Conditionally Optimal Algorithms for Generalized Büchi 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 re...

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Veröffentlicht in:arXiv.org
Hauptverfasser: Chatterjee, Krishnendu, Dvořák, Wolfgang, Henzinger, Monika, Loitzenbauer, Veronika
Format: Paper
Sprache:Englisch
Veröffentlicht: Ithaca Cornell University Library, arXiv.org 20.07.2016
Schlagworte:
Algorithms
Apexes
Boolean algebra
Combinatorial analysis
Games
Graph theory
Graphs
Lower bounds
Multiplication
Objectives
Program verification (computers)
Run time (computers)
ISSN:2331-8422
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Zusammenfassung: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 \(n\) vertices, \(m\) edges, and generalized B\"uchi objectives with \(k\) conjunctions. First, we present an algorithm with running time \(O(k \cdot n^2)\), improving the previously known \(O(k \cdot n \cdot m)\) and \(O(k^2 \cdot n^2)\) 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 \(k_1\) conjunctions in the antecedent and \(k_2\) conjunctions in the consequent, and present an \(O(k_1 \cdot k_2 \cdot n^{2.5})\)-time algorithm, improving the previously known \(O(k_1 \cdot k_2 \cdot n \cdot m)\)-time algorithm for \(m > n^{1.5}\).
Bibliographie:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:2331-8422
DOI:10.48550/arxiv.1607.05850