2024/04/08 by Bringmann, Karl, Dürr, Anita, Polak, Adam
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2404.05681
We present a pseudopolynomial-time algorithm for the Knapsack problem that has running time \widetildeO(n + t√pmax), where n is the number of items, t is the knapsack capacity, and pmax is the maximum item profit. This improves over the \widetildeO(n + t pmax)-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 \widetildeO(n1.5)-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 \widetildeO(n + OPT√wmax), \widetildeO(n + (nwmaxpmax)1/3t2/3), and \widetildeO(n + (nwmaxpmax)1/3 OPT2/3), where OPT is the optimal total profit and wmax is the maximum item weight.