Computer Science

We study Boolean networks which are simple spatial models of the highly conserved Delta-Notch system. The models assume the inhibition of Delta in each cell by Notch in the same cell, and the activation of Notch in presence of Delta in surrounding cells. We consider fully asynchronous dynamics over undirected graphs representing the neighbour relation between cells. In this framework, one can show that all attractors are fixed points for the system, independently of the neighbour relation, for instance by using known properties of simplified versions of the models, where only one species per cell is defined. The fixed points correspond to the so-called fine-grained "patterns" that emerge in discrete and continuous modelling of lateral inhibition. We study the reachability of fixed points, giving a characterisation of the trap spaces and the basins of attraction for both the full and the simplified models. In addition, we use a characterisation of the trap spaces to investigate the robustness of patterns to perturbations. The results of this qualitative analysis can complement and guide simulation-based approaches, and serve as a basis for the investigation of more complex mechanisms.

We show that the number of length-n words over a k-letter alphabet having no even palindromic prefix is the same as the number of length-n unbordered words, by constructing an explicit bijection between the two sets. A slightly different but analogous result holds for those words having no odd palindromic prefix. Using known results on borders, we get an asymptotic enumeration for the number of words having no even (resp., odd) palindromic prefix . We obtain an analogous result for words having no nontrivial palindromic prefix. Finally, we obtain similar results for words having no square prefix, thus proving a 2013 conjecture of Chaffin, Linderman, Sloane, and Wilks.

A fundamental problem in network science is the normalization of the topological or physical distance between vertices, that requires understanding the range of variation of the unnormalized distances. Here we investigate the limits of the variation of the physical distance in linear arrangements of the vertices of trees. In particular, we investigate various problems on the sum of edge lengths in trees of a fixed size: the minimum and the maximum value of the sum for specific trees, the minimum and the maximum in classes of trees (bistar trees and caterpillar trees) and finally the minimum and the maximum for any tree. We establish some foundations for research on optimality scores for spatial networks in one dimension.

Budget Minimization is a scheduling problem with precedence constraints, i.e., a scheduling problem on a partially ordered set of jobs (N,⊴) . A job j∈N is available for scheduling, if all jobs i∈N with i⊴j are completed. Further, each job j∈N is assigned real valued costs c j , which can be negative or positive. A schedule is an ordering j 1 ,…, j |N| of all jobs in N . The budget of a schedule is the external investment needed to complete all jobs, i.e., it is max l∈{0,…,|N|} ∑ 1≤k≤l c j k . The goal is to find a schedule with minimum budget. Rafiey et al. (2015) showed that Budget Minimization is NP-hard following from a reduction from a molecular folding problem. We extend this result and prove that it is NP-hard to α(N) -approximate the minimum budget even on bipartite partial orders. We present structural insights that lead to arguably simpler algorithms and extensions of the results by Rafiey et al. (2015). In particular, we show that there always exists an optimal solution that partitions the set of jobs and schedules each subset independently of the other jobs. We use this structural insight to derive polynomial-time algorithms that solve the problem to optimality on series-parallel and convex bipartite partial orders.

A k -colouring of a graph G is an assignment of at most k colours to the vertices of G so that adjacent vertices are assigned different colours. The reconfiguration graph of the k -colourings, R k (G) , is the graph whose vertices are the k -colourings of G and two colourings are joined by an edge in R k (G) if they differ in colour on exactly one vertex. For a k -colourable graph G , we investigate the connectivity and diameter of R k+1 (G) . It is known that not all weakly chordal graphs have the property that R k+1 (G) is connected. On the other hand, R k+1 (G) is connected and of diameter O( n 2 ) for several subclasses of weakly chordal graphs such as chordal, chordal bipartite, and P 4 -free graphs. We introduce a new class of graphs called OAT graphs that extends the latter classes and in fact extends outside the class of weakly chordal graphs. OAT graphs are built from four simple operations, disjoint union, join, and the addition of a clique or comparable vertex. We prove that if G is a k -colourable OAT graph then R k+1 (G) is connected with diameter O( n 2 ) . Furthermore, we give polynomial time algorithms to recognize OAT graphs and to find a path between any two colourings in R k+1 (G) .

A procedure called \textit{graph burning} was introduced to facilitate the modelling of spread of an alarm, a social contagion, or a social influence or emotion on graphs and networks. Graph burning runs on discrete time-steps (or rounds). At each step t , first (a) an unburned vertex is burned (as a \textit{fire source}) from "outside", and then (b) the fire spreads to vertices adjacent to the vertices which are burned till step t−1 . This process stops after all the vertices of G have been burned. The aim is to burn all the vertices in a given graph in minimum time-steps. The least number of time-steps required to burn a graph is called its \textit{burning number}. The less the burning number is, the faster a graph can be burned. Burning a general graph optimally is an NP-Complete problem. It has been proved that optimal burning of path forests, spider graphs, and trees with maximum degree three is NP-Complete. We study the \textit{graph burning problem} on several sub-classes of \textit{geometric graphs}. We show that burning interval graphs (Section 7.1, Theorem 7.1), permutation graphs (Section 7.2, Theorem 7.2) and disk graphs (Section 7.3, Theorem 7.3) optimally is NP-Complete. In addition, we opine that optimal burning of general graphs (Section 9.2, Conjecture 9.1) cannot be approximated better than 3-approximation factor.

We introduce the rendezvous game with adversaries. In this game, two players, {\sl Facilitator} and {\sl Disruptor}, play against each other on a graph. Facilitator has two agents, and Disruptor has a team of k agents located in some vertices of the graph. They take turns in moving their agents to adjacent vertices (or staying). Facilitator wins if his agents meet in some vertex of the graph. The goal of Disruptor is to prevent the rendezvous of Facilitator's agents. Our interest is to decide whether Facilitator can win. It appears that, in general, the problem is PSPACE-hard and, when parameterized by k , co-W[2]-hard. Moreover, even the game's variant where we ask whether Facilitator can ensure the meeting of his agents within ? steps is co-NP-complete already for ?=2 . On the other hand, for chordal and P 5 -free graphs, we prove that the problem is solvable in polynomial time. These algorithms exploit an interesting relation of the game and minimum vertex cuts in certain graph classes. Finally, we show that the problem is fixed-parameter tractable parameterized by both the graph's neighborhood diversity and ? .

We study the problem of finding contraction orderings on tensor networks for physical simulations using a syncretic abstract data type, the contraction-tree , and explain its connection to temporal and spatial measures of tensor contraction computational complexity (nodes express time; arcs express space). We have implemented the Ratcatcher of Seymour and Thomas for determining the carving-width of planar networks, in order to offer experimental evidence that this measure of spatial complexity makes a generally effective heuristic for limiting their total contraction time.

Pattern avoiding machines were introduced recently by Claesson, Cerbai and Ferrari as a particular case of the two-stacks in series sorting device. They consist of two restricted stacks in series, ruled by a right-greedy procedure and the stacks avoid some specified patterns. Some of the obtained results have been further generalized to Cayley permutations by Cerbai, specialized to particular patterns by Defant and Zheng, or considered in the context of functions over the symmetric group by Berlow. In this work we study pattern avoiding machines where the first stack avoids a pair of patterns of length 3 and investigate those pairs for which sortable permutations are counted by the (binomial transform of the) Catalan numbers and the Schröder numbers.

Boolean nested canalizing functions (NCFs) have important applications in molecular regulatory networks, engineering and computer science. In this paper, we study their certificate complexity. For both Boolean values b∈{0,1} , we obtain a formula for b -certificate complexity and consequently, we develop a direct proof of the certificate complexity formula of an NCF. Symmetry is another interesting property of Boolean functions and we significantly simplify the proofs of some recent theorems about partial symmetry of NCFs. We also describe the algebraic normal form of s -symmetric NCFs. We obtain the general formula of the cardinality of the set of n -variable s -symmetric Boolean NCFs for s=1,…,n . In particular, we enumerate the strongly asymmetric Boolean NCFs.