Linear Pseudo-Polynomial Factor Algorithm for Automaton Constrained Tree Knapsack Problem
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.
KeywordsKnapsack problem Dynamic programming Tree automaton
We thank the anonymous reviewers for their helpful comments.
- 1.Backurs, A., Indyk, P., Schmidt, L.: Better approximations for tree Sparsity in nearly-linear time. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 2215–2229 (2017)Google Scholar
- 3.Chekuri, C., Pál, M.: A recursive greedy algorithm for walks in directed graphs. In: Proceedings of the 46th Annual Symposium on Foundations of Computer Science (FOCS 2005), vol. 2005, pp. 245–253 (2005)Google Scholar
- 5.Comon, H., et al.: Tree automata techniques and applications (2007)Google Scholar
- 6.Cygan, M., Mucha, M., Węgrzycki, K., Włodarczyk, M.: On problems equivalent to (min,+)-convolution. arXiv:1702.07669 (2017)
- 7.Hirao, T., Yoshida, Y., Nishino, M., Yasuda, N., Nagata, M.: Single-document summarization as a tree knapsack problem. In: Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing (EMNLP 2013), pp. 1515–1520 (2013)Google Scholar
- 8.Hochbaum, D.S., Pathria, A.: Node-optimal connected k-subgraphs. University of California, Berkeley, Technical report (1994)Google Scholar
- 10.Ishihata, M., Maehara, T., Rigaux, T.: Algorithmic meta-theorems for monotone submodular maximization. arXiv:1807.04575 (2018)
- 13.Lawler, E.L.: Fast approximation algorithms for knapsack problems. In: Proceedings of the 18th Annual Symposium on Foundations of Computer Science (FOCS 1977), vol. 4(4), pp. 339–357 (1977)Google Scholar