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

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 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, this algorithm has a quadratic pseudo-polynomial factor in its complexity because of the maxplus convolution. In this study, we propose a new dynamic programming technique, called heavy-light recursive dynamic programming, to obtain 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 complexity as the original problem.

Average Sensitivity of Dynamic Programming

“…) as a function of x, and the second inequality is from |V | ≥ 2 and ǫ < 0.05. We obtain the claim by combining ( 8), ( 9), (10), and (11).…”

“…Here, we describe the intuition behind our reduction from the RNA folding problem to the MWC problem. Our reduction is inspired by a pseudo-polynomial time algorithm for solving constrained knapsack problems on trees [11], in which they reduced the dependency of the time complexity on the weight limit from quadratic to linear. Before getting into details, we formally define the RNA folding problem.…”

Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem

“…The straightforward evaluation is that for each v ∈ V(T ), the values wbcs(v, * ) are computed by a dynamic programming algorithm, which runs in O(p v W 2 ) time, where p v is the number of children of v. Therefore, the overall running time is upper bounded by O(nW 2 ). To improve the quadratic dependency of W, we can exploit the heavy-light recursive dynamic programming technique [17]. They proved that, given a tree whose vertex contains an item and each item has a weight and a value, the problem, called tree constrained knapsack problems, of maximizing the total value of items that induces a subtree subject to the condition that the total weight is upper bounded by a given budget can be solved in O(n log 3 W) = O(n 1.585 W) time, where W is the total weight of items.…”