Abstract
In this paper, we study the maximum bounded connected bipartition problem: given a vertex-weighted connected graph \(G=(V,E;w)\) and an upper bound B, the vertex set V is partitioned into two subsets \((V_1,V_2)\) such that the total weight of the two subgraphs induced by \(V_1\) and \(V_2\) is maximized, and these subgraphs are connected, where the weight of a subgraph is the minimum of B and the total weight of all vertices in the subgraph. In this paper, we prove that this problem is NP-hard even when G is a completed graph, a grid graph with only 3 rows or an interval graph, and we present an \(\frac{8}{7}\)-approximation algorithm. In particular, we present a \(\frac{14}{13}\)-approximation algorithm for the case of grid graphs, and we present a fully polynomial-time approximation scheme for the case of interval graphs.
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Acknowledgements
We thank the two anonymous reviewers for their valuable comments and constructive suggestions, which helped us significantly improve the quality of our work.
Funding
Funding was provided by Natural Science Foundation of China (Grant no. 12071417) and (Grant no. 62262069), the Project for Innovation Team(Cultivation) of Yunnan Province [No. 202005AE160006], and the Open Project Program of Yunnan Key Laboratory of Intelligent Systems and Computing [No. ISC22Z03].
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Liu, X., Li, Y., Li, W. et al. Combinatorial approximation algorithms for the maximum bounded connected bipartition problem. J Comb Optim 45, 51 (2023). https://doi.org/10.1007/s10878-022-00981-9
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DOI: https://doi.org/10.1007/s10878-022-00981-9