Redundancy in Logic II: 2CNF and Horn Propositional Formulae

We report complexity results about redundancy of formulae in 2CNF form. We first consider the problem of checking redundancy and show some algorithms that are slightly better than the trivial one. We then analyze problems related to finding irredundant equivalent subsets (I.E.S.) of a given set. The concept of cyclicity proved to be relevant to the complexity of these problems. Some results about Horn formulae are also shown.Comment: Corrected figures on Theorem 10; added and modified some reference

Regular Languages meet Prefix Sorting

Indexing strings via prefix (or suffix) sorting is, arguably, one of the most successful algorithmic techniques developed in the last decades. Can indexing be extended to languages? The main contribution of this paper is to initiate the study of the sub-class of regular languages accepted by an automaton whose states can be prefix-sorted. Starting from the recent notion of Wheeler graph [Gagie et al., TCS 2017]-which extends naturally the concept of prefix sorting to labeled graphs-we investigate the properties of Wheeler languages, that is, regular languages admitting an accepting Wheeler finite automaton. Interestingly, we characterize this family as the natural extension of regular languages endowed with the co-lexicographic ordering: when sorted, the strings belonging to a Wheeler language are partitioned into a finite number of co-lexicographic intervals, each formed by elements from a single Myhill-Nerode equivalence class. Moreover: (i) We show that every Wheeler NFA (WNFA) with $n$ states admits an equivalent Wheeler DFA (WDFA) with at most $2n-1-|\Sigma|$ states that can be computed in $O(n^3)$ time. This is in sharp contrast with general NFAs. (ii) We describe a quadratic algorithm to prefix-sort a proper superset of the WDFAs, a $O(n\log n)$-time online algorithm to sort acyclic WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. By contribution (i), our algorithms can also be used to index any WNFA at the moderate price of doubling the automaton's size. (iii) We provide a minimization theorem that characterizes the smallest WDFA recognizing the same language of any input WDFA. The corresponding constructive algorithm runs in optimal linear time in the acyclic case, and in $O(n\log n)$ time in the general case. (iv) We show how to compute the smallest WDFA equivalent to any acyclic DFA in nearly-optimal time.Comment: added minimization theorems; uploaded submitted version; New version with new results (W-MH theorem, linear determinization), added author: Giovanna D'Agostin

Subset sum problems with digraph constraints

We introduce and study optimization problems which are related to the well-known Subset Sum problem. In each new problem, a node-weighted digraph is given and one has to select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints called digraph constraints and maximality need to be satisfied. The digraph constraint imposes that a node must belong to the solution if at least one of its predecessors is in the solution. An alternative of this constraint says that a node must belong to the solution if all its predecessors are in the solution. The maximality constraint ensures that no superset of a feasible solution is also feasible. The combination of these constraints provides four problems. We study their complexity and present some approximation results according to the type of input digraph, such as directed acyclic graphs and oriented trees

Algorithms, haplotypes and phylogenetic networks

Preface. Before I started my PhD in computational biology in 2005, I had never even heard of this term. Now, almost four years later, I think I have some idea of what is meant by it. One of the goals of my PhD was to explore different topics within computational biology and to see where the biggest opportunities for discrete/combinatorial mathematicians could be found. Roughly speaking, the first two years of my PhD I focussed mainly on problems related to haplotyping and genome rearrangements and the last two years on phylogenetic networks. I must say I really enjoyed learning so much about both mathematics and biology. It was especially amazing to learn how exact, theoretical mathematics can be used to solve complex, practical problems from biology. The topics I studied clearly show how extremely useful mathematics can be for biology. But I also learned that there are many more interesting topics in computational biology than the ones that I could study so far. The number of opportunities for discrete mathematicians is absolutely immense. I did not include my studies on genome rearrangements in this thesis, because my most interesting results [Hur07a; Hur07b] are not directly related to biology. This work is nevertheless interesting to mathematicians and I recommend them to read it. I can certainly conclude that also in this field there is a vast number of opportunities for mathematicians and that the topic genome rearrangements provides numerous beautiful mathematical problems. I could never have written this thesis without a great amount of help from many different people. I want to thank my supervisors Leen Stougie and Judith Keijsper for guiding me, for helping me, for correcting my mistakes, for supplying ideas and for the enjoyable time I had while working with them. I also want to thank the Dutch BSIK/BRICKS project for funding my research and Gerhard Woeginger for giving me the opportunity to work in his group and being my second promotor. I want to thank Jens Stoye and Julia Zakotnik for the work we did together and for the great time I had in Bielefeld. I want to thank Ferry Hagen and Teun Boekhout for helping me to make my work relevant for "real" biology. I also want to thank John Tromp, Rudi Cilibrasi, Cor Hurkens and all others I worked with during my PhD. I want to thank Erik de Vink and Mike Steel for reading and commenting my thesis. I want to thank my colleagues from the Combinatorial Optimisation group at the Technische Universiteit Eindhoven for the pleasant working conditions and the fun things we did besides work. I especially want to thank Matthias Mnich, not only a great colleague but also a good friend, for all his ideas, his humour and our good and fruitful cooperation. I also want to thank Steven Kelk. I must say that I was very lucky to work with Steven during my PhD. He introduced me to problems, had an enormous amount of ideas, found the critical mistakes in my proofs and made my PhD a success both in terms of results and in terms of enjoying work. Finally, I want to thank Conno Hendriksen and Bas Heideveld for assisting me during my PhD defence and I want to thank them and all my other friends and family for helping me with everything in my life but research