A faster algorithm for Vertex Cover parameterized by solution size

Abstract

We describe a new algorithm for vertex cover with runtime $O^*(1.25284^k)$, where $k$ is the size of the desired solution and $O^*$ hides polynomial factors in the input size. This improves over previous runtime of $O^*(1.2738^k)$ due to Chen, Kanj, & Xia (2010) standing for more than a decade. The key to our algorithm is to use a potential function which simultaneously tracks $k$ as well as the optimal value $λ$ of the vertex cover LP relaxation. This approach also allows us to make use of prior algorithms for Maximum Independent Set in bounded-degree graphs and Above-Guarantee Vertex Cover. The main step in the algorithm is to branch on high-degree vertices, while ensuring that both $k$ and $μ= k - λ$ are decreased at each step. There can be local obstructions in the graph that prevent $μ$ from decreasing in this process; we develop a number of novel branching steps to handle these situations.

Figures (12)

• Figure 1: In the subgraph $G_0$, the vertex $x$ becomes isolated.
• Figure 2: In the subgraph $G - u$, the vertex $v$ has degree two.
• Figure 3: In the subgraph $G - X$, there is an isolated vertex, hidden in the "shadow" of $X$. Here $\mathop{\mathrm{shad}}\nolimits(X) = -1$.
• Figure 4: A funnel (here, a 3-triangle) $u$ before (left) and after (right) applying P3
• Figure 5: A kite $u,x,y,z$ before (left) and after (right) applying P3

Theorems & Definitions (119)

• Theorem 1: fpt-lp
• Theorem 2
• Theorem 3
• Proposition 4
• Proposition 5
• Lemma 6
• Proposition 8
• Proposition 10
• Proposition 11
• Lemma 13