In this paper, we consider a class of scheduling problems that are among the fundamentaloptimization problems in operations research. More specifically, we deal with a particularversion called job shop scheduling with unit length tasks. Using theresults of Hromkovič, Mömke, Steinhöfel, and Widmayer presented in their work JobShop Scheduling with Unit Length Tasks: Bounds and Algorithms, we analyze theproblem setting for 2 jobs with an unequal number of tasks. We contribute a deterministicalgorithm which achieves a vanishing delay in certain cases and a randomized algorithmwith a competitive ratio tending to 1. Furthermore, we investigate the problem with 3 jobsand we construct a randomized online algorithm which also has a competitive ratio tendingto 1.

Let F3,3 be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. We show that for all n ≥ 6, the maximum number of edges in an F3,3-free 3-graph on n vertices is . This sharpens results of Zhou [9] and of the second author and Rödl [7].

An infinite permutation α is a linear ordering of N. We study propertiesof infinite permutations analogous to those of infinite words, and show some resemblancesand some differences between permutations and words. In this paper, we try to extend topermutations the notion of automaticity. As we shall show, the standard definitions whichare equivalent in the case of words are not equivalent in the context of permutations. Weinvestigate the relationships between these definitions and prove that they constitute achain of inclusions. We also construct and study an automaton generating the Thue-Morsepermutation.

The fixed point submonoid of an endomorphism of a free product of a free monoid andcyclic groups is proved to be rational using automata-theoretic techniques. Maslakova’sresult on the computability of the fixed point subgroup of a free group automorphism isgeneralized to endomorphisms of free products of a free monoid and a free group which areautomorphisms of the maximal subgroup.

Multiwords are words in which a single symbol can be replaced by a nonempty set of symbols. They extend the notion of partial words. A word w is certain in a multiword M if it occurs in every word that can be obtained by selecting one single symbol among the symbols provided in each position of M. Motivated by a problem on incomplete databases, we investigate a variant of the pattern matching problem which is to decide whether a word w is certain in a multiword M. We study the language CERTAIN(w) of multiwords in which w is certain. We show that this regular language is aperiodic for three large families of words. We also show its aperiodicity in the case of partial words over an alphabet with at least three symbols.

For an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes.

We study the avoidance of Abelian powers of words and consider three reasonablegeneralizations of the notion of Abelian power to fractional powers. Our main goal is tofind an Abelian analogue of the repetition threshold, i.e., a numericalvalue separating k-avoidable and k-unavoidable Abelianpowers for each size k of the alphabet. We prove lower bounds for theAbelian repetition threshold for large alphabets and all definitions of Abelian fractionalpower. We develop a method estimating the exponential growth rate of Abelian-power-freelanguages. Using this method, we get non-trivial lower bounds for Abelian repetitionthreshold for small alphabets. We suggest that some of the obtained bounds are the exactvalues of Abelian repetition threshold. In addition, we provide upper bounds for thegrowth rates of some particular Abelian-power-free languages.