Approximating Minimum k -Section in Trees with Linear Diameter

Fernandes, Cristina G.; Schmidt, Tina Janne; Taraz, Anusch

doi:10.1016/j.endm.2015.07.013

Cited by 2 publications

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“…Due to Proposition (14a) this implies that ∈ and ≠ , which contradicts Proposition (17d). Hence, and are disjoint and, with (9), it follows that | | ≤ ∕2 as desired. For the tree decomposition ( ′ ,  ′ ) of [ ] required for Option (2) of Theorem 13, we can use the restriction of ( , ) to ′ ∶= and [ ], which is indeed a tree decomposition of [ ] of width at most − 1.…”

Section: Case 2amentioning

confidence: 91%

“…by applying the techniques used in the proof of Theorem 3 directly to the considered tree instead of working with a tree decomposition, see [19]. Furthermore, the bounds and algorithms presented here can be extended to k-sections, where the vertex set of a graph has to be partitioned into k sets for some integer k while minimizing the number of edges between these sets, see [8,9]. Moreover, extensions to bisections and k-sections in trees with weighted vertices have been considered, see [12].…”

Section: 3mentioning

confidence: 99%

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On minimum bisection and related cut problems in trees and tree‐like graphs

Fernandes

Schmidt

Taraz

2018

Journal of Graph Theory

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Minimum bisection denotes the NP‐hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. It is intuitively clear that graphs with a somewhat linear structure are easy to bisect, and therefore our aim is to relate the minimum bisection width of a bounded‐degree graph G to a parameter that measures the similarity between G and a path. First, for trees, we use the diameter and show that the minimum bisection width of every tree T on n vertices satisfies MinBis (T)≤8nΔ(T)/ diam (T). Second, we generalize this to arbitrary graphs with a given tree decomposition (T,X) and give an upper bound on the minimum bisection width that depends on how close (T,X) is to a path decomposition. Moreover, we show that a bisection satisfying our general bound can be computed in time proportional to the encoding length of the tree decomposition when the latter is provided as input.

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Section: Case 2amentioning

confidence: 91%

Section: 3mentioning

confidence: 99%