5 research outputs found
Improved Algorithms for Parity and Streett objectives
The computation of the winning set for parity objectives and for Streett
objectives in graphs as well as in game graphs are central problems in
computer-aided verification, with application to the verification of closed
systems with strong fairness conditions, the verification of open systems,
checking interface compatibility, well-formedness of specifications, and the
synthesis of reactive systems. We show how to compute the winning set on
vertices for (1) parity-3 (aka one-pair Streett) objectives in game graphs in
time and for (2) k-pair Streett objectives in graphs in time
. For both problems this gives faster algorithms for dense
graphs and represents the first improvement in asymptotic running time in 15
years
Conditionally Optimal Algorithms for Generalized B\"uchi Games
Games on graphs provide the appropriate framework to study several central
problems in computer science, such as the verification and synthesis of
reactive systems. One of the most basic objectives for games on graphs is the
liveness (or B\"uchi) objective that given a target set of vertices requires
that some vertex in the target set is visited infinitely often. We study
generalized B\"uchi objectives (i.e., conjunction of liveness objectives), and
implications between two generalized B\"uchi objectives (known as GR(1)
objectives), that arise in numerous applications in computer-aided
verification. We present improved algorithms and conditional super-linear lower
bounds based on widely believed assumptions about the complexity of (A1)
combinatorial Boolean matrix multiplication and (A2) CNF-SAT. We consider graph
games with vertices, edges, and generalized B\"uchi objectives with
conjunctions. First, we present an algorithm with running time , improving the previously known and worst-case bounds. Our algorithm is optimal for dense graphs under (A1).
Second, we show that the basic algorithm for the problem is optimal for sparse
graphs when the target sets have constant size under (A2). Finally, we consider
GR(1) objectives, with conjunctions in the antecedent and
conjunctions in the consequent, and present an -time algorithm, improving the previously known -time algorithm for
Semiring Provenance for B\"uchi Games: Strategy Analysis with Absorptive Polynomials
This paper presents a case study for the application of semiring semantics
for fixed-point formulae to the analysis of strategies in B\"uchi games.
Semiring semantics generalizes the classical Boolean semantics by permitting
multiple truth values from certain semirings. Evaluating the fixed-point
formula that defines the winning region in a given game in an appropriate
semiring of polynomials provides not only the Boolean information on who wins,
but also tells us how they win and which strategies they might use. This is
well-understood for reachability games, where the winning region is definable
as a least fixed point. The case of B\"uchi games is of special interest, not
only due to their practical importance, but also because it is the simplest
case where the fixed-point definition involves a genuine alternation of a
greatest and a least fixed point.
We show that, in a precise sense, semiring semantics provide information
about all absorption-dominant strategies -- strategies that win with minimal
effort, and we discuss how these relate to positional and the more general
persistent strategies. This information enables further applications such as
game synthesis or determining minimal modifications to the game needed to
change its outcome. Lastly, we discuss limitations of our approach and present
questions that cannot be immediately answered by semiring semantics.Comment: Full version of a paper submitted to GandALF 202
Fine-Grained Complexity: Exploring Reductions and their Properties
Η σχεδίαση αλγορίθμων αποτελεί ένα απο τα κύρια θέματα ενδιαφέροντος για τον τομέα της Πληροφορικής. Παρά τα πολλά αποτελέσματα σε ορισμένους τομείς, η προσέγγιση αυτή έχει πετύχει κάποια πρακτικά αδιέξοδα που έχουν αποδειχτεί προβληματικά στην πρόοδο του τομέα. Επίσης, οι κλασικές πρακτικές Υπολογιστικής Πολυπλοκότητας δεν ήταν σε θέση να παρακάμψουν αυτά τα εμπόδια. Η κατανόηση της δυσκολίας του κάθε προβλήματος δεν είναι τετριμμένη. Η Ραφιναρισμένη Πολυπλοκότητα παρέχει νέες προ-οπτικές για τα κλασικά προβλήματα, με αποτέλεσμα σταθερούς δεσμούς μεταξύ γνωστών εικασιών στην πολυπλοκότητα και την σχεδίαση αλγορίθμων. Χρησιμεύει επίσης ως εργα-λείο για να αποδείξει τα υπο όρους κατώτατα όρια για προβλήματα πολυωνυμικής χρονικής πολυπλοκότητας, ένα πεδίο που έχει σημειώσει πολύ λίγη πρόοδο μέχρι τώρα. Οι δημοφι-λείς υποθέσεις/παραδοχές όπως το SETH, το OVH, το 3SUM, και το APSP, δίνουν πολλά φράγματα που δεν έχουν ακόμα αποδειχθεί με κλασικές τεχνικές και παρέχουν μια νέα κατανόηση της δομής και της εντροπίας των προβλημάτων γενικά. Σκοπός αυτής της εργασίας είναι να συμβάλει στην εδραίωση του πλαισίου για αναγωγές από κάθε εικασία και να διερευνήσει την διαρθρωτική διαφορά μεταξύ των προβλημάτων σε κάθε περίπτωση.Algorithmic design has been one of the main subjects of interest for Computer science. While very effective in some areas, this approach has been met with some practical dead ends that have been very problematic in the progress of the field. Classical Computational Complexity practices have also not been able to bypass these blocks. Understanding the hardness of each problem is not trivial. Fine-Grained Complexity provides new perspectives on classic problems, resulting to solid links between famous conjectures in Complexity, and Algorithmic design. It serves as a tool to prove conditional lower bounds for problems with polynomial time complexity, a field that had seen very little progress until now. Popular conjectures such as SETH, k-OV, 3SUM, and APSP, imply many bounds that have yet to be proven using classic techniques, and provide a new understanding of the structure and entropy of problems in general. The aim of this thesis is to contribute towards solidifying the framework for reductions from each conjecture, and to explore the structural difference between the problems in each cas