A Bad Example for Jain’s Iterative Rounding Theorem for the Cover Small Cuts Problem

Abstract

The Cover Small Cuts problem is a key problem in Network Design. In the Cover Small Cuts problem, we are given a capacitated (undirected) graph G = (V,E,u) and a threshold value λ, as well as a set of links L with end-nodes in V and a non-negative cost for each link 𝓁 ∈ L; the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than λ is covered by a link.
Jain’s iterative rounding theorem is a well-known result in the area of approximation algorithms and, more broadly, in combinatorial optimization. The theorem asserts that LP relaxations of several problems in network design and combinatorial optimization have the following key property: for every basic feasible solution x there exists a variable x_e that has value at least a constant (e.g., x_e ≥ 1/2).
We construct an example showing that this property fails to hold for the standard LP relaxation of the Cover Small Cuts problem. This indicates that the polyhedron of feasible solutions to the LP (for Cover Small Cuts) differs in an essential way from the polyhedrons associated with several problems in combinatorial optimization. Moreover, our example shows that a direct application of Jain’s iterative rounding algorithm does not give an O(1) approximation algorithm for Cover Small Cuts.
We mention that Bansal et al. [Ishan Bansal et al., 2024] showed that the WGMV primal-dual algorithm, due to Williamson et al. [David P. Williamson et al., 1995], applied to the same standard LP relaxation achieves approximation ratio 16 for the Cover Small Cuts problem. That is, the WGMV primal-dual algorithm, applied to the same LP relaxation, finds an integer solution of cost ≤ 16 times the optimal value of the LP.