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 \alpha(G) vertex-disjoint paths, where \alpha(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 \alpha(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 \alpha(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 \alpha(G) - k paths. A key subroutine in our algorithm is an FPT algorithm, parameterized by \alpha(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.

翻译：Gallai和Milgram的经典定理（1960年）断言：对于任意图G，其顶点集可被划分为至多α(G)条顶点不相交的路径，其中α(G)表示G中最大独立集的规模。Gallai-Milgram定理的证明是构造性的，并产生了一个多项式时间算法，能够计算出用至多α(G)条顶点不相交路径对G的覆盖。本文针对无向图证明了Gallai-Milgram定理的如下算法扩展：判定一个无向图能否被少于α(G)-k条顶点不相交路径覆盖的问题，在以k为参数时是固定参数可处理（FPT）的。更精确地说，我们提出一个算法，对于n个顶点的图G和整数参数k≥1，该算法在2^{k^{O(k^4)}}·n^{O(1)}时间内运行，并输出G的一个路径覆盖P。此外，该算法：- 要么正确报告P是最小规模的路径覆盖，- 要么在输出P的同时，输出一个规模为|P|+k的独立集，以证明P至多包含α(G)-k条路径。我们算法中的一个关键子程序是一个以α(G)为参数的FPT算法，用于判定G是否包含哈密顿路径。这一结果具有独立意义——在我们的工作之前，即使是对于独立数至多为3的图，也尚未知存在判定哈密顿性的多项式时间算法。此外，我们所发展的技术适用于无向图中的一系列广泛问题，包括哈密顿回路、路径覆盖、最大连通子图以及拓扑子式包含。我们证明，在以图的独立数为参数时，所有这些问题都是FPT的。值得注意的是，独立数这一参数化方式描述了图的稠密性，这与参数复杂性研究中通常关注描述图稀疏性的参数（如树宽或顶点覆盖）的主流方向有所不同。