Path Cover, Hamiltonicity, and Independence Number: An FPT Perspective

TL;DR

The following algorithmic extension of Gallai--Milgram's theorem for undirected graphs is proved: determining whether an undirected graph can be covered by fewer than \alpha(G) - k vertex-disjoint paths is fixed-parameter tractable (FPT) when parameterized by k.

Abstract

The classic theorem of Gallai and Milgram (1960) asserts that for every graph G, the vertex set of G can be partitioned into at most α(G) vertex-disjoint paths, where α(G) is the maximum size of an independent set in G. The proof of Gallai--Milgram's theorem is constructive and yields a polynomial-time algorithm that computes a covering of G by at most α(G) vertex-disjoint paths. We prove the following algorithmic extension of Gallai--Milgram's theorem for undirected graphs: determining whether an undirected graph can be covered by fewer than α(G) - k vertex-disjoint paths is fixed-parameter tractable (FPT) when parameterized by k. More precisely, we provide an algorithm that, for an n-vertex graph G and an integer parameter k \ge 1, runs in time 2^{k^{O(k^4)}} \cdot n^{O(1)}, and outputs a path cover P of G. Furthermore, it: - either correctly reports that P is a minimum-size path cover, - or outputs, together with P, an independent set of size |P| + k certifying that P contains at most α(G) - k paths. A key subroutine in our algorithm is an FPT algorithm, parameterized by α(G), for deciding whether G contains a Hamiltonian path. This result is of independent interest -- prior to our work, no polynomial-time algorithm for deciding Hamiltonicity was known even for graphs with independence number at most 3. Moreover, the techniques we develop apply to a wide array of problems in undirected graphs, including Hamiltonian Cycle, Path Cover, Largest Linkage, and Topological Minor Containment. We show that all these problems are FPT when parameterized by the independence number of the graph. Notably, the independence number parameterization, which describes graph's density, departs from the typical flow of research in parameterized complexity, which focuses on parameters describing graph's sparsity, like treewidth or vertex cover.

Figures (3)

• Figure 1: Example of a three-path transformation. In part (a), $P_2$ is a usual path, while $P_1$ and $P_3$ are special paths. Dashed lines represent some of the absent edges. Part (b) represents the same paths connected by edges required for the transformation. Part (c) shows how new paths are formed. $P_1$ and $P_3$ become usual paths, while $P_2$ can become special or even disappear completely.
• Figure 2: Illustration of transformations performed in (a) \ref{['rrule:merge_special_paths']} and (b) \ref{['rrule:two_specials_to_one']}.. Paths before the transformations are horizontal. Paths after the transformations are highlighted in bold and have pairwise-distinct colors. Dashed lines represent absence of an edge. Snake lines represent paths instead of single edges.
• Figure 3: An illustration to the proof of Lemma \ref{['lemma:model_sep']}: A graph $G$ with a TM-model of $H$, the graph $M_W$ of the cut descriptor corresponding to the model, and the graph $H$, the model of which is sought. The images of the vertices of $H$ in $G$ and $M_W$ that are fixed by $f$ and $f_W$, respectively, are indicated by the matching node shapes.

Theorems & Definitions (38)

• Theorem 1
• Definition 1: TM-model and TM-embedding
• Definition 2: List TM-embedding
• Theorem 2
• Proposition 1
• Theorem 2
• Lemma 1
• proof
• Claim 1
• Lemma 2