Conditionally Optimal Algorithms for Generalized Büchi Games
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 k1 conjunctions in the antecedent and k2 conjunctions in the consequent, and present an O(k1 · k2 · n^2.5)-time algorithm, improving the previously known O(k1 · k2 · n · m)-time algorithm for m > n^1.5.
|Paper in Conference Proceedings or in Workshop Proceedings (Full Paper in Proceedings)|
|41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)|
|Theory and Applications of Algorithms|
|August 22-26, 2016|
|41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}|