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

TL;DR

The paper challenges Jain's Iterative Rounding Theorem for the Cover Small Cuts problem by constructing a capacitated graph instance where the LP relaxation admits a basic solution with all positive variables equal to $1/k$ (for even $k\ge 4$). It develops a two-family cut structure (nested N_i and Q_j) and a links graph decomposed into $k$ s–t paths, and proves that the incidence matrix of small cuts and links has full rank by linking to a circulant matrix $A^{PQ}$ with $\det(A^{PQ})=k/2$. Consequently, Jain's iterative rounding cannot yield a constant-factor approximation for Cover Small Cuts, although existing primal-dual approaches achieve such guarantees. The results illuminate fundamental differences in LP polyhedra for Cover Small Cuts and delineate the limits of iterative rounding in network-design problems.

Abstract

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 solution $x$ there exists a variable $x_e$ that has value at least a constant (e.g., $x_e\geq\frac12$). We construct an example showing that this property fails to hold for the Cover Small Cuts problem. In this problem, we are given an undirected, capacitated 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 $\ell\in L$; 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. This indicates that the polyhedron of feasible solutions to the LP relaxation (of 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. (Algorithmica 2024) present an $O(1)$ approximation algorithm for Cover Small Cuts based on the primal-dual method of Williamson et al. (Combinatorica 1995).

Figures (2)

• Figure 1: The capacitated graph for $k=4$. The small cuts are given by the nested sets $N_1,\dots,N_7$ (inidicated by dashed lines) and the sets $Q_1,Q_2,Q_3$ (indicated by ovals).
• Figure 2: The links graph for $k=4$. The links partition into $s,t$-paths $P_1,\dots,P_4$, and the links of each of these $s,t$-paths is indicated by a distinct colour. Thus, the links of $P_1$ are $\ell_1,\ell_5,\ell_7$, the links of $P_2$ are $\ell_2,\ell_8,\ell_9$, the links of $P_3$ are $\ell_3,\ell_6,\ell_{10}$, and $P_4$ has the link $\ell_4$. The small cuts are given by the nested sets $N_1,\dots,N_7$ (inidicated by dashed lines) and the sets $Q_1,Q_2,Q_3$ (indicated by ovals).

Theorems & Definitions (18)

• Proposition 1
• Theorem 2: Jain Jain01
• Theorem 3
• Proposition 3
• Proposition 3
• Lemma 4
• Claim 5
• proof
• Claim 6
• proof