(1 − 𝜀 )-Approximation of Knapsack in Nearly Quadratic Time

Knapsack is one of the most fundamental problems in theoretical computer science. In the (1 − є)-approximation setting, although there is a fine-grained lower bound of (n + 1 / є) 2 − o(1) based on the (min, +)-convolution hypothesis ([K'unnemann, Paturi and Stefan Schneider, ICALP 2017] and [Cygan, Mucha, Wegrzycki and Wlodarczyk, 2017]), the best algorithm is randomized and runs in Õ(n + (1/є)11/5/2Ω(√log(1/є))) time [Deng, Jin and Mao, SODA 2023], and it remains an important open problem whether an algorithm with a running time that matches the lower bound (up to a sub-polynomial factor) exists. We answer the question positively by showing a deterministic (1 − є)-approximation scheme for knapsack that runs in Õ(n + (1 / є) 2) time. We first extend a known lemma in a recursive way to reduce the problem to n є-additive approximation for n items with profits in [1, 2). Then we give a simple efficient geometry-based algorithm for the reduced problem.