Linear Pseudo-Polynomial Factor Algorithm for Automaton Constrained Tree Knapsack Problem

TL;DR

This paper introduces a new dynamic programming method called heavy-light recursive DP that reduces the pseudo-polynomial factor from quadratic to linear for automaton constrained tree knapsack problems, enabling more efficient solutions.

Contribution

The paper presents a novel heavy-light recursive dynamic programming technique that achieves linear pseudo-polynomial factors, improving the efficiency of solving automaton constrained tree knapsack problems.

Findings

Achieves linear pseudo-polynomial time complexity.

Enables efficient solutions for problems with small capacities or profits.

Extends to the k-subtree version problem with similar complexity.

Abstract

The automaton constrained tree knapsack problem is a variant of the knapsack problem in which the items are associated with the vertices of the tree, and we can select a subset of items that is accepted by a top-down tree automaton. If the capacities or the profits of items are integers, the problem can be solved in pseudo-polynomial time using the dynamic programming algorithm. However, the natural implementation of this algorithm has a quadratic pseudo-polynomial factor in its complexity because of the max-plus convolution. In this study, we propose a new dynamic programming technique, called \emph{heavy-light recursive dynamic programming}, to obtain pseudo-polynomial time algorithms having linear pseudo-polynomial factors in the complexity. Such algorithms can be used for solving the problems with polynomially small capacities/profits efficiently, and used for deriving efficient fully polynomial-time approximation schemes. We also consider the $k$-subtree version problem that finds $k$ disjoint subtrees and a solution in each subtree that maximizes total profit under a budget constraint. We show that this problem can be solved in almost the same order as the original problem.