351 research outputs found

    Arc-Disjoint Paths and Trees in 2-Regular Digraphs

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    An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected spanning subdigraph of D in which every vertex x != s has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the complexity of the following two problems: 1) Given a 2-regular digraph DD, decide if it contains two arc-disjoint branchings B^+_u, B^-_v. 2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u such that D remains connected after removing the arcs of B^+_u. Both problems are NP-complete for general digraphs. We prove that the first problem remains NP-complete for 2-regular digraphs, whereas the second problem turns out to be polynomial when we do not prescribe the root in advance. We also prove that, for 2-regular digraphs, the latter problem is in fact equivalent to deciding if DD contains two arc-disjoint out-branchings. We generalize this result to k-regular digraphs where we want to find a number of pairwise arc-disjoint spanning trees and out-branchings such that there are k in total, again without prescribing any roots.Comment: 9 pages, 7 figure

    Directed Hamiltonicity and Out-Branchings via Generalized Laplacians

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    We are motivated by a tantalizing open question in exact algorithms: can we detect whether an nn-vertex directed graph GG has a Hamiltonian cycle in time significantly less than 2n2^n? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant 0<λ<10<\lambda<1 and prime pp we can count the Hamiltonian cycles modulo p⌊(1−λ)n3p⌋p^{\lfloor (1-\lambda)\frac{n}{3p}\rfloor} in expected time less than cnc^n for a constant c<2c<2 that depends only on pp and λ\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∗(3n−α(G))O^*(3^{n-\alpha(G)}) time and polynomial space, where α(G)\alpha(G) is the size of the maximum independent set in GG. In particular, this yields an O∗(3n/2)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)O^*(2^k)-time randomized algorithm for detecting out-branchings with at least kk 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 kk-Leaf problem, based on a non-standard monomial detection problem

    Independent branchings in acyclic diagraphs

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    AbstractLet D be a finite directed acyclic multigraph and t be a vertex of D such that for each other vertex x of D, there are n pairwise openly disjoint paths in D from x to t. It is proved that there exist n spanning trees B1,…,Bn in D directed toward t such that for each vertex x ≠ t of D, the n paths from x to t in B1,…,Bn are pairwise openly disjoint
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