Chatterjee, Krishnendu ;
Dvorák, Wolfgang ;
Henzinger, Monika ;
Loitzenbauer, Veronika
Conditionally Optimal Algorithms for Generalized Büchi Games
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 computeraided verification. We present improved algorithms and conditional superlinear lower bounds based on widely believed assumptions about the complexity of (A1) combinatorial Boolean matrix multiplication and (A2) CNFSAT. 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) worstcase 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}.
BibTeX  Entry
@InProceedings{chatterjee_et_al:LIPIcs:2016:6440,
author = {Krishnendu Chatterjee and Wolfgang Dvor{\'a}k and Monika Henzinger and Veronika Loitzenbauer},
title = {{Conditionally Optimal Algorithms for Generalized B{\"u}chi Games}},
booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
pages = {25:125:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770163},
ISSN = {18688969},
year = {2016},
volume = {58},
editor = {Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6440},
URN = {urn:nbn:de:0030drops64403},
doi = {10.4230/LIPIcs.MFCS.2016.25},
annote = {Keywords: generalized B{\"u}chi objective, GR(1) objective, conditional lower bounds, graph games, graph algorithms, computeraided verification}
}
19.08.2016
Keywords: 

generalized Büchi objective, GR(1) objective, conditional lower bounds, graph games, graph algorithms, computeraided verification 
Seminar: 

41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Issue date: 

2016 
Date of publication: 

19.08.2016 