Parameterized Complexity of Synchronization and Road Coloring

First, we close the multivariate analysis of a canonical problem concerning short reset words (SYN), as it was started by Fernau et al. (2013). Namely, we prove that the problem, parameterized by the number of states, does not admit a polynomial kernel unless the polynomial hierarchy collapses. Second, we consider a related canonical problem concerning synchronizing road colorings (SRCP). Here we give a similar complete multivariate analysis. Namely, we show that the problem, parameterized by the number of states, admits a polynomial kernel and we close the previous research of restrictions to particular values of both the alphabet size and the maximum word length

Complexity of Road Coloring with Prescribed Reset Words

By the Road Coloring Theorem (Trahtman, 2008), the edges of any aperiodic directed multigraph with a constant out-degree can be colored such that the resulting automaton admits a reset word. There may also be a need for a particular reset word to be admitted. For certain words it is NP-complete to decide whether there is a suitable coloring of a given multigraph. We present a classification of all words over the binary alphabet that separates such words from those that make the problem solvable in polynomial time. We show that the classification becomes different if we consider only strongly connected multigraphs. In this restricted setting the classification remains incomplete.Comment: To be presented at LATA 201

On the Synchronizing Probability Function and the Triple Rendezvous Time for Synchronizing Automata

Cerny's conjecture is a longstanding open problem in automata theory. We study two different concepts, which allow to approach it from a new angle. The first one is the triple rendezvous time, i.e., the length of the shortest word mapping three states onto a single one. The second one is the synchronizing probability function of an automaton, a recently introduced tool which reinterprets the synchronizing phenomenon as a two-player game, and allows to obtain optimal strategies through a Linear Program. Our contribution is twofold. First, by coupling two different novel approaches based on the synchronizing probability function and properties of linear programming, we obtain a new upper bound on the triple rendezvous time. Second, by exhibiting a family of counterexamples, we disprove a conjecture on the growth of the synchronizing probability function. We then suggest natural follow-ups towards Cernys conjecture.Comment: A preliminary version of the results has been presented at the conference LATA 2015. The current ArXiv version includes the most recent improvement on the triple rendezvous time upper bound as well as formal proofs of all the result

On the synchronization of finite state automata

Abstract: We study some problems related to the synchronization of finite state automata and the Cˇerny’s conjecture. We focus on the synchronization of small sets of states, and more specifically on the synchronization of triples. We argue that it is the most simple synchronization scenario that exhibits the intricacies of the original Cˇerny’s scenario (all states synchronization). Thus, we argue that it is complex enough to be interesting, and tractable enough to be studied via algo- rithmic tools. We use those tools to establish a long list of facts related to those issues. We observe that planar automata seems to be representative of the synchroniz- ing behavior of deterministic finite state automata. Moreover, we present strong evidence suggesting the importance of planar automata in the study of Cˇerny’s conjecture. We also study synchronization games played on planar automata. We prove that recognizing the planar games that can be won by the synchronizer is a co-NP hard problem. We prove some additional results indicating that pla- nar games are as hard as nonplanar games. Those results amount to show that planar automata are representative of the intricacies of automata synchronization.Doctorad

An Experimental Comparison of Partitioning Strategies in Distributed Graph Processing

In this paper, we study the problem of choosing among partitioning strategies in distributed graph processing systems.To this end, we evaluate and characterize both the performance and resource usage of different partitioning strategies under various popular distributed graph processing systems, applications, input graphs, and execution environments. Through our experiments, we found that no single partitioning strategy is the best fit for all situations, and that the choice of partitioning strategy has a significant effect on resource usage and application run-time. Our experiments demonstrate that the choice of partitioning strategy depends on (1) the degree distribution of input graph, (2) the type and duration of the application, and (3) the cluster size. Based on our results, we present rules of thumb to help users pick the best partitioning strategy for their particular use cases. We present results from each system, as well as from all partitioning strategies implemented in one common system (PowerLyra).Ope

Most Classic Problems Remain NP-hard on Relative Neighborhood Graphs and their Relatives

Proximity graphs have been studied for several decades, motivated by applications in computational geometry, geography, data mining, and many other fields. However, the computational complexity of classic graph problems on proximity graphs mostly remained open. We now study 3-Colorability, Dominating Set, Feedback Vertex Set, Hamiltonian Cycle, and Independent Set on the proximity graph classes relative neighborhood graphs, Gabriel graphs, and relatively closest graphs. We prove that all of the problems remain NP-hard on these graphs, except for 3-Colorability and Hamiltonian Cycle on relatively closest graphs, where the former is trivial and the latter is left open. Moreover, for every NP-hard case we additionally show that no $2^{o(n^{1/4})}$-time algorithm exists unless the ETH fails, where n denotes the number of vertices

Relating Structure and Power: Comonadic Semantics for Computational Resources

Combinatorial games are widely used in finite model theory, constraint satisfaction, modal logic and concurrency theory to characterize logical equivalences between structures. In particular, Ehrenfeucht-Fraisse games, pebble games, and bisimulation games play a central role. We show how each of these types of games can be described in terms of an indexed family of comonads on the category of relational structures and homomorphisms. The index k is a resource parameter which bounds the degree of access to the underlying structure. The coKleisli categories for these comonads can be used to give syntax-free characterizations of a wide range of important logical equivalences. Moreover, the coalgebras for these indexed comonads can be used to characterize key combinatorial parameters: tree-depth for the Ehrenfeucht-Fraisse comonad, tree-width for the pebbling comonad, and synchronization-tree depth for the modal unfolding comonad. These results pave the way for systematic connections between two major branches of the field of logic in computer science which hitherto have been almost disjoint: categorical semantics, and finite and algorithmic model theory.Comment: To appear in Proceedings of Computer Science Logic 201

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum