Overview
Tomita Clique Algorithm is a branch and bound method for maximum clique. It works by using greedy coloring bounds to prune maximum-clique search.
SpecializedSource checked
Tomita Clique Algorithm
Tomita Clique Algorithm addresses maximum clique by using greedy coloring bounds to prune maximum-clique search.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Optimization
- Difficulty
- Advanced
- Family
- Branch and Bound
- Tier
- Specialized
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 0
- Relationships
- 3
- Code example
- Not ready
- Visualization
- Ready
Core Concepts
maximum cliquebranch and boundcolor refinementgreedy choiceoptimization
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 Tomita Clique Algorithm when solving maximum clique and you want to benefit from using greedy coloring bounds to prune maximum-clique search.
- Choose it when O(2^n) average/worst-case time and O(2^n) memory match your workload profile.
- Prefer it over Branch-and-Bound Maximum Clique 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 using greedy coloring bounds to prune maximum-clique search.
- Maintains key invariants across maximum clique, branch and bound, color refinement.
Use Cases
- Production and research workloads requiring maximum clique via using greedy coloring bounds to prune maximum-clique search.
- Branch and Bound pipelines where O(2^n) time and O(2^n) space fit system constraints.
- Engineering comparisons against Branch-and-Bound Maximum Clique 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 using greedy coloring bounds to prune maximum-clique search.
- 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 Tomita Clique Algorithm invariants
- Explanation
- Tomita Clique Algorithm processes the input by using greedy coloring bounds to prune maximum-clique search, 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 using greedy coloring bounds to prune maximum-clique search 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
Tomita Clique Algorithm operates in O(2^n) time and O(2^n) space within the Branch and Bound family. Often compared with Branch-and-Bound Maximum Clique and Bron-Kerbosch Algorithm.
Sources (1)
What each reference supports for this entry.
Tomita Maximum Clique
PaperReference for branch-and-bound maximum clique search with coloring-based upper bounds.
Open sourceExplore Next
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Relationships
View full relationship pageAlternative toBranch-and-Bound Maximum Clique
Tomita Clique Algorithm and Branch-and-Bound Maximum Clique are practical alternatives within maximum clique, with different assumptions and tradeoffs.
Conceptual neighborBron-Kerbosch Algorithm
Tomita Clique Algorithm and Bron-Kerbosch Algorithm share a concrete branch and bound mechanism for maximum clique.
Related Algorithms
Tags
OptimizationGreedy