A linear-time algorithm for finding an edge-partition with max-min ratio at most two

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

doi:10.1016/j.dam.2012.11.009

Cited by 4 publications

(2 citation statements)

References 11 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%

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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Section: Related Workmentioning

confidence: 96%

A Little Charity Guarantees Fair Connected Graph Partitioning

Caragiannis

Micha

Shah

2022

AAAI

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“…This result also directly yields a 3-approximation for the problem Max-Min Balanced Connected Edge Partition; the graph dual of Max-Min BCP that searches for an edge partition instead of a vertex partition. Note that this is also the first constant factor approximation for this problem on general instances, since the 2-approximations in [BES19] and [CWC13] both only hold for the restriction max e∈E w(e) ≤ w (G) 2k on the weights. Further, in the process of deriving the algorithm that gives our main result, we obtain an intermediate result which in particular shows how to compute a (q-)weighted crown decomposition in O (|E| |V |).…”

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confidence: 99%

Balanced Crown Decomposition for Connectivity Constraints

Casel¹,

Friedrich²,

Issac³

et al. 2020

Preprint

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We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved kernelization and approximation algorithms for a variety of such problems.In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a 3-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component.Further, we show how to compute our balanced crown decomposition very efficiently which as an intermediate result yields an improved running time in O(|V | |E|) for the well-known (Weighted) q-Expansion Lemma.

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Approximate envy-freeness in graphical cake cutting

Yuen,

Suksompong

2024

Discrete Applied Mathematics

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A linear-time algorithm for finding an edge-partition with max-min ratio at most two

Cited by 4 publications

References 11 publications

A Little Charity Guarantees Fair Connected Graph Partitioning

A Little Charity Guarantees Fair Connected Graph Partitioning

Balanced Crown Decomposition for Connectivity Constraints

Approximate envy-freeness in graphical cake cutting

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