Overview
Knapsack FPTAS is a approximation algorithms method for knapsack approximation. It works by scaling item values so dynamic programming runs in polynomial time.
ExtendedSource checked
Knapsack FPTAS
Knapsack FPTAS addresses knapsack approximation by scaling item values so dynamic programming runs in polynomial time.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Optimization
- Difficulty
- Intermediate
- Family
- Approximation Algorithms
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 3
- Formulas
- 0
- Relationships
- 6
- Code example
- Ready
- Visualization
- Ready
Core Concepts
knapsack approximationapproximation algorithmsdynamic programmingoptimization
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n)
Average
O(n log n)
Worst
O(n^2)
Space
O(n)
When to Use
- Use Knapsack FPTAS when solving knapsack approximation and you want to benefit from scaling item values so dynamic programming runs in polynomial time.
- Choose it when O(n log n) average/worst-case time and O(n) memory match your workload profile.
- Prefer it over 0/1 Knapsack Dynamic Programming or Meet-in-the-Middle Knapsack when the problem structure favors knapsack approximation.
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 knapsack approximation within Approximation Algorithms workloads (O(n log n) time, O(n) space).
- Works by scaling item values so dynamic programming runs in polynomial time.
- Maintains key invariants across knapsack approximation, approximation algorithms, dynamic programming.
Use Cases
- Production and research workloads requiring knapsack approximation via scaling item values so dynamic programming runs in polynomial time.
- Approximation Algorithms pipelines where O(n log n) time and O(n) space fit system constraints.
- Engineering comparisons against 0/1 Knapsack Dynamic Programming and Meet-in-the-Middle Knapsack when tuning knapsack approximation.
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 knapsack approximation instance, objective function, and initial candidate state or active search tree.
- 2.Evaluate candidate feasibility and objective bounds (knapsack approximation, approximation algorithms).
- 3.Advance the search by scaling item values so dynamic programming runs in polynomial time.
- 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 knapsack approximation test case
- Output
- Best known feasible solution satisfying Knapsack FPTAS invariants
- Explanation
- Knapsack FPTAS processes the input by scaling item values so dynamic programming runs in polynomial time, tracking knapsack approximation and approximation algorithms to produce the verified knapsack approximation 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 scaling item values so dynamic programming runs in polynomial time on a small knapsack approximation example.
Step 1 of 5: Formulate the knapsack approximation instance, objective function, and initial candidate state or active search tree.
optimization-mechanismNotes
Knapsack FPTAS operates in O(n log n) time and O(n) space within the Approximation Algorithms family. Often compared with 0/1 Knapsack Dynamic Programming and Meet-in-the-Middle Knapsack.
Sources (3)
What each reference supports for this entry.
The Design of Approximation Algorithms
BookApproximation algorithms reference for Knapsack FPTAS's guarantee, reduction, or approximation-analysis model.
Open sourceKnapsack Problems
BookKnapsack reference for Knapsack FPTAS's exact, approximate, branch-and-bound, or split-enumeration formulation.
Ibarra-Kim Knapsack FPTAS
PaperOriginal reference for fast approximation algorithms for 0-1 knapsack and subset-sum variants.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Approximates0/1 Knapsack Dynamic Programming
Optimization · Core
Knapsack FPTAS gives an approximate or heuristic answer for the problem family associated with 0/1 Knapsack Dynamic Programming.
Compare with thisRelationships
View full relationship pageApproximates0/1 Knapsack Dynamic Programming
Knapsack FPTAS gives an approximate or heuristic answer for the problem family associated with 0/1 Knapsack Dynamic Programming.
ApproximatesMeet-in-the-Middle Knapsack
Knapsack FPTAS gives an approximate or heuristic answer for the problem family associated with Meet-in-the-Middle Knapsack.
ApproximatesBranch-and-Bound Knapsack
Knapsack FPTAS gives an approximate or heuristic answer for the problem family associated with Branch-and-Bound Knapsack.
Related Algorithms
Tags
OptimizationDynamic ProgrammingApproximation