Abstract
A function \(f: 2^V \rightarrow \mathbb {R}\) on a finite set V is posimodular if \(f(X)+f(Y) \ge f(X{\setminus } Y)+f(Y{\setminus } X)\), for all \(X,Y\subseteq V\). Posimodular functions often arise in combinatorial optimization such as undirected cut functions. We consider the problem of finding a nonempty subset X minimizing f(X), when the posimodular function f is given by oracle access. We show that posimodular function minimization requires exponential time, contrasting with the polynomial solvability of submodular function minimization that forms another generalization of cut functions. On the other hand, the problem is fixed-parameter tractable in terms of the size D of the image (or range) of f. In more detail, we show that \(\varOmega (2^{0.32n} T_f)\) time is necessary and \(O(2^{0.92n}T_f)\) sufficient, where \(T_f\) denotes the time for one function evaluation and \(n = |V|\). When the image of f is \(D=\{0,1,\ldots ,d\}\) for integer d, \(O(2^{1.218d}nT_f)\) time is sufficient. We can also generate all sets minimizing f in time \(2^{O(d)} n^2 T_f\). Finally, we also consider the problem of maximizing a given posimodular function, showing that it requires at least \(2^{n-1}T_f\) time in general, while it has time complexity \(\varTheta ({n \atopwithdelims ()d-1}T_f)\) when \(D=\{0,1,\ldots , d\}\) is the image of f, for integer \(d=O(n^{1/4})\).
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Acknowledgements
We are most grateful for suggestions from anonymous reviewers that allowed us to improve the time complexity of Algorithm 3 and simplify several arguments. We thank S. Fujishige, M. Grötschel, and S. Tanigawa for their helpful comments. This research was partially supported by the Scientific Grant-in-Aid from Ministry of Education, Culture, Sports, Science and Technology of Japan.
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An extended abstract of this article was presented in Proceedings of the 15th International Symposium on Algorithms and Data Structures (WADS 2017) [7]
This research was partially supported by Icelandic Research Fund Grants 152679-05 and 174484-05, MEXT KAKENHI Grant Numbers JP24106002, JSPS KAKENHI Grant Numbers JP25280004, JP26280001, JP16K00001 and 20K11699, and JST CREST Grant Number JPMJCR1402, Japan.
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Halldórsson, M.M., Ishii, T., Makino, K. et al. Posimodular Function Optimization. Algorithmica 84, 1107–1131 (2022). https://doi.org/10.1007/s00453-021-00910-y
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DOI: https://doi.org/10.1007/s00453-021-00910-y