Complexity of Road Coloring with Prescribed Reset Words

By the Road Coloring Theorem (Trahtman, 2008), the edges of any aperiodic directed multigraph with a constant out-degree can be colored such that the resulting automaton admits a reset word. There may also be a need for a particular reset word to be admitted. For certain words it is NP-complete to decide whether there is a suitable coloring of a given multigraph. We present a classification of all words over the binary alphabet that separates such words from those that make the problem solvable in polynomial time. We show that the classification becomes different if we consider only strongly connected multigraphs. In this restricted setting the classification remains incomplete.Comment: To be presented at LATA 201

On the Number of Synchronizing Colorings of Digraphs

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have different colors. Such a coloring corresponds naturally to an automaton. The road coloring theorem states that every primitive digraph has a synchronizing coloring. In the present paper we study how many synchronizing colorings can exist for a digraph with $n$ vertices. We performed an extensive experimental investigation of digraphs with small number of vertices. This was done by using our dedicated algorithm exhaustively enumerating all small digraphs. We also present a series of digraphs whose fraction of synchronizing colorings is equal to $1-1/k^d$, for every $d \ge 1$ and the number of vertices large enough. On the basis of our results we state several conjectures and open problems. In particular, we conjecture that $1-1/k$ is the smallest possible fraction of synchronizing colorings, except for a single exceptional example on 6 vertices for $k=2$.Comment: CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1

Primitive digraphs with large exponents and slowly synchronizing automata

We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. All these automata are tightly related to primitive digraphs with large exponent.Comment: 23 pages, 11 figures, 3 tables. This is a translation (with a slightly updated bibliography) of the authors' paper published in Russian in: Zapiski Nauchnyh Seminarov POMI [Kombinatorika i Teorija Grafov. IV], Vol. 402, 9-39 (2012), see ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v402/p009.pdf Version 2: a few typos are correcte

A linear bound on the k-rendezvous time for primitive sets of NZ matrices

A set of nonnegative matrices is called primitive if there exists a product of these matrices that is entrywise positive. Motivated by recent results relating synchronizing automata and primitive sets, we study the length of the shortest product of a primitive set having a column or a row with k positive entries, called its k-rendezvous time (k-RT}), in the case of sets of matrices having no zero rows and no zero columns. We prove that the k-RT is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We provide two upper bounds on the k-RT: the second is an improvement of the first one, although the latter can be written in closed form. We then report numerical results comparing our upper bounds on the k-RT with heuristic approximation methods.Comment: 27 pages, 10 figur

Parameterized Complexity of Synchronization and Road Coloring

First, we close the multivariate analysis of a canonical problem concerning short reset words (SYN), as it was started by Fernau et al. (2013). Namely, we prove that the problem, parameterized by the number of states, does not admit a polynomial kernel unless the polynomial hierarchy collapses. Second, we consider a related canonical problem concerning synchronizing road colorings (SRCP). Here we give a similar complete multivariate analysis. Namely, we show that the problem, parameterized by the number of states, admits a polynomial kernel and we close the previous research of restrictions to particular values of both the alphabet size and the maximum word length

Attainable Values of Reset Thresholds

An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. The reset threshold is the length of the shortest such word. We study the set RT_n of attainable reset thresholds by automata with n states. Relying on constructions of digraphs with known local exponents we show that the intervals [1, (n^2-3n+4)/2] and [(p-1)(q-1), p(q-2)+n-q+1], where 2 n, gcd(p,q)=1, belong to RT_n, even if restrict our attention to strongly connected automata. Moreover, we prove that in this case the smallest value that does not belong to RT_n is at least n^2 - O(n^{1.7625} log n / log log n). This value is increased further assuming certain conjectures about the gaps between consecutive prime numbers. We also show that any value smaller than n(n-1)/2 is attainable by an automaton with a sink state and any value smaller than n^2-O(n^{1.5}) is attainable in general case. Furthermore, we solve the problem of existence of slowly synchronizing automata over an arbitrarily large alphabet, by presenting for every fixed size of the alphabet an infinite series of irreducibly synchronizing automata with the reset threshold n^2-O(n)

On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs

We tackle the problem of the randomized generation of slowly synchronizing deterministic automata (DFAs) by generating random primitive sets of matrices. We show that when the randomized procedure is too simple the exponent of the generated sets is O(n log n) with high probability, thus the procedure fails to return DFAs with large reset threshold. We extend this result to random nondeterministic automata (NDFAs) by showing, in particular, that a uniformly sampled NDFA has both a 2-directing word and a 3-directing word of length O(n log n) with high probability. We then present a more involved randomized algorithm that manages to generate DFAs with large reset threshold and we finally leverage this finding for exhibiting new families of DFAs with reset threshold of order $\Omega(n^2/4)$.Comment: 31 pages, 9 figures. arXiv admin note: text overlap with arXiv:1805.0672

The Synchronizing Probability Function for Primitive Sets of Matrices

Motivated by recent results relating synchronizing DFAs and primitive sets, we tackle the synchronization process and the related longstanding \v{C}ern\'{y} conjecture by studying the primitivity phenomenon for sets of nonnegative matrices having neither zero-rows nor zero-columns. We formulate the primitivity process in the setting of a two-player probabilistic game and we make use of convex optimization techniques to describe its behavior. We develop a tool for approximating and upper bounding the exponent of any primitive set and supported by numerical results we state a conjecture that, if true, would imply a quadratic upper bound on the reset threshold of a new class of automata.Comment: 24 pages, 9 figures. Submitted to DLT 2018 Special Issu