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.
@InProceedings{chlamtac_et_al:LIPIcs.APPROX/RANDOM.2020.41, author = {Chlamt\'{a}\v{c}, Eden and Kolman, Petr}, 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 = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.41}, URN = {urn:nbn:de:0030-drops-126446}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.41}, annote = {Keywords: Approximation Algorithms, Length Bounded Cuts, Cut-Flow Duality, Rounding of Linear Programms} }
Feedback for Dagstuhl Publishing