[Model] Knapsack

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Motivation

One of Karp's 21 NP-complete problems; foundational in combinatorial optimization, cryptography, and resource allocation. Has direct reductions to ILP and QUBO.

Definition

Name: Knapsack
Reference: Karp, 1972; Kellerer, Pferschy & Pisinger, Knapsack Problems, 2004

Given $n$ items, each with weight $w_i$ and value $v_i$, and a capacity $C$, find a subset $S \subseteq {0, \ldots, n-1}$ such that $\sum_{i \in S} w_i \leq C$, maximizing $\sum_{i \in S} v_i$.

Variables

• Count: $n$ (one variable per item)
• Per-variable domain: binary ${0, 1}$
• Meaning: $x_i = 1$ if item $i \in S$ (selected)

Schema (data type)

Type name: Knapsack
Variants: 0-1 (binary), bounded, unbounded

Field	Type	Description
weights	Vec<i64>	Item weights $w_0, \ldots, w_{n-1}$
values	Vec<i64>	Item values $v_0, \ldots, v_{n-1}$
capacity	i64	Knapsack capacity $C$

Problem Size

Metric	Expression	Description
num_items	$n$	Number of items

Complexity

• Decision complexity: NP-complete (Karp, 1972)
• Best known exact algorithm: $O(nC)$ pseudo-polynomial dynamic programming; $O(2^{n/2})$ via meet-in-the-middle
• Approximation: FPTAS exists — $(1-\varepsilon)$-approximation in $O(n^2 / \varepsilon)$ time
• References: Karp 1972; Ibarra & Kim 1975 (FPTAS)

Extra Remark

Knapsack is weakly NP-hard (admits pseudo-polynomial algorithms), unlike strongly NP-hard problems such as TSP. It is a special case of ILP with a single constraint. The subset-sum problem is the special case where $v_i = w_i$. Knapsack-based cryptosystems (Merkle-Hellman) were historically important in public-key cryptography.

How to solve

• It can be solved by (existing) bruteforce.
• It can be solved by reducing to integer programming, through a new Knapsack → ILP rule issue (to be filed).

Bruteforce: enumerate all $2^n$ subsets, check capacity constraint, return maximum value.

Example Instance

$n = 4$ items, capacity $C = 7$:

Item	Weight	Value	Value/Weight
0	2	3	1.50
1	3	4	1.33
2	4	5	1.25
3	5	7	1.40

Feasible subsets:

Subset	Weight	Value
${0, 3}$	7	10
${1, 2}$	7	9
${0, 2}$	6	8
${0, 1}$	5	7
${3}$	5	7

Optimal: $S = {0, 3}$, $x = (1,0,0,1)$, weight $= 7$, value $= 10$.

Note: greedy by value/weight ratio picks item 0 (1.50) then item 3 (1.40), which happens to be optimal here, but greedy is not optimal in general.

[isPANN]

Closing as duplicate — this problem is covered by the Garey & Johnson extraction (P18 Knapsack, G&J Chapter 3, Section 3.2.1). The extracted reference issue in references/Garey&Johnson/issues/models/P18_knapsack.md provides a comprehensive definition from the original source.

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Issue Quality Check — Model

Check	Result	Details
Usefulness	✅ Pass	Novel problem, Karp #18, concrete use cases, planned reductions
Non-trivial	✅ Pass	Distinct weighted subset selection with capacity constraint
Correctness	✅ Pass	All three references verified; complexity claims accurate
Well-written	✅ Pass	All sections complete, symbols consistent, example fully worked

Overall: 4 passed, 0 failed, 0 warnings

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Usefulness

Problem does not exist in the codebase (pred show Knapsack fails). Knapsack is #18 in Karp's 21 NP-complete problems and a foundational problem in combinatorial optimization. Motivation is strong: cryptography, resource allocation, and connection to existing problems (ILP, QUBO). Planned reduction rules (Knapsack → ILP) ensure it won't be an orphan node. Brute-force solver path identified. Pass.

Non-trivial

Knapsack is a weighted subset selection problem with a capacity constraint — genuinely distinct from all 23 existing problems. The closest existing problem is BinPacking (which minimizes the number of bins needed), but Knapsack has fundamentally different structure: single capacity constraint, maximize value, items have both weight and value. ILP is a generalization but Knapsack has specific structure worthy of its own model. Pass.

Correctness

Karp, 1972 — "Reducibility Among Combinatorial Problems." DOI verified. Knapsack is indeed problem #18, shown NP-complete via Exact Cover → Knapsack. ✓

Kellerer, Pferschy & Pisinger, 2004 — "Knapsack Problems", Springer. DOI verified. The definitive textbook on knapsack problems. ✓

Ibarra & Kim, 1975 — "Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems", JACM 22(4), 463–468. DOI verified. Introduced the FPTAS for knapsack. ✓

Complexity claims verified:

• NP-complete (Karp 1972) ✓
• O(nC) pseudo-polynomial DP — well-known, correct ✓
• O(2^{n/2}) meet-in-the-middle — Horowitz & Sahni (1974), correct ✓
• FPTAS in O(n²/ε) — Ibarra & Kim 1975 ✓
• Weakly NP-hard — correct (pseudo-polynomial algorithm exists) ✓
• Subset-sum is special case where v_i = w_i — correct ✓

Cross-check against references.md: Knapsack is listed as Karp problem #18, with reduction from Exact Cover. Consistent. ✓

Pass.

Well-written

All required sections present and substantive:

• Motivation ✓ — combinatorial optimization, cryptography, resource allocation
• Name ✓ — "Knapsack" follows convention (no prefix needed, like "MaxCut", "QUBO")
• Reference ✓ — three citations, all with DOIs
• Definition ✓ — input (items with weights/values, capacity), feasibility ($\sum w_i \leq C$), objective (maximize $\sum v_i$)
• Variables ✓ — $n$ binary variables, $x_i = 1$ if item selected
• Schema ✓ — type, fields, variants mentioned
• Complexity ✓ — decision, exact, and approximation bounds with citations
• How to solve ✓ — brute-force checked, ILP planned
• Example ✓ — 4 items, multiple feasible subsets, optimal S = {0,3} with value 10

Symbol consistency: $n$, $w_i$, $v_i$, $C$, $S$, $x_i$ all defined before use and consistent across sections. ✓

Example quality: non-trivial (4 items, exercises capacity constraint), optimal solution verified, insightful note about greedy not being optimal in general. ✓

Problem Size metric num_items is well-defined. For declare_variants!, the best exact complexity is $O(2^{n/2})$ (meet-in-the-middle), which translates to complexity string "2^(0.5 * num_items)".

Pass.