Arc-Disjoint Paths and Trees in 2-Regular Digraphs

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 $D$, 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 $D$ 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

A vizing-type theorem for matching forests

A well known Theorem of Vizing states that one can colour the edges of a graph by $\Delta +\alpha$ colours, such that edges of the same colour form a matching. Here, $\Delta$ denotes the maximum degree of a vertex, and $\alpha$ the maximum multiplicity of an edge in the graph. An analogue of this Theorem for directed graphs was proved by Frank. It states that one can colour the arcs of a digraph by $\Delta +\alpha$ colours, such that arcs of the same colour form a branching. For a digraph, $\Delta$ denotes the maximum indegree of a vertex, and $\alpha$ the maximum multiplicity of an arc

Non ambiguous structures on 3-manifolds and quantum symmetry defects

The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic oriented cusped $3$-manifolds can be split in a "symmetrization" factor and a "reduced" state sum. We show that these factors are invariants on their own, that we call "symmetry defects" and "reduced QHI", provided the manifolds are endowed with an additional "non ambiguous structure", a new type of combinatorial structure that we introduce in this paper. A suitably normalized version of the symmetry defects applies to compact $3$-manifolds endowed with $PSL_2(\mathbb{C})$-characters, beyond the case of cusped manifolds. Given a manifold $M$ with non empty boundary, we provide a partial "holographic" description of the non-ambiguous structures in terms of the intrinsic geometric topology of $\partial M$. Special instances of non ambiguous structures can be defined by means of taut triangulations, and the symmetry defects have a particularly nice behaviour on such "taut structures". Natural examples of taut structures are carried by any mapping torus with punctured fibre of negative Euler characteristic, or by sutured manifold hierarchies. For a cusped hyperbolic $3$-manifold $M$ which fibres over $S^1$, we address the question of determining whether the fibrations over a same fibered face of the Thurston ball define the same taut structure. We describe a few examples in detail. In particular, they show that the symmetry defects or the reduced QHI can distinguish taut structures associated to different fibrations of $M$. To support the guess that all this is an instance of a general behaviour of state sum invariants of 3-manifolds based on some theory of 6j-symbols, finally we describe similar results about reduced Turaev-Viro invariants.Comment: 58 pages, 32 figures; exposition improved, ready for publicatio

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 $2^n$? 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 $c^n$ 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 $O^*(2^k)$-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