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oai:drops-oai.dagstuhl.de:12644 2024-03-06T10:51:20Z ddc:004 open_access

How to Cut a Ball Without Separating: Improved Approximations for Length Bounded Cut Chlamtáč, Eden Kolman, Petr Approximation Algorithms Length Bounded Cuts Cut-Flow Duality Rounding of Linear Programms 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Eden Chlamtáč and Petr Kolman 2020 Is Part Of LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2020.41 urn:nbn:de:0030-drops-126446 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.41 eng https://creativecommons.org/licenses/by/3.0/legalcode