Abstract
Given a vertex-weighted connected graph \(G = (V, E, w(\cdot ))\), the maximally balanced connected graph k-partition (k-BGP) seeks to partition the vertex set V into k non-empty parts such that the subgraph induced by each part is connected and the weights of these k parts are as balanced as possible. When the concrete objective is to maximize the minimum (to minimize the maximum, respectively) weight of the k parts, the problem is denoted as max–min k-BGP (min–max k-BGP, respectively), and has received much study since about four decades ago. On general graphs, max–min k-BGP is strongly NP-hard for every fixed \(k \ge 2\), and remains NP-hard even for the vertex uniformly weighted case; when k is part of the input, the problem is denoted as max–min BGP, and cannot be approximated within 6/5 unless P \(=\) NP. In this paper, we study the tripartition problems from approximation algorithms perspective and present a 3/2-approximation for min–max 3-BGP and a 5/3-approximation for max–min 3-BGP, respectively. These are the first non-trivial approximation algorithms for 3-BGP, to our best knowledge.
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Notes
I.e., obtained by moving the vertex \(u_i\) from \(V_2\) to \(V_1\).
\(G'\) is obtained by removing all the non-cut vertices of V(H) and all the edges of E(H) from the input graph G.
Note that the weights of these three parts are not sorted yet. Also, since \(G[V {\setminus } \{v_1\}]\) is connected, the bipartition \(\{V_1, V_2\}\) can be trivially obtained.
\(G'\) is obtained by removing all the non-cut vertices of V(H) and all the edges of E(H) from the input graph G.
\(G'\) is obtained by removing all the non-cut vertices of \(V(H_1) \cup V(H_2)\) and all the edges of \(E(H_1) \cup E(H_2)\) from the input graph G.
These two cut vertices were assigned the weight of the same connected component.
Note that \(G[V {\setminus } \{v_1, v_2\}]\) could be connected and, if so then \(V_3 = V {\setminus } \{v_1, v_2\}\).
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Acknowledgements
GC, YC and AZ are supported by the NSFC Grants 11971139, 11771114 and 11571252; YC and AZ are also supported by the CSC Grants 201508330054 and 201908330090, respectively. ZZC is supported in part by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan, under Grant No. 18K11183. GL is supported by the NSERC Canada.
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Chen, G., Chen, Y., Chen, ZZ. et al. Approximation algorithms for the maximally balanced connected graph tripartition problem. J Comb Optim 44, 1753–1773 (2022). https://doi.org/10.1007/s10878-020-00544-w
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DOI: https://doi.org/10.1007/s10878-020-00544-w