A Parameterized Perspective on $P_2$-Packings

}We study (vertex-disjoint) $P_2$-packings in graphs under a parameterized perspective. Starting from a maximal $P_2$-packing $\p$ of size $j$ we use extremal arguments for determining how many vertices of $\p$ appear in some $P_2$-packing of size $(j+1)$. We basically can ‘reuse’ $2.5j$ vertices. We also present a kernelization algorithm that gives a kernel of size bounded by $7k$. With these two results we build an algorithm which constructs a $P_2$-packing of size $k$ in time $\Oh^*(2.482^{3k})$.

💡 Research Summary

The paper investigates the vertex‑disjoint P₂‑packing problem – selecting the maximum number of length‑2 paths (three vertices, two edges) that do not share vertices – from a parameterized perspective, where the parameter k denotes the desired packing size. After introducing the problem as a natural generalization of matching (P₁‑packing) and recalling its NP‑completeness, the authors review related work, notably the 15k‑kernel and O⁎(3.403^{3k}) algorithm by Prieto and Sloper, and the O⁎(4.61^{3k}) algorithm for 3‑set packing by Liu et al.

The first major contribution is a combinatorial “reuse theorem”. Let P be a maximal P₂‑packing of size j. If a packing of size j + 1 exists, then there is such a packing Q that contains at least 2.5 j of the 3 j vertices of P. The proof proceeds by defining a hierarchy of optimal packings: Q(1) maximizes the number of common edges with P, and among those Q(2) maximizes the number of common edges overall. Using notions of “foldable” paths, “mid‑points” and “end‑points”, the authors show that each path of P either contributes two vertices to Q or can be paired with two distinct paths of Q, guaranteeing the 2.5 j bound. This improves on the earlier 3‑set packing bound of 2 j reusable elements.

The second major contribution is a linear‑time kernelization yielding a kernel with at most 7k vertices. Starting from a maximal packing P of size t < k, the graph G \ P consists only of isolated vertices (Q₀) and isolated edges (Q₁). Two reduction rules are applied repeatedly:

1. If a path in P has two vertices each adjacent to distinct Q₀‑vertices, replace them to reduce Q₀ by two and increase Q₁ by one.
2. If a path in P has two vertices each adjacent to distinct Q₁‑edges, replace them to reduce Q₁ by two and increase the size of P by one. Each rule can be applied at most O(n) and O(k) times respectively, so the process terminates in polynomial time. When no rule applies, the numbers satisfy |Q₀| ≤ 2k − 3 and |Q₁| ≤ k − 1, implying a total vertex bound of 3t + |Q₀| + 2|Q₁| ≤ 7k − 8.

To handle the remaining instances, the authors employ crown decompositions. A “double crown” exists when an independent set I satisfies |I| ≥ 2|N(I)|, and a “fat crown” exists when a collection of disjoint K₂’s J satisfies |J| ≥ |N(J)|. Lemmas guarantee that such crowns can be found in linear time. Theorem 1 shows that if |Q₀| exceeds 2k − 3 a double crown can be extracted, while Theorem 2 shows that if |Q₁| exceeds k − 1 a fat crown can be extracted. Removing the crown reduces the problem size while preserving the existence of a size‑k packing, leading to the 7k kernel.

After kernelization, the algorithm proceeds by dynamic programming. Starting from the maximal packing of size j, the reuse theorem guarantees that at least 2.5 j vertices of the current packing can be kept when extending to size j + 1. This dramatically shrinks the search space. Repeating the extension up to k yields a running time of O⁎(2.482^{3k}), which improves upon the previous best O⁎(3.403^{3k}) and O⁎(4.61^{3k}) bounds.

The paper also derives a dual result for the total edge‑cover problem: because a total edge cover of size k can be interpreted as a P₂‑packing, the same kernelization yields a kernel with at most 1.5k vertices. Lower bounds are discussed: any kernel for P₂‑packing must contain at least 3k vertices, and a kernel for total edge cover smaller than (7/6)k would imply P = NP.

In summary, the authors combine extremal combinatorial analysis, refined crown‑based kernelization, and a novel reuse argument to obtain a significantly smaller kernel (7k) and a faster fixed‑parameter algorithm (O⁎(2.482^{3k})) for the P₂‑packing problem, advancing the state of the art in parameterized graph packing.