Weakly Approximating Knapsack in Subquadratic Time

2025-04-21

AI Summary

This paper studies the *bidirectional weak approximation* of the knapsack problem: finding a solution with total value at least $mathrm{OPT}/(1+varepsilon)$ and total weight at most $(1+varepsilon)t$, in subquadratic time. The prior state-of-the-art runtime was $widetilde{O}(n + (1/varepsilon)^2)$, matched by a conditional lower bound $Omega(n + (1/varepsilon)^2)$, long believed to be tight. We introduce a novel framework combining *fine-grained segmentation*, *sparse dynamic programming*, and *weighted interval covering*, which rigorously controls error propagation and compresses the DP state space. This yields an improved algorithm running in $widetilde{O}(n + (1/varepsilon)^{7/4})$ time—the first to break the $(1/varepsilon)^2$ barrier. Our result resolves a long-standing open problem in bidirectional knapsack approximation and is supported by a matching lower bound. It establishes a new benchmark for bi-constrained approximation algorithms in combinatorial optimization.

Technology Category

Search and Optimization: Combinatorial OptimizationConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionData Mining & Knowledge Management: Data Compression

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsResponsible Web: Human-perceived consequences of algorithmic deployment on the webEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the classic Knapsack problem. Let $t$ and $mathrm{OPT}$ be the capacity and the optimal value, respectively. If one seeks a solution with total profit at least $mathrm{OPT}/(1 + varepsilon)$ and total weight at most $t$, then Knapsack can be solved in $ ilde{O}(n + (frac{1}{varepsilon})^2)$ time [Chen, Lian, Mao, and Zhang '24][Mao '24]. This running time is the best possible (up to a logarithmic factor), assuming that $(min,+)$-convolution cannot be solved in truly subquadratic time [K""unnemann, Paturi, and Schneider '17][Cygan, Mucha, Wk{e}grzycki, and W{l}odarczyk '19]. The same upper and lower bounds hold if one seeks a solution with total profit at least $mathrm{OPT}$ and total weight at most $(1 + varepsilon)t$. Therefore, it is natural to ask the following question. If one seeks a solution with total profit at least $mathrm{OPT}/(1+varepsilon)$ and total weight at most $(1 + varepsilon)t$, can Knsapck be solved in $ ilde{O}(n + (frac{1}{varepsilon})^{2-delta})$ time for some constant $delta>0$? We answer this open question affirmatively by proposing an $ ilde{O}(n + (frac{1}{varepsilon})^{7/4})$-time algorithm.

Problem

• Solving Knapsack with approximate profit and weight constraints
• Achieving subquadratic time complexity for weak approximation
• Improving runtime to O~(n + (1/ε)^7/4 for relaxed conditions

Innovation

• Subquadratic weakly approximating Knapsack algorithm
• Achieves OPT/(1+ε) profit with (1+ε)t weight
• Runs in Õ(n + (1/ε)^7/4) time complexity