knapsack-problems

What it does

The skill presents an expert-driven framework for solving knapsack optimization problems and resource allocation. It categorizes problem types (0/1, bounded, unbounded, multiple, multidimensional, quadratic) and provides mathematical formulations, algorithms, and example implementations to maximize value under capacity constraints.

How it works

• It first classifies the problem into a knapsack variant (e.g., 0/1 Knapsack, Bounded Knapsack, Unbounded Knapsack, Multidimensional Knapsack, Multiple Knapsack, Quadratic Knapsack).
• For each variant, it states the decision variables, objective (maximize value), and relevant constraints (e.g., weight, volume, or multiple resources).
• It includes algorithmic approaches and concrete code blocks: dynamic programming for 0/1 (with a detailed Python implementation), a space-optimized DP variant, branch-and-bound for larger problems, a greedy approximation, a bounded knapsack approach via expanding items and solving as 0/1, unbounded knapsack with backtracking, and a multidimensional greedy approach.
• For the Multidimensional Knapsack, it outlines a greedy + local search method with a note on NP-hardness.
• For the Multiple Knapsack, it provides a skeleton that imports a linear programming tool (e.g., from pulp) to handle assignment across knapsacks.

When to use it

Use this skill when the user asks to solve knapsack optimization problems, select constrained items, or maximize value with limited capacity. It also directs usage when the user mentions variants like 0/1 knapsack, bounded, unbounded, multiple, multidimensional, value-based packing, capacity-constrained selection, or resource allocation with limits. It also points to related areas (bin packing and general optimization) for broader contexts.

What it can touch

• Code blocks labeled with function definitions and algorithms for:
  • knapsack_01_dynamic
  • knapsack_01_space_optimized
  • knapsack_branch_and_bound
  • knapsack_greedy_approximation
  • bounded_knapsack
  • unbounded_knapsack
  • multidimensional_knapsack
  • multiple_knapsack (example scaffolding using pulp)
• It relies on standard Python data structures and mathematical formulations to compute optimal or approximate solutions.

Caveats

• It notes NP-hardness for the multidimensional knapsack and suggests greedy + local search as an approximate approach.
• The multiple knapsack section uses an external tool (pulp) as part of the implementation scaffolding; licensing or installation considerations would apply outside the skill text.
• The skill provides sample code and formulations but does not guarantee optimality for all problem sizes, particularly for NP-hard variants.