No Sublogarithmic-time Approximation Scheme for Bipartite Vertex Cover

K"onig’s theorem states that on bipartite graphs the size of a maximum matching equals the size of a minimum vertex cover. It is known from prior work that for every \epsilon > 0 there exists a constant-time distributed algorithm that finds a (1+\epsilon)-approximation of a maximum matching on 2-coloured graphs of bounded degree. In this work, we show—somewhat surprisingly—that no sublogarithmic-time approximation scheme exists for the dual problem: there is a constant \delta > 0 so that no randomised distributed algorithm with running time o(\log n) can find a (1+\delta)-approximation of a minimum vertex cover on 2-coloured graphs of maximum degree 3. In fact, a simple application of the Linial–Saks (1993) decomposition demonstrates that this lower bound is tight. Our lower-bound construction is simple and, to some extent, independent of previous techniques. Along the way we prove that a certain cut minimisation problem, which might be of independent interest, is hard to approximate locally on expander graphs.

💡 Research Summary

The paper investigates the distributed‑computing complexity of two classic problems on bipartite graphs: maximum matching and minimum vertex cover. By König’s theorem these problems share the same optimal value, yet their approximability under locality constraints diverges sharply. Prior work showed that for any ε > 0 a constant‑time (O(1) rounds) randomized distributed algorithm can compute a (1+ε)‑approximation of a maximum matching on bounded‑degree, 2‑coloured graphs. The authors prove that the dual problem—approximating a minimum vertex cover—behaves fundamentally differently.

Specifically, they exhibit a constant δ > 0 such that no randomized distributed algorithm running in o(log n) rounds can guarantee a (1+δ)‑approximation on 2‑coloured graphs of maximum degree three. The lower bound is established via a two‑step construction. First, a variant of the Linial–Saks (1993) decomposition partitions the input graph into small‑diameter clusters separated by a sparse cut. Within each cluster, local information is insufficient to decide whether a vertex belongs to an optimal cover, forcing any sub‑logarithmic algorithm to make errors. Second, the authors embed a cut‑minimisation problem on expander graphs into this clustered structure. Because expanders have high connectivity despite a small edge boundary, any algorithm that only sees a limited neighbourhood cannot accurately estimate the global cut value, and this hardness transfers to the vertex‑cover problem.

The paper also shows that the lower bound is tight. By applying the original Linial–Saks decomposition, one can design an O(log n)‑round algorithm that achieves a (1+δ)‑approximation, establishing Θ(log n) as the exact distributed complexity for a constant‑factor approximation of minimum vertex cover on such graphs.

Beyond the primary result, the authors highlight that the cut‑minimisation problem they introduce may be of independent interest, as its local hardness on expanders could impact other areas such as network design, community detection, and distributed optimisation. The construction is relatively simple and, to a notable extent, independent of earlier techniques, suggesting a new template for proving locality‑based lower bounds for other graph‑theoretic optimisation problems.