edges
• Complexity: Depends on internal matching algorithm
• For bipartite graphs, typical maximum matching complexity is O(E√V)

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7. is_matching – Matching Validation

Description
Checks whether a given set of edges (or mapping) is a valid matching in the graph.

A valid matching must satisfy:

• Every edge in the matching exists in the graph
• No two matching edges share a vertex
• Each vertex belongs to at most one matching edge

Product Value

• Provides pre‑ or post‑validation for matching results
• Prevents duplicate assignment of an object
• Suitable for validating manual input, external system output, or model‑generated pairings

Typical Scenarios

• Validate task allocation results
• Check matching system output
• Detect if anyone has been double‑paired
• Verify that paired edges actually exist
• Test matching algorithms and verify results

Input Notes

• matching can be an edge set or a node‑to‑node mapping
• If a mapping, it should be consistent bidirectionally, e.g., matching[u] == v and matching[v] == u

Applicability & Characteristics

• Graph type: Undirected
• Output: bool
• Complexity: Linear in matching size
• Often used together with matching generation algorithms

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8. greedy_color – Greedy Vertex Coloring

Description
Assigns a colour (integer) to each vertex so that adjacent vertices receive different colours.

Basic idea:

• Traverse vertices in a specified order
• For each vertex, assign the smallest positive integer not used by any already‑colored neighbour
• Return a mapping from vertex to colour

Product Value

• Converts conflict relationships into grouping solutions
• Supports scheduling, resource allocation, frequency assignment under constraints
• Fast, intuitive, and easy to productise

Typical Scenarios

• Exam scheduling
• Meeting / task conflict resolution
• Frequency assignment
• Compiler register allocation
• Mutually exclusive grouping of users or devices
• Isolated grouping of risk objects

Key Parameters

• strategy: Vertex ordering strategy for greedy coloring. Common choices:
  • largest_first
  • random_sequential
  • smallest_last
  • independent_set
  • connected_sequential_bfs
  • connected_sequential_dfs
  • saturation_largest_first (DSATUR)
• interchange: Whether to enable colour interchange optimisation to try to reduce the number of colours

Applicability & Characteristics

• Graph type: Undirected / Directed (treated as undirected conflicts)
• Output: dict[node → color_id]
• Complexity: Typically O(V + E), depends on strategy
• Greedy coloring does not guarantee the minimum number of colours but is efficient and stable

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9. equitable_color – Equitable Vertex Coloring

Description
Assigns colours to vertices using a given number of colours, ensuring adjacent vertices have different colours and the sizes of colour classes differ by at most one (balanced groups).

Equitable coloring focuses on:

• Conflict‑free assignment (adjacent vertices different)
• Balanced colour class sizes

Product Value

• Adds load balancing to conflict‑free grouping
• Suitable for scenarios requiring “fair distribution” or “balanced load”
• More appropriate than plain greedy coloring when resource capacities are similar

Typical Scenarios

• Balancing the number of examinees per session in exam scheduling
• Balancing load across task groups
• Balanced frequency / channel assignment
• User bucketing for experiments
• Batched processing of risk lists with balanced sizes

Key Parameters

• num_colors: Number of colours to use
• Sufficiently many colours must be available, otherwise the constraints may be unsatisfiable

Applicability & Characteristics

• Graph type: Typically undirected
• Output: dict[node → color_id]
• Features: Emphasises group size balance compared to greedy_color
• Suitable for scenarios with both conflict constraints and load balancing requirements

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10. chromatic_polynomial – Chromatic Polynomial

Description
Computes the chromatic polynomial of the graph. The chromatic polynomial P(G, k) counts the number of proper vertex colourings of G using at most k colours (adjacent vertices receive different colours).

Product Value

• Characterises the coloring complexity of a graph combinatorially
• Analyses the number of feasible colourings for different numbers of colours
• Suitable for theoretical analysis, structural evaluation, and feasibility studies of conflict networks

Typical Scenarios

• Analysing number of colouring schemes
• Evaluating the complexity of conflict graphs
• Feasibility analysis for scheduling / grouping
• Graph theory education and research
• Combinatorial structure analysis of small graphs

Applicability & Characteristics

• Graph type: Typically undirected
• Output: Symbolic polynomial expression
• Note: Computing the chromatic polynomial is generally more expensive than greedy coloring; suitable for small graphs or analytical tasks
• Difference from greedy_color:
  • greedy_color produces one concrete colouring
  • chromatic_polynomial gives the number of proper colourings for each k

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V. Recommended Usage Guide

1. One‑to‑One Pairing and Matching

• Maximise profit: max_weight_matching
• Minimise cost: min_weight_matching
• Quickly obtain a feasible pairing: maximal_matching
• Validate a pairing: is_matching

2. Bipartite Full Matching

• Minimum‑cost perfect matching (covering one side): minimum_weight_full_matching
• Fast bipartite matching: eppstein_matching
• Assign jobs / tasks / resources: prefer minimum_weight_full_matching

3. Covering All Vertices

• Minimum edge cover: min_edge_cover
• When every vertex must appear at least once: min_edge_cover
• Construct minimal edge subset covering all vertices: min_edge_cover

4. Conflict‑Free Grouping and Scheduling

• Adjacent vertices must be in different groups: greedy_color
• For stable, commonly used heuristics: largest_first
• To use fewer colours: try saturation_largest_first (DSATUR)
• To balance group sizes: equitable_color

5. Coloring Complexity Analysis

• Need a concrete grouping: greedy_color
• Need balanced grouping: equitable_color
• Need to count number of proper colourings: chromatic_polynomial

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VI. Typical Questions That Can Be Directly Answered

• “How to maximise total profit under one‑to‑one constraints?”
• “How to optimally match candidates to jobs?”
• “How to pair with minimum total cost?”
• “Give me a quick conflict‑free pairing.”
• “Is this pairing valid? Is anyone double‑paired?”
• “How to achieve a minimum‑cost full matching in a bipartite graph?”
• “How to quickly compute a matching in a bipartite graph?”
• “How to cover all vertices with the fewest edges?”
• “How to ensure every vertex participates in at least one connection?”
• “How to assign conflicting tasks to different groups?”
• “How to schedule exams so that conflicting students are not in the same session?”
• “How to keep group sizes balanced when grouping under conflicts?”
• “Which coloring strategy uses fewer colours on this graph?”
• “How many proper colourings does this conflict graph have for k colours?”
• “What is the chromatic polynomial of this graph?”

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VII. Operator List

No.	Operator Name	Description
1	max_weight_matching	Maximum weight matching
2	min_weight_matching	Minimum weight matching
3	maximal_matching	Maximal matching
4	minimum_weight_full_matching	Minimum weight full matching (in bipartite graphs)
5	eppstein_matching	Eppstein bipartite matching
6	min_edge_cover	Minimum edge cover
7	is_matching	Matching validation
8	greedy_color	Greedy vertex coloring
9	equitable_color	Equitable vertex coloring
10	chromatic_polynomial	Chromatic polynomial computation