Conditionally optimal algorithms for generalized Büchi Games Conference Paper

Author(s): Chatterjee, Krishnendu; Dvorák, Wolfgang; Henzinger, Monika; Loitzenbauer, Veronika
Title: Conditionally optimal algorithms for generalized Büchi Games
Title Series: LIPIcs
Affiliation IST Austria
Abstract: Games on graphs provide the appropriate framework to study several central problems in computer science, such as verification and synthesis of reactive systems. One of the most basic objectives for games on graphs is the liveness (or Büchi) objective that given a target set of vertices requires that some vertex in the target set is visited infinitely often. We study generalized Büchi objectives (i.e., conjunction of liveness objectives), and implications between two generalized Büchi 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 Büchi objectives with k conjunctions. First, we present an algorithm with running time O(k*n^2), improving the previously known O(k*n*m) and O(k^2*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 k_2 n^{2.5})-time algorithm, improving the previously known O(k_1*k_2*n*m)-time algorithm for m > n^{1.5}.
Keywords: Computer-aided verification; Graph games; Graph algorithms; Conditional lower bounds; Objective
Conference Title: MFCS: Mathematical Foundations of Computer Science (SG)
Volume: 58
Conference Dates: August 22-26, 2016
Conference Location: Kraków, POL
Publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik  
Date Published: 2016-01-01
Start Page: Article Number: 25
Copyright Statement: CC BY
Sponsor: K. C., M. H., and W. D. are partially supported by the Vienna Science and Technology Fund (WWTF) through project ICT15-003. K. C. is partially supported by the Austrian Science Fund (FWF) NFN Grant No S11407-N23 (RiSE/SHiNE) and an ERC Start grant (279307
DOI: 10.4230/LIPIcs.MFCS.2016.25
Open access: yes (OA journal)
IST Austria Authors