A tight example for approximation ratio 5 for covering small cuts by the primal-dual method

In the Small Cuts Cover problem we seek to cover by a min-cost edge-set the set family of cuts of size/capacity $<k$ of a graph. Recently, Simmons showed that the primal-dual algorithm of Williamson, Goemans, Mihail, and Vazirani achieves approximation ratio $5$ for this problem, and asked whether this bound is tight. We will answer this question positively, by providing an example in which the ratio between the solution produced by the primal-dual algorithm and the optimum is arbitrarily close to $5$.

💡 Research Summary

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The paper addresses the Small Cuts Cover problem, a network design task in which one is given a graph G = (V, E), a set L of potential links (edges) each with a non‑negative cost, and an integer k ≥ 1. A cut S ⊂ V is called small if the number of edges crossing the cut, denoted d(S) = |δ(S)|, is strictly less than k. The objective is to select a minimum‑cost subset J ⊆ L that “covers” every small cut, i.e., for each small cut S there exists at least one selected link with exactly one endpoint in S.

Earlier work showed that the primal‑dual algorithm originally introduced by Williamson, Goemans, Mihail, and Vazirani (WGMV) yields a 5‑approximation for this problem. However, it was unknown whether the factor 5 is merely an artifact of the analysis or whether there exist instances on which the algorithm actually performs that badly. The present paper resolves this open question affirmatively by constructing a family of instances on which the algorithm’s solution cost approaches five times the optimum.

Algorithmic background.
The standard linear programming relaxation (P) for covering a family F of sets (here the small cuts) is
 min ∑{e∈E} c_e x_e subject to ∑{e∈δ(S)} x_e ≥ 1 ∀ S∈F, x_e ≥ 0.
Its dual (D) is
 max ∑{S∈F} y_S subject to ∑{S∋e} y_S ≤ c_e ∀ e∈E, y_S ≥ 0.
A link e is tight when the corresponding dual inequality holds with equality. The WGMV primal‑dual algorithm works in two phases. In Phase 1 it repeatedly identifies the cores (inclusion‑minimal uncovered sets) of the currently uncovered family, raises the dual variables of all cores uniformly until some edge becomes tight, and adds all newly tight edges to the solution set J. Phase 2 is a reverse‑delete step that processes the edges in reverse order of insertion and discards any edge whose removal does not uncover any set.

Construction of the hard instance.
The authors first describe a small “base” gadget (Figure 1) parameterised by integers q ≥ 1 and k ≥ 2q + 1. The gadget contains three “red” links (tx, ay, yr) and two “blue” links (tb, rz). The cost of a link equals the number of non‑white vertices it connects; consequently, ay and tb have cost 1, while all other links have cost 2. The small cuts of the gadget are explicitly enumerated: singletons {t} and {r}, a three‑vertex set C = {x, y, z}, and four additional cuts A, X, Y, each of size k − 1 or less. Lemma 2 proves that the family of cores consists exactly of {t}, {r}, and C, while Lemma 3 shows that each colour class (red or blue) forms a minimal feasible solution.

To amplify the gap, the gadget is replicated p times (p ≥ 2) and the copies are “glued” together along three axis vertices r, b, z. All copies of r are merged into a single vertex r, similarly for b and z, and the multiplicities of the r‑b and b‑z edges are adjusted so that the overall capacity remains consistent with the parameter k. The resulting graph has 4p + 3 vertices. The set of small cuts now consists of:

• the p singleton cuts {t_i},
• the p copies of C (which together form a larger set C = {z} ∪ { x_i, y_i | i=1…p }),
• all cuts A_i = {t_i, a_i},
• the cuts X_i = {a_i, y_i} and Y_i = {y_i, r}.

Lemma 4 establishes that the cores of the whole instance are exactly {r}, the p singletons {t_i}, and the large set C. Lemma 5 proves that the red link family
 L_red = { t_i‑x_i, a_i‑y_i, y_i‑r | i=1…p }
and the blue link family
 L_blue = { t_i‑b | i=1…p } ∪ { r‑z }
are each inclusion‑minimal feasible solutions.

Behaviour of the primal‑dual algorithm.
When the WGMV algorithm is run on this instance, Phase 1 raises the dual variables of all p + 2 cores (the p singletons, r, and C) to value 1. At that moment every red link becomes tight (each red link is incident to exactly one core of value 1), so all red links are added to J. None of the blue links become tight during Phase 1. In Phase 2 the algorithm processes the edges in reverse order; because each blue link is redundant with respect to the already selected red links, all blue edges are removed, leaving only the red set L_red. Consequently the algorithm’s output has total cost

 cost(J) = 5p (2·p red links of cost 2 plus p red links of cost 1).

The optimal solution, however, is simply the blue family L_blue, whose cost is

 opt = p + 2 (p links of cost 1 plus the r‑z link of cost 2).

Thus the approximation ratio achieved on this instance is

 cost(J) / opt = 5p / (p + 2),

which tends to 5 as p → ∞. By choosing q = 1 and p ≈ ⌊(k − 1)/2⌋ the authors obtain the largest possible gap for a given k, yielding ratios of the form 5·(k − 1)/(k + 3) for odd k and 5·(k − 2)/(k + 2) for even k.

Implications and conclusions.
The construction demonstrates that the 5‑approximation bound proved for the WGMV primal‑dual algorithm is tight: there exist arbitrarily large instances on which the algorithm’s solution is within an arbitrarily small ε of five times the optimum. The paper also notes that by slightly increasing the costs of the blue links (adding a tiny ε) one can force the algorithm to ignore the blue edges even in Phase 1, reinforcing the tightness argument.

While this result settles the optimality of the 5‑approximation for the standard primal‑dual approach, it does not rule out the possibility of better ratios using fundamentally different techniques or stronger linear programming relaxations. The authors remark that iterative‑rounding methods have already been shown unsuitable for this problem (see reference