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We revisit the standard 0-1 knapsack problem. The latest polynomial-time approximation scheme by Rhee (2015) with approximation factor 1+eps has running time near O (n+ (1/eps) ^5/2) (ignoring polylogarithmic factors), and is randomized. We present a simpler algorithm which achieves the same result and is deterministic. With more effort, our ideas can actually lead to an improved time bound near O (n + (1/eps) ^12/5), and still further improvements for small n.
Timothy M. Chan (2018) studied this question.