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, it can be solved in pseudo-polynomial time by the dynamic programming algorithm. However, 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 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.
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Notes
- 1.
For simplicity, we only consider the case in which the weights are integers. The same result is obtained when the profits are integers.
- 2.
The additive term is naturally O(dC); however, it is separated and included in the recursive terms.
- 3.
Our definition of the heavy edge is slightly different to the original one: In [17], (u, v) is said to be “heavy” if \(2 \times \mathrm {size}(v) > \mathrm {size}(u)\), where \(\mathrm {size}(v)\) is the number of descendants of v. Thus, their heavy edge is always our heavy edge, but the converse is not. In particular, in their definition, any internal vertex has at most one heavy edge, but in our definition, any internal vertex has exactly one heavy edge.
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We thank the anonymous reviewers for their helpful comments.
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Kumabe, S., Maehara, T., Sin’ya, R. (2019). Linear Pseudo-Polynomial Factor Algorithm for Automaton Constrained Tree Knapsack Problem. In: Das, G., Mandal, P., Mukhopadhyaya, K., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2019. Lecture Notes in Computer Science(), vol 11355. Springer, Cham. https://doi.org/10.1007/978-3-030-10564-8_20
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