Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing

We present a pseudopolynomial-time algorithm for the Knapsack problem that has running time Õ(n + t√{p_{max}}), where n is the number of items, t is the knapsack capacity, and p_{max} is the maximum item profit. This improves over the Õ(n + t p_{max})-time algorithm based on the convolution and prediction technique by Bateni et al. (STOC 2018). Moreover, we give some evidence, based on a strengthening of the Min-Plus Convolution Hypothesis, that our running time might be optimal. Our algorithm uses two new technical tools, which might be of independent interest. First, we generalize the Õ(n^{1.5})-time algorithm for bounded monotone min-plus convolution by Chi et al. (STOC 2022) to the rectangular case where the range of entries can be different from the sequence length. Second, we give a reduction from general knapsack instances to balanced instances, where all items have nearly the same profit-to-weight ratio, up to a constant factor. Using these techniques, we can also obtain algorithms that run in time Õ(n + OPT√{w_{max}}), Õ(n + (nw_{max}p_{max})^{1/3}t^{2/3}), and Õ(n + (nw_{max}p_{max})^{1/3} OPT^{2/3}), where OPT is the optimal total profit and w_{max} is the maximum item weight.