When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.41
URN: urn:nbn:de:0030-drops-126446
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12644/
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How to Cut a Ball Without Separating: Improved Approximations for Length Bounded Cut

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Abstract

The Minimum Length Bounded Cut problem is a natural variant of Minimum Cut: given a graph, terminal nodes s,t and a parameter L, find a minimum cardinality set of nodes (other than s,t) whose removal ensures that the distance from s to t is greater than L. We focus on the approximability of the problem for bounded values of the parameter L.
The problem is solvable in polynomial time for L ≤ 4 and NP-hard for L ≥ 5. The best known algorithms have approximation factor ⌈ (L-1)/2⌉. It is NP-hard to approximate the problem within a factor of 1.17175 and Unique Games hard to approximate it within Ω(L), for any L ≥ 5. Moreover, for L = 5 the problem is 4/3-ε Unique Games hard for any ε > 0.
Our first result matches the hardness for L = 5 with a 4/3-approximation algorithm for this case, improving over the previous 2-approximation. For 6-bounded cuts we give a 7/4-approximation, improving over the previous best 3-approximation. More generally, we achieve approximation ratios that always outperform the previous ⌈ (L-1)/2⌉ guarantee for any (fixed) value of L, while for large values of L, we achieve a significantly better ((11/25)L+O(1))-approximation.
All our algorithms apply in the weighted setting, in both directed and undirected graphs, as well as for edge-cuts, which easily reduce to the node-cut variant. Moreover, by rounding the natural linear programming relaxation, our algorithms also bound the corresponding bounded-length flow-cut gaps.

BibTeX - Entry

```@InProceedings{chlamt_et_al:LIPIcs:2020:12644,
author =	{Eden Chlamt{\'a}č and Petr Kolman},
title =	{{How to Cut a Ball Without Separating: Improved Approximations for Length Bounded Cut}},
booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
pages =	{41:1--41:17},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-164-1},
ISSN =	{1868-8969},
year =	{2020},
volume =	{176},
editor =	{Jaros{\l}aw Byrka and Raghu Meka},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},