Henrik Björklund

A Discrete Subexponential Algorithm for Parity Games

We suggest a new randomized algorithm for solving parity games with worst case time complexity roughly min(O(n3 ? (n/k+ 1)k), 2O(?n log n), where n is the number of vertices and k the number of colors of the game. This is comparable with the previously known algorithms when the number of colors is small. However, the subexponential bound is an advantage when the number of colors is large, k = ?(n1/2+?).

Memoryless determinacy of parity and mean payoff games: a simple proof

We give a simple, direct, and constructive proof of memoryless determinacy for parity and mean payoff games. First, we prove by induction that the finite duration versions of these games, played until some vertex is repeated, are determined and both players have memoryless winning strategies. In contrast to the proof of Ehrenfeucht and Mycielski, Internat. J. Game Theory, 8 (1979) 109-113, our proof does not refer to the infinite-duration versions. Second, we show that memoryless determinacy straightforwardly generalizes to infinite duration versions of parity and mean payoff games.

Conjunctive query containment over trees

The complexity of containment and satisfiability of conjunctive queries over finite, unranked, labeled trees is studied with respect to the axes Child, NextSibling, their transitive and reflexive closures, and Following. For the containment problem a trichotomy is presented, classifying the problems as in PTIME, coNP-complete, or Π2P -complete. For the satisfiability problem most problems are classified as either in PTIME or NP-complete.

Combinatorial structure and randomized subexponential algorithms for infinite games

The complexity of solving infinite games, including parity, mean payoff, and simple stochastic, is an important open problem in verification, automata, and complexity theory. In this paper, we develop an abstract setting for studying and solving such games, based on function optimization over certain discrete structures. We introduce new classes of recursively local-global (RLG) and partial recursively local-global (PRLG) functions, and show that strategy evaluation functions for simple stochastic, mean payoff, and parity games belong to these classes.In this setting, we suggest randomized subexponential algorithms appropriate for RLG-and PRLG-function optimization. We show that the subexponential algorithms for combinatorial linear programming, due to Kalai and Matousek, Sharir, Welzl, can be adapted for optimizing the RLG-and PRLG-functions.

The Tractability Frontier for NFA Minimization

We essentially show that minimizing finite automata is NP-hard as soon as one deviates from the class of deterministic finite automata. More specifically, we show that minimization is NP-hard for all finite automata classes that subsume the class that is unambiguous, allows at most one state qwith a non-deterministic transition for at most one alphabet symbol a, and is allowed to visit state qat most once in a run. Furthermore, this result holds even for automata that only accept finite languages.

Bounded depth data trees

A data tree is a tree where each node has a label from a finite set, and a data value from a possibly infinite set. We consider data trees whose depth is bounded beforehand. By developing an appropriate automaton model, we show that under this assumption various formalisms, including a two variable first-order logic and a subset of XPath, have decidable emptiness problems.

Complexity of Model Checking by Iterative Improvement: The Pseudo-Boolean Framework

We present several new algorithms as well as new lower and upper bounds for optimizing functions underlying infinite games pertinent to computer-aided verification.

The tractability frontier for NFA minimization

We prove that minimizing finite automata is NP-hard for almost all classes of automata that extend the class of deterministic finite automata. More specifically, we show that minimization is NP-hard for all finite automata classes that subsume the class of @dNFAs which accept strings of length at most three. Here, @dNFAs are the finite automata that are unambiguous, allow at most one state q with a non-deterministic transition for at most one alphabet symbol a, and are allowed to visit state q at most once in a run. As a corollary, we also obtain that the same result holds for all finite automata classes that subsume that class of finite automata that are unambiguous, have at most two initial states, and accept strings of length at most two.

Validity of Tree Pattern Queries with Respect to Schema Information

We prove that various containment and validity problems for tree pattern queries with respect to a schema are EXPTIME-complete. When one does not require the root of a tree pattern query to match the root of a tree, validity of a non-branching tree pattern query with respect to a Relax NG schema or W3C XML Schema is already EXPTIME-hard when the query does not branch and uses only child axes. These hardness results already hold when the alphabet size is fixed. Validity with respect to a DTD is proved to be EXPTIME-hard already when the query only uses child axes and is allowed to branch only once.

Recognizing shuffled languages

Language models that use interleaving, or shuffle, operators have applications in various areas of computer science, including system verification, plan recognition, and natural language processing. We study the complexity of the membership problem for such models, i.e., how difficult it is to determine if a string belongs to a language or not. In particular, we investigate how interleaving can be introduced into models that capture the context-free languages.