A tight bound on the min-ratio edge-partitioning problem of a tree

Chu, An-Chiang; Wu, Bang Ye; Wang, Hung-Lung; Chao, Kun-Mao

doi:10.1016/j.dam.2010.05.014

Cited by 4 publications

(2 citation statements)

References 9 publications

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“…For the special case of trees, this problem was introduced by Wu et al (2007), who proved the existence of 3-balanced and (2 − 1/n)-proportional edge partitions; note that this is without any edge exclusions. Later, Dye (2009) improved the balancedness approximation to 2 for n ∈ {2, 3, 4}, Chu et al (2010) extended this result to all values of n, and Chu, Wu, and Chao (2013) showed how to achieve this in linear time even when the edges are weighted. In Section 5, we make an connection between edge partitions of trees with no edge exclusions and node partitions of general graphs with at most n − 1 node exclusions, allowing us to leverage the above results to obtain upper bounds for our problem.…”

Section: Related Workmentioning

confidence: 96%

“…However, it turns out that there exist reasonably balanced and proportional edge partitions of a tree that do not require any edge exclusions. Theorem 6 (Chu et al 2010). For any n 2, every tree admits a connected n-edge-partition that is 2-balanced and, hence, (2 − 1/n)-proportional, and such a partition can be computed in polynomial time.…”

Section: Upper Bounds For Higher Nmentioning

confidence: 99%

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A Little Charity Guarantees Fair Connected Graph Partitioning

Caragiannis

Micha

Shah

2022

AAAI

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Motivated by fair division applications, we study a fair connected graph partitioning problem, in which an undirected graph with m nodes must be divided between n agents such that each agent receives a connected subgraph and the partition is fair. We study approximate versions of two fairness criteria: \alpha-proportionality requires that each agent receive a subgraph with at least (1/\alpha)*m/n nodes, and \alpha-balancedness requires that the ratio between the sizes of the largest and smallest subgraphs be at most \alpha. Unfortunately, there exist simple examples in which no partition is reasonably proportional or balanced. To circumvent this, we introduce the idea of charity. We show that by "donating" just n-1 nodes, we can guarantee the existence of 2-proportional and almost 2-balanced partitions (and find them in polynomial time), and that this result is almost tight. More generally, we chart the tradeoff between the size of charity and the approximation of proportionality or balancedness we can guarantee.

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