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

The classic theorem of Gallai and Milgram (1960) generalizes several fundamental results in Graph Theory, such as Dilworth's theorem on posets and K\H{o}nig's theorem on matchings in bipartite graphs. The theorem 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 the Gallai-Milgram theorem is constructive and yields a polynomial-time algorithm that computes a covering of G by at most \alpha(G) vertex-disjoint paths. While the Gallai-Milgram theorem is tight, it was not known prior to our work whether deciding if a graph G could be covered by fewer than \alpha(G) vertex-disjoint paths can be done in polynomial time. We resolve this question by proving the following algorithmic extension of the Gallai-Milgram theorem for undirected graphs: There is an algorithm that, for an n-vertex graph G and an integer parameter k \ge 1, runs in time 2^{2^{O(k^4\log{k})}} \cdot n^{O(1)} and outputs a path cover P of G together with - a correct conclusion that P is a minimum-size path cover, or - an independent set of size |P|+k, certifying that P contains at most \alpha(G) - k paths. The proof of our algorithmic extension of the Gallai-Milgram theorem is non-trivial and builds on several novel algorithmic ideas. One of the key subroutines 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 three. Moreover, the algorithmic techniques we develop apply to a wide array of problems in undirected graphs, including Hamiltonian Cycle, Path Cover, Largest Linkage, and Topological Minor Containment.