How to cut a ball without separating: Improved approximations for length bounded cut

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 d(L− 1)/2e. 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 d(L− 1)/2e 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.