Conceitos essenciais
The authors present a deterministic fully polynomial-time approximation scheme (FPTAS) for the classic Knapsack problem that runs in nearly quadratic time, which is essentially the best possible under the conjecture that (min, +)-convolution has no truly subquadratic-time algorithm.
Resumo
The authors investigate the classic Knapsack problem and propose a deterministic FPTAS that runs in e^O(n + (1/ε)^2) time. Prior to this work, the best known FPTAS had a running time of e^O(n + (1/ε)^11/5).
The key technical contributions are:
Establishing a "robust" proximity result that allows the approximation to be performed efficiently for a sequence of different knapsack capacities, rather than a single capacity.
Leveraging the proximity result and additive combinatorics techniques to design a uniform dynamic programming approach that can reuse computational results across different capacity intervals.
Employing a rescaling technique to further improve the running time of the dynamic programming.
The authors show that their FPTAS is essentially tight, as the Knapsack problem has no O((n + 1/ε)^2-δ)-time FPTAS for any constant δ > 0, conditioned on the conjecture that (min, +)-convolution has no truly subquadratic-time algorithm.