(1-ε)-Approximation of Knapsack in Nearly Quadratic Time
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Knapsack is one of the most fundamental problems in theoretical computer science. In the (1 - ϵ)-approximation setting, although there is a fine-grained lower bound of (n + 1 / ϵ) ^ 2 - o(1) based on the (min, +)-convolution hypothesis ([Künnemann, Paturi and Stefan Schneider, ICALP 2017] and [Cygan, Mucha, Wegrzycki and Wlodarczyk, 2017]), the best algorithm is randomized and runs in Õ(n + (1 / ϵ) ^ 11/5) 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 problem positively by showing a deterministic (1 - ϵ)-approximation scheme for knapsack that runs in Õ(n + (1 / ϵ) ^ 2) time. We first extend a known lemma in a recursive way to reduce the problem to n ϵ-additve approximation for n items. Then we give a simple efficient geometry-based algorithm for the reduced problem.
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