Chatterjee, Krishnendu and Dvorák, Wolfgang and Henzinger, Monika and Loitzenbauer, Veronika
(2016)
*Conditionally optimal algorithms for generalized Büchi Games.*
In: MFCS: International Symposium on Mathematical Foundations of Computer Science , August 22-26, 2016, Krakow, POL.

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## 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}.

Item Type: | Conference or Workshop Item (Paper) |
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Uncontrolled Keywords: | Computer-aided verification; graph games; graph algorithms; conditional lower bounds; objective |

Subjects: | 000 Computer science, knowledge & general works > 000 Computer science, knowledge & systems 000 Computer science, knowledge & general works > 000 Computer science, knowledge & systems > 004 Data processing & computer science 000 Computer science, knowledge & general works > 000 Computer science, knowledge & systems > 006 Special computer methods |

Research Group: | Chatterjee Group |

SWORD Depositor: | Sword Import User |

Depositing User: | Sword Import User |

Date Deposited: | 02 Mar 2017 07:43 |

Last Modified: | 02 Mar 2017 07:43 |

URI: | https://repository.ist.ac.at/id/eprint/779 |

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