Strong inapproximability of the shortest reset word

The \v{C}ern\'y conjecture states that every $n$-state synchronizing automaton has a reset word of length at most $(n-1)^2$. We study the hardness of finding short reset words. It is known that the exact version of the problem, i.e., finding the shortest reset word, is NP-hard and coNP-hard, and complete for the DP class, and that approximating the length of the shortest reset word within a factor of $O(\log n)$ is NP-hard [Gerbush and Heeringa, CIAA'10], even for the binary alphabet [Berlinkov, DLT'13]. We significantly improve on these results by showing that, for every $\epsilon>0$, it is NP-hard to approximate the length of the shortest reset word within a factor of $n^{1-\epsilon}$. This is essentially tight since a simple $O(n)$-approximation algorithm exists.Comment: extended abstract to appear in MFCS 201

Checking Whether an Automaton Is Monotonic Is NP-complete

An automaton is monotonic if its states can be arranged in a linear order that is preserved by the action of every letter. We prove that the problem of deciding whether a given automaton is monotonic is NP-complete. The same result is obtained for oriented automata, whose states can be arranged in a cyclic order. Moreover, both problems remain hard under the restriction to binary input alphabets.Comment: 13 pages, 4 figures. CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_2

Fluid passage-time calculation in large Markov models

Recent developments in the analysis of large Markov models facilitate the fast approximation of transient characteristics of the underlying stochastic process. So-called fluid analysis makes it possible to consider previously intractable models whose underlying discrete state space grows exponentially as model components are added. In this work, we show how fluid approximation techniques may be used to extract passage-time measures from performance models. We focus on two types of passage measure: passage-times involving individual components; as well as passage-times which capture the time taken for a population of components to evolve. Specifically, we show that for models of sufficient scale, passage-time distributions can be well approximated by a deterministic fluid-derived passage-time measure. Where models are not of sufficient scale, we are able to generate approximate bounds for the entire cumulative distribution function of these passage-time random variables, using moment-based techniques. Finally, we show that for some passage-time measures involving individual components the cumulative distribution function can be directly approximated by fluid techniques

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