文章基本信息
- 标题：How to Cut a Ball Without Separating: Improved Approximations for Length Bounded Cut
- 本地全文：下载
- 作者：Eden Chlamt{'a}Ä ; Petr Kolman
- 期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
- 电子版ISSN：1868-8969
- 出版年度：2020
- 卷号：176
- 页码：41:1-41:17
- DOI：10.4230/LIPIcs.APPROX/RANDOM.2020.41
- 出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
- 摘要：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 âO^ (L-1)/2âO. 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 âO^ (L-1)/2âO 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.
- 关键词：Approximation Algorithms; Length Bounded Cuts; Cut-Flow Duality; Rounding of Linear Programms
