Document

# How to Cut a Ball Without Separating: Improved Approximations for Length Bounded Cut

## File

LIPIcs.APPROX-RANDOM.2020.41.pdf
• Filesize: 1.37 MB
• 17 pages

## Cite As

Eden Chlamtáč and Petr Kolman. How to Cut a Ball Without Separating: Improved Approximations for Length Bounded Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.41

## 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Approximation algorithms analysis
##### Keywords
• Approximation Algorithms
• Length Bounded Cuts
• Cut-Flow Duality
• Rounding of Linear Programms

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. J. Adámek and V. Koubek. Remarks on flows in network with short paths. Commentationes Mathematicae Universitatis Carolinae, 12(4):661-667, 1971.
2. K. Altmanová, P. Kolman, and J. Voborník. On polynomial-time combinatorial algorithms for maximum l-bounded flow. In Proc. of Algorithms and Data Structures Symposium (WADS), pages 14-27, 2019.
3. G. Baier, T. Erlebach, A. Hall, E. Köhler, P. Kolman, O. Pangrác, H. Schilling, and M. Skutella. Length-bounded cuts and flows. ACM Trans. Algorithms, 7(1):4:1-4:27, 2010. Preliminary version in Proc. of ICALP, 2006.
4. M. O. Ball, B. L. Golden, and R. V. Vohra. Finding the most vital arcs in a network. Operations Research Letters, 8(2):73-76, 1989.
5. A. Bar-Noy, S. Khuller, and B. Schieber. The complexity of finding most vital arcs and nodes. Technical Report CS-TR-3539, Univ. of Maryland, Dept. of Computer Science, November 1995. URL: ftp://ftp.cs.umd.edu/pub/papers/papers/3539/3539.ps.Z.
6. C. Bazgan, A. Nichterlein, and R. Niedermeier. A refined complexity analysis of finding the most vital edges for undirected shortest paths. In Proc. of Algorithms and Complexity - 9th International Conference (CIAC), volume 9079 of LNCS, pages 47-60, 2015.
7. J. A. Bondy and U. S. R. Murty. Graph Theory with Applications. North Holland, New York, 1976.
8. I. Dinur and S. Safra. On the hardness of approximating minimum vertex cover. Annals of Mathematics, 162(1):439-485, 2005. URL: https://doi.org/10.4007/annals.2005.162.439.
9. P. Dvořák and D. Knop. Parametrized complexity of length-bounded cuts and multi-cuts. In Proc. of 12 Annual Conference on Theory and Applications of Models of Computation (TAMC), pages 441-452, 2015.
10. T. Fluschnik, D. Hermelin, A. Nichterlein, and R. Niedermeier. Fractals for kernelization lower bounds. SIAM J. Discrete Math, 32(1):656-681, 2018.
11. P. A. Golovach and D. M. Thilikos. Paths of bounded length and their cuts: Parameterized complexity and algorithms. Discrete Optimization, 8(1):72-86, 2011.
12. S. Khot, D. Minzer, and M. Safra. Pseudorandom sets in grassmann graph have near-perfect expansion. In 59th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 592-601, 2018. URL: https://doi.org/10.1109/FOCS.2018.00062.
13. S. Khot and O. Regev. Vertex cover might be hard to approximate to within 2. Journal of Computer and System Sciences, 74(3):335-349, 2008. Computational Complexity 2003. URL: https://doi.org/10.1016/j.jcss.2007.06.019.
14. P. Kolman. On algorithms employing treewidth for l-bounded cut problems. Journal of Graph Algorithms and Applications, 22(2):177-191, 2018. URL: https://doi.org/10.7155/jgaa.00462.
15. E. Lee. Improved hardness for cut, interdiction, and firefighter problems. In Proc. of 44rd International Colloquium on Automata, Languages, and Programming (ICALP), 2017.
16. L. Lovász, V. Neumann-Lara, and M. D. Plummer. Mengerian theorems for paths of bounded length. Periodica Mathematica Hungarica, 9:269-276, 1978.
17. A. R. Mahjoub and S. T. McCormick. Max flow and min cut with bounded-length paths: complexity, algorithms, and approximation. Math. Program, 124(1-2):271-284, 2010.
18. F. Pan and A. Schild. Interdiction problems on planar graphs. Discrete Applied Mathematics, 198:215-231, 2016.
X

Feedback for Dagstuhl Publishing