Overview
Branch-and-Bound Maximum Clique is a branch and bound method for maximum clique. It works by branching on candidate vertices while bounding clique size with coloring.
ExtendedSource checked
Branch-and-Bound Maximum Clique
Branch-and-Bound Maximum Clique addresses maximum clique by branching on candidate vertices while bounding clique size with coloring.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Optimization
- Difficulty
- Intermediate
- Family
- Branch and Bound
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 3
- Formulas
- 0
- Relationships
- 4
- Code example
- Ready
- Visualization
- Ready
Core Concepts
maximum cliquebranch and boundcolor refinementoptimization
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n)
Average
O(2^n)
Worst
O(2^n)
Space
O(2^n)
When to Use
- Use Branch-and-Bound Maximum Clique when solving maximum clique and you want to benefit from branching on candidate vertices while bounding clique size with coloring.
- Choose it when O(2^n) average/worst-case time and O(2^n) memory match your workload profile.
- Prefer it over Tomita Clique Algorithm or Bron-Kerbosch Algorithm when the problem structure favors maximum clique.
When Not to Use
- Avoid when exact optimality is required but the method is heuristic or approximate.
- Avoid when constraints are poorly defined or cheap feasibility checks are unavailable.
What It Does
- Computes maximum clique within Branch and Bound workloads (O(2^n) time, O(2^n) space).
- Works by branching on candidate vertices while bounding clique size with coloring.
- Maintains key invariants across maximum clique, branch and bound, color refinement.
Use Cases
- Production and research workloads requiring maximum clique via branching on candidate vertices while bounding clique size with coloring.
- Branch and Bound pipelines where O(2^n) time and O(2^n) space fit system constraints.
- Engineering comparisons against Tomita Clique Algorithm and Bron-Kerbosch Algorithm when tuning maximum clique.
Inputs
- Objective, constraints, and candidate variables
- Problem-specific costs or feasibility rules
Outputs
- Best known feasible solution
- Optimality or approximation result when available
Pseudocode
- 1.Formulate the maximum clique instance, objective function, and initial candidate state or active search tree.
- 2.Evaluate candidate feasibility and objective bounds (maximum clique, branch and bound).
- 3.Advance the search by branching on candidate vertices while bounding clique size with coloring.
- 4.Prune suboptimal branches, propagate unit/conflict clauses, or update incumbent best solutions.
- 5.Return the optimal or best-found feasible assignment once convergence or exhaustion is reached.
Canonical Example
- Input
- Objective, constraints, and candidate variables configured for a representative maximum clique test case
- Output
- Best known feasible solution satisfying Branch-and-Bound Maximum Clique invariants
- Explanation
- Branch-and-Bound Maximum Clique processes the input by branching on candidate vertices while bounding clique size with coloring, tracking maximum clique and branch and bound to produce the verified maximum clique result.
Implementation Notes
- State the objective and feasibility checks in code before adding pruning.
- Keep incumbent solutions separate from bounds.
- Add tests for infeasible and exactly-tied cases.
Common Pitfalls
- Assuming a heuristic is exact without checking its guarantee.
- Using weak bounds that make search behave like brute force.
- Forgetting infeasible, empty, or degenerate constraint cases.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow branching on candidate vertices while bounding clique size with coloring on a small maximum clique example.
Step 1 of 5: Formulate the maximum clique instance, objective function, and initial candidate state or active search tree.
optimization-mechanismNotes
Branch-and-Bound Maximum Clique operates in O(2^n) time and O(2^n) space within the Branch and Bound family. Often compared with Tomita Clique Algorithm and Bron-Kerbosch Algorithm.
Sources (3)
What each reference supports for this entry.
Handbook of Combinatorial Optimization
BookCombinatorial optimization reference for Branch-and-Bound Maximum Clique's exact, heuristic, or branch-and-bound search model.
The Algorithm Design Manual
BookAlgorithm catalog support for Branch-and-Bound Maximum Clique's problem context, known alternatives, and practical selection tradeoffs.
Open sourceMaximum Clique Branch and Bound Survey
PaperReference for branch-and-bound maximum clique search and coloring-based pruning strategies.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Alternative toBron-Kerbosch Algorithm
Graph Traversal · Core
Branch-and-bound maximum clique and Bron-Kerbosch both explore clique search spaces with different pruning goals.
Compare with thisRelationships
View full relationship pageUsesBranch and Bound
Maximum clique search uses branch-and-bound pruning to discard candidate sets that cannot improve the incumbent clique.
Alternative toBron-Kerbosch Algorithm
Branch-and-bound maximum clique and Bron-Kerbosch both explore clique search spaces with different pruning goals.
Conceptual neighborTomita Clique Algorithm
Tomita's maximum clique algorithm is a refined branch-and-bound clique search using coloring bounds.
Related Algorithms
Tags
Optimization