A Group-Based Resource Allocation Model for the Fractional Knapsack Problem
Updated 8 h ago · first seen 11 Sept 2026
paper_01M294G6RXNBV916E3JK6XS23P
- Published
- 11 Sept 2026
- T1 · 8 h ago
- arXiv
- 2609.06470
- T1 · 8 h ago
- Category
- cs.DS
- T1 · 8 h ago
Abstract
-cross Abstract: To solve the fractional knapsack problem, Dantzig's greedy rule orders items according to their value-to-cost ratio. This ordering introduces priority issues. An arbitrarily small perturbation to the input can change the allocation if the budget is exhausted between two items with very similar ratios. To mitigate that problem, we introduce a two-stage rule. We group items sharing attributes within a radius $\delta$. These groups are then evaluated in descending order of ratio, and divide their group's budget share without further ranking. Consider a group featuring an aggregate capacity $U_G$, unit costs contained in $[w^-,w^+]$, and a representative value $\widehat{v}$. The group's loss compared to the exact optimum is bounded by $\widehat{v}\, U_G\frac{w^+-w^-}{w^++w^-}+\varepsilon_v U_G$, in which $\varepsilon_v$ limits the group's internal value variation. Moreover, for any group size, this harmonic factor remains tight. The overall loss becomes restricted to the single budget-binding group whenever the grouping remains order-compatible; thus, groups containing at most $K$ items suffer a per-item loss of $\mathcal{O}(\frac{K}{n})$. Should group ratio intervals exhibit an overlap of at most $\omega$, an additive term $\omega C$ degrades this bound. Within the separation margin between adjacent groups, the grouped allocation remains Lipschitz continuous with respect to cost data, exhibiting a modulus of $\frac{K}{w_{\min}}$. Computing this allocation takes $\mathcal{O}(n+m\log m+|\Gamma|\log|\Gamma|)$ time given $m$ groups and a boundary group $\Gamma$. Alternatively, the time complexity drops to $\mathcal{O}(n+m\log m)$ if a linear-time selection method identifies the boundary group's allocation.
Authors 1
Abhinaba Chakraborty
Specification
- Official page
Source:arXiv (Atom API + RSS)T1observed 8 h agohigh
- Arxiv announce type
- replace
Source:arXiv (Atom API + RSS)T1observed 8 h agohigh
- arXiv id
- 2609.06470
Source:arXiv (Atom API + RSS)T1observed 8 h agohigh
- Categories
- cs.DS, cs.CL
Source:arXiv (Atom API + RSS)T1observed 8 h agohigh
Source:arXiv (Atom API + RSS)T1observed 8 h agohigh
- Primary category
- cs.DS
Source:arXiv (Atom API + RSS)T1observed 8 h agohigh
- Published
- 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 8 h agohigh
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T19
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8 h ago
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- Authors
- Abhinaba Chakraborty
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Claim history · Abstract
Abstractabstract1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| -cross Abstract: To solve the fractional knapsack problem, Dantzig's greedy rule orders items according to their value-to-cost ratio. This ordering introduces priority issues. An arbitrarily small perturbation to the input can change the allocation if the budget is exhausted between two items with very similar ratios. To mitigate that problem, we introduce a two-stage rule. We group items sharing attributes within a radius $\delta$. These groups are then evaluated in descending order of ratio, and divide their group's budget share without further ranking. Consider a group featuring an aggregate capacity $U_G$, unit costs contained in $[w^-,w^+]$, and a representative value $\widehat{v}$. The group's loss compared to the exact optimum is bounded by $\widehat{v}\, U_G\frac{w^+-w^-}{w^++w^-}+\varepsilon_v U_G$, in which $\varepsilon_v$ limits the group's internal value variation. Moreover, for any group size, this harmonic factor remains tight. The overall loss becomes restricted to the single budget-binding group whenever the grouping remains order-compatible; thus, groups containing at most $K$ items suffer a per-item loss of $\mathcal{O}(\frac{K}{n})$. Should group ratio intervals exhibit an overlap of at most $\omega$, an additive term $\omega C$ degrades this bound. Within the separation margin between adjacent groups, the grouped allocation remains Lipschitz continuous with respect to cost data, exhibiting a modulus of $\frac{K}{w_{\min}}$. Computing this allocation takes $\mathcal{O}(n+m\log m+|\Gamma|\log|\Gamma|)$ time given $m$ groups and a boundary group $\Gamma$. Alternatively, the time complexity drops to $\mathcal{O}(n+m\log m)$ if a linear-time selection method identifies the boundary group's allocation. | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Claims are temporal and append-only: a new observation closes the previous claim (valid_to) instead of overwriting it. Conflicting claims from different sources are kept side by side and flagged — never averaged. Methodology →
New paper: A Group-Based Resource Allocation Model for the Fractional Knapsack Problem
arxiv
| Source | Document | Type | Tier | Last observed | Snapshots |
|---|---|---|---|---|---|
| arXiv (Atom API + RSS) | rss.arxiv.org/rss/cs.CL | feed | T1· Official | 22 min ago | 1 |
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