Sum-of-Max Partition under a Knapsack Constraint

TL;DR

This paper introduces an efficient linear-time algorithm for a sequence partition problem with weight and parameter constraints, and explores its NP-completeness on trees with specialized algorithms for certain cases.

Contribution

It presents a novel $O(n)$ time algorithm for the sum-of-max partition problem under a knapsack constraint, improving over previous dynamic programming solutions.

Findings

The $O(n)$ algorithm is simple to implement.

The tree partition problem is NP-complete.

Algorithms for tree cases with unit and integer weights are provided.

Abstract

Sequence partition problems arise in many fields, such as sequential data analysis, information transmission, and parallel computing. In this paper, we study the following partition problem variant: given a sequence of $n$ items $1,\ldots,n$, where each item $i$ is associated with weight $w_i$ and another 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 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 the tree is NP-complete and we present an $O(w_0 n^2)$ time ($O(w_0^2n^2)$ time, respectively) algorithm for the unit weight (integer weight, respectively) case.