Simpler Algorithms for Knapsack, Subset Sum, and Min-Plus Convolution

Abstract

We simplify algorithms for Knapsack, output-sensitive Subset Sum, and near-convex min-plus convolution. For bounded Knapsack, we give a deterministic algorithm using $O(N+W^2\log^3(W+2))$ arithmetic and comparison operations, where $W$ is the maximum item weight and $N$ counts input records with binary-encoded multiplicities. Following Bringmann's approach, we partition the items and bound the weight added or removed within each part when correcting a greedy solution to an optimum. These bounds keep the dynamic-programming tables small. The same analysis gives the corresponding bound with maximum profit in place of weight. For multiple-choice Knapsack, we give a randomized $\widetilde O(N+w^2\min\{r,w\})$ algorithm, where $N$ counts alternatives, $r$ counts classes, and $w$ is the maximum within-class weight range. Randomly grouping classes exploits cancellation between positive and negative weight changes. For Subset Sum of $n$ nonnegative integer vectors in any fixed dimension $d$, we obtain expected time $\widetilde O(n+s\sqrt n)$, where $s$ counts attainable sums in the target box. With high probability, all sums are returned within the same bound. This improves the $\widetilde O(n+s n^{d/(d+1)})$ bound of Bringmann, Fischer, and Nakos for $d>1$. The key step computes a sumset inside a box without generating the potentially much larger unrestricted sumset. Finally, we simplify the $\widetilde O(N(D+1))$ algorithm for min-plus convolution of integer arrays of total length $N$, where $D$ is the sum of their maximum deviations above convex arrays. The deviations can change the minimizing pairs substantially, but restrict relevant candidate values to short intervals.