Vertex Cover Problem Parameterized Above and Below Tight Bounds

We study the well-known Vertex Cover problem parameterized above and below tight bounds. We show that two of the parameterizations (both were suggested by Mahajan, Raman and Sikdar, J. Computer and System Sciences, 75(2):137–153, 2009) are fixed-parameter tractable and two other parameterizations are W[1]-hard (one of them is, in fact, W[2]-hard).

💡 Research Summary

The paper investigates four parameterizations of the classic Vertex Cover problem that are defined relative to tight bounds on the optimum solution, rather than the traditional “size‑k” parameterization. The motivation is that in many applications one is interested in how far a solution can deviate from a known bound (e.g., a matching‑based upper bound) rather than its absolute size. Building on the suggestions of Mahajan, Raman, and Sikdar (2009), the authors formalize two “below‑bound” and two “above‑bound” variants and determine their fixed‑parameter tractability status.

The first variant asks whether there exists a vertex cover whose size is at most τ(G) − k, where τ(G) denotes the size of a minimum vertex cover. By first computing a maximum matching ν(G) (which can be done in polynomial time) and exploiting König’s theorem (τ(G) = ν(G) for bipartite graphs, and τ(G) ≤ 2·ν(G) in general), the authors reduce the instance to a kernel of size O(k²). They then apply a bounded‑search‑tree algorithm that branches on the remaining uncovered edges, achieving a running time of O*(2^{O(k)}). This shows that the “below‑τ” problem is fixed‑parameter tractable (FPT).

The second variant is “above‑matching”: does a vertex cover of size at most ν(G) + k exist? Since ν(G) is a readily computable lower bound, the same kernelization technique can be used. After extracting a maximum matching, the graph is partitioned into matched edges and the residual subgraph. The residual part is small (again O(k²) vertices) and can be solved by exhaustive search within the same O*(2^{O(k)}) bound. Hence this “above‑matching” parameterization is also FPT.

The third variant flips the direction: is there a vertex cover of size at most τ(G) + k? Here the parameter measures how much larger a solution may be than the optimum. The authors prove W