Directed Hamiltonicity and Out-Branchings via Generalized Laplacians

We are motivated by a tantalizing open question in exact algorithms: can we detect whether an n-vertex directed graph G has a Hamiltonian cycle in time significantly less than 2? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant 0 < λ < 1 and prime p we can count the Hamiltonian cycles modulo pb(1−λ) n 3p c in expected time less than c for a constant c < 2 that depends only on p and λ. Such an algorithm was previously known only for the case of counting modulo two [Björklund and Husfeldt, FOCS 2013]. 2. We show that we can detect a Hamiltonian cycle in O∗(3n−α(G)) time and polynomial space, where α(G) is the size of the maximum independent set in G. In particular, this yields an O∗(3n/2) time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013]. Our algorithms are based on the algebraic combinatorics of “incidence assignments” that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix–Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix–Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an O∗(2k)-time randomized algorithm for detecting out-branchings with at least k internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Björklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed k-Leaf problem, based on a non-standard monomial detection problem. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.1 Combinatorics, G.2.2 Graph Theory