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.
References
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)
Bremner, D., et al.: Necklaces, convolutions, and X + Y. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 160–171. Springer, Heidelberg (2006). https://doi.org/10.1007/11841036_17
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)
Cho, G., Shaw, D.X.: A depth-first dynamic programming algorithm for the tree knapsack problem. J. Comput. 9(4), 431–438 (1997)
Comon, H., et al.: Tree automata techniques and applications (2007)
Cygan, M., Mucha, M., Węgrzycki, K., Włodarczyk, M.: On problems equivalent to (min,+)-convolution. arXiv:1702.07669 (2017)
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)
Hochbaum, D.S., Pathria, A.: Node-optimal connected k-subgraphs. University of California, Berkeley, Technical report (1994)
Ibarra, O., Kim, C.: Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22(4), 463–468 (1975)
Ishihata, M., Maehara, T., Rigaux, T.: Algorithmic meta-theorems for monotone submodular maximization. arXiv:1807.04575 (2018)
Johnson, D.S., Niemi, K.: On knapsacks, partitions, and a new dynamic programming technique for trees. Mathe. Oper. Res. 8(1), 1–14 (1983)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-24777-7
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)
Lukes, J.A.: Efficient algorithm for the partitioning of trees. IBM J. Res. Dev. 18(3), 217–224 (1974)
Van der Merwe, D., Hattingh, J.M.: Tree knapsack approaches for local access network design. Eur. J. Oper. Res. 174(3), 1968–1978 (2006)
Pferschy, U., Schauer, J.: The knapsack problem with conflict graphs. J. Graph Algorithms Appl. 13(2), 233–249 (2009)
Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)
Thatcher, J.W., Wright, J.B.: Generalized finite automata theory with an application to a decision problem of second-order logic. Mathe. Syst. Theor. 2(1), 57–81 (1968)
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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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