[Paper Review] An O(n2) time algorithm for alternating Büchi games

Krishnendu Chatterjee, Monika Henzinger|arXiv (Cornell University)|Jan 1, 2011

Formal Methods in Verification29 references10 citations

TL;DR

This paper presents an O(n²) time algorithm for solving alternating Büchi games, breaking the long-standing eO(n·m) complexity barrier. The approach uses a novel hierarchical graph decomposition and dynamic progress measure maintenance, enabling the first dynamic algorithm for edge insertions/deletions with O(n) amortized time per operation, and improving bounds for related problems like MDPs and end-component decomposition.

ABSTRACT

Computing the winning set for Büchi objectives in alternating games on graphs is a central problem in computer aided verification with a large number of applications. The long standing best known upper bound for solving the problem is ̃O(n·m), where n is the number of vertices and m is the number of edges in the graph. We are the first to break the ̃O(n·m) boundary by presenting a new technique that reduces the running time to O(n2). This bound also leads to O(n2) time algorithms for computing the set of almost-sure winning vertices for Büchi objectives (1) in alternating games with probabilistic transitions (improving an earlier bound of O(n·m)), (2) in concurrent graph games with constant actions (improving an earlier bound of O(n3)), and (3) in Markov decision processes (improving for m > n4/3 an earlier bound of O(min(m1.5, m·n2/3)). We also show that the same technique can be used to compute the maximal end-component decomposition of a graph in time O(n2), which is an improvement over earlier bounds for m > n4/3. Finally, we show how to maintain the winning set for Büchi objectives in alternating games under a sequence of edge insertions or a sequence of edge deletions in O(n) amortized time per operation. This is the first dynamic algorithm for this problem.

Motivation & Objective

• To break the eO(n·m) time complexity barrier for solving alternating Büuchi games, a central problem in formal verification.
• To develop the first dynamic algorithm for maintaining the winning set under edge insertions or deletions in alternating Büuchi games.
• To improve time bounds for related problems: Markov decision processes, concurrent games with constant actions, and probabilistic games with B"uchi objectives.
• To provide an O(n²) algorithm for maximal end-component decomposition, improving over prior O(min(m¹·⁵, m·n²/³)) bounds for m > n⁴/³.
• To unify and extend dynamic graph algorithm techniques to the setting of alternating games with B"uchi objectives.

Proposed method

• Introduces a hierarchical graph decomposition technique to efficiently identify and remove non-winning vertices in alternating Büchi games.
• Employs a dynamic progress measure data structure that maintains the least fix-point of the coLift operator under edge updates.
• Uses a queue-based incremental algorithm to propagate changes in progress measures after edge insertions, with amortized O(n) time per operation.
• Maintains specialized lists for vertices in V₂ (player 2) to track neighbors with equal or one greater progress measure, enabling efficient updates.
• Applies the Even-Shiloach-style dynamic reachability technique to the progress measure framework, generalizing it to alternating games.
• Computes the initial progress measure via fix-point iteration starting from zero, using a similar incremental procedure to reach the least fix-point.

Experimental results

Research questions

• RQ1 Can the O(n·m) time complexity for alternating Büchi games be broken using new algorithmic techniques?
• RQ2 Is it possible to design a dynamic algorithm that maintains the winning set under edge insertions or deletions in alternating Büuchi games with sublinear amortized time per update?
• RQ3 Can the proposed method be generalized to improve time bounds for related problems such as MDPs, concurrent games, and end-component decomposition?
• RQ4 How does the hierarchical decomposition technique compare to classical repeated reachability computation in terms of efficiency and scalability?
• RQ5 What is the theoretical and practical impact of using dynamic progress measures on the complexity of solving B"uchi objectives in game graphs?

Key findings

• The paper presents the first O(n²) time algorithm for alternating Büchi games, breaking the eO(n·m) complexity barrier for all n and m.
• The algorithm achieves O(n) amortized time per edge insertion or deletion, making it the first dynamic algorithm for this problem.
• The maximal end-component decomposition problem is solved in O(n²) time, improving over the previous O(min(m¹·⁵, m·n²/³)) bound for m > n⁴/³.
• The set of almost-sure winning vertices in Markov decision processes with B"uchi objectives is computed in O(n²) time, improving the prior O(min(m¹·⁵, m·n²/³)) bound for m > n⁴/³.
• The set of almost-sure winning vertices in concurrent games with constant actions and B"uchi objectives is computed in O(n²) time, improving the prior O(n³) worst-case bound.
• The algorithm for simple stochastic games with B"uchi objectives runs in O(n²) time, improving the previous eO(n·m) bound via linear reduction from such games to alternating Büchi games.