Sum-of-Max Partition under a Knapsack Constraint

Sequence partition problems arise in many fields, such as sequential data analysis, information transmission, and parallel computing. In this paper we study the following variant of partition problem: Given a sequence of $n$ items $1,\ldots,n$, where each item $i$ is associated with a weight $w_i$ and a parameter $s_i$, partition the sequence into several consecutive subsequences, so that the total weight of each subsequence is no more than a threshold $w_0$, and the sum of the largest $s_i$ in each subsequence is minimized. This problem admits a straightforward solution based on dynamic programming, which costs $O(n^2)$ time and can be improved to $O(n\log n)$ time easily. Our main contribution is an $O(n)$ time algorithm, which is nontrivial yet easy to implement. We also study the corresponding tree partition problem. We prove that the problem on tree is NP-complete and we present an $O(w_0^2 n^2)$ time algorithm for the unit weight case.