Our paper is on indeterminate strings, which are important for their applicability in bioinformatics. (They have been considered, for example, in Christodoulakis 2015 and Helling 2017.)
An interesting feature of indeterminate strings is the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimal alphabet size of indeterminates representing any given undirected graph corresponds to the size of the minimal clique cover of this graph. This paper first considers a related problem proposed in Helling 2017: characterize $latex\Theta_n(m)$, which is the size of the largest possible minimal clique cover (i.e., an exact upper bound), and hence alphabet size of the corresponding indeterminate, of any graph on vertices and edges. We provide improvements to the known upper bound for . Helling 2017 also presents an algorithm which finds clique covers in polynomial time. We build on this result with an original heuristic for vertex sorting which significantly improves their algorithm’s results, particularly in dense graphs.
This work was the result of building on Helling 2017 (see this post) and of a year of research undertaken by Ryan McIntyre under my (Michael Soltys) supervision at the California State University Channel Islands.
Very happy that our paper Computing covers from matchings with permutations, with Ariel Fernández and Ryszard Janicki, is going to appear in the International Journal of Computer Applications.
We present a matrix permutation algorithm for computing a minimal vertex cover from a maximal matching in a bipartite graph. Our algorithm is linear time and linear space, and provides an interesting perspective on a well known problem. Unlike most algorithms, it does not use the concept of alternating paths, and it is formulated entirely in terms of combinatorial operations on a binary matrix. The algorithm relies on permutations of rows and columns of a 0-1 matrix which encodes a bipartite graph together with its maximal matching. This problem has many important applications such as network switches which essentially compute maximal matchings between their incoming and outgoing ports.
Abstract: In this study, we provide mathematical and practice-driven justification for using [0, 1] normalization of inconsistency indicators in pairwise comparisons. The need for normalization, as well as problems with the lack of normalization, are presented. A new type of paradox of infinity is described.
Accepted for publication in the International Journal of Approximate Reasoning, April 2017.
A new paper:On normalization of inconsistency indicators in pairwise comparisons, by W.W. Koczkodaj, J.-P. Magnot, J. Mazurek, J.F. Peters, H. Rakhshani, M. Soltys, D. Strzałka, J. Szybowski and A. Tozzi.
Abstract: In this study, we provide mathematical and practice-driven justification for using [0,1] normalization of inconsistency indicators in pairwise comparisons. The need for normalization, as well as problems with the lack of normalization, are presented. A new type of paradox of infinity is described.
In a new paper, Square-free strings over alphabet lists, my PhD student Neerja Mhaskar and I, solve an open problem that was posed in A new approach to non repetitive sequences, by Jaroslaw Grytczuk, Jakub Kozik, and Pitor Micek, in arXiv:1103.3809, December 2010.
The problem can be stated as follows: Given an alphabet list $L=L_1,\ldots,L_n$, where $|L_i|=3$ and $0 \leq i \leq n$, can we always find a square-free string, $W=W_1W_2 \ldots W_n$, where $W_i\in L_i$? We show that this is indeed the case. We do so using an “offending suffix” characterization of forced repetitions, and a counting, non-constructive, technique. We discuss future directions related to finding a constructive solution, namely a polytime algorithm for generating square-free words over such lists.
I wrote a paper about finite games which I presented at Computability in Europe in Athens, 2009. Now it turns out, that the Vienna school of Economics, Wirtschaftsuniversität Wien, has been citing it repeatedly in the last few months, in particular Aurélien Fichet de Clairfontaine. It is very satisfying to see research being picked up by other areas!
Abstract: A word is non-repetitive if it does not contain a subword of the form vv. Given a list of alphabets L = L1, L2, . . . , Ln, we investigate the question of generating non-repetitive words w = w1w2 . . . wn, such that the symbol wi is a letter in the alphabet Li. This problem has been studied by several authors (e.g., [GKM10], [Sha09]), and it is a natural extension of the original problem posed and solved by A. Thue. While we do not solve the problem in its full generality, we show that such strings exist over many classes of lists. We also suggest techniques for tackling the problem, ranging from online algorithms, to combinatorics over 0-1 matrices, and to proof complexity. Finally, we show some properties of the extension of the problem to abelian squares.
Abstract: We show that shuffle, the problem of determining whether a string w can be composed from an order preserving shuffle of strings x and y, is not in AC0, but it is in AC1. The fact that shuffle is not in AC0 is shown by a reduction of parity to shuffle and invoking the seminal result of Furst et al., while the fact that it is in AC1 is implicit in the results of Mansfield. Together, the two results provide a lower and upper bound on the complexity of this combinatorial problem. We also explore an interesting relationship between graphs and the shuffle problem, namely what types of graphs can be represented with strings exhibiting the anti-Monge condition.