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Directed Hamiltonicity and Out-Branchings via Generalized Laplacians (1607.04002v2)

Published 14 Jul 2016 in cs.DS

Abstract: 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 $2n$? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant $0<\lambda<1$ and prime $p$ we can count the Hamiltonian cycles modulo $p{\lfloor (1-\lambda)\frac{n}{3p}\rfloor}$ in expected time less than $cn$ for a constant $c<2$ that depends only on $p$ and $\lambda$. Such an algorithm was previously known only for the case of counting modulo two [Bj\"orklund and Husfeldt, FOCS 2013]. 2. We show that we can detect a Hamiltonian cycle in $O*(3{n-\alpha(G)})$ time and polynomial space, where $\alpha(G)$ is the size of the maximum independent set in $G$. In particular, this yields an $O*(3{n/2})$ 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 [Bj\"orklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed $k$-Leaf problem, based on a non-standard monomial detection problem.

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