Joaquim Gabarró

MALLBA: a library of skeletons for combinatorial optimisation

The MALLBA project tackles the resolution of combinatorial optimization problems using algorithmic skeletons implemented in C++. mallba offers three families of generic resolution methods: exact, heuristic and hybrid. Moreover, for each resolution method, MALLBA provides three different implementations: sequential, parallel for local area networks, and parallel for wide area networks (currently under development). This paper explains the architecture of the MALLBA library, presents some of its skeletons, and offers several computational results to show the viability of the approach.

Deciding bisimilarity isP-complete

In finite labelled transition systems the problems of deciding strong bisimilarity, observation equivalence and observation congruence areP-complete under many—oneNC-reducibility. As a consequence, algorithms for automated analysis of finite state systems based on bisimulation seem to be inherently sequential in the following sense: the design of anNC algorithm to solve any of these problems will require an algorithmic breakthrough, which is exceedingly hard to achieve.

Efficient parallel LAN/WAN algorithms for optimization: the MALLBA project

The MALLBA project tackles the resolution of combinatorial optimization problems using generic algorithmic skeletons implemented in C++. A skeleton in the MALLBA library implements an optimization method in one of the three families of generic optimization techniques offered: exact, heuristic and hybrid. Moreover, for each of those methods, MALLBA provides three different implementations: sequential, parallel for Local Area Networks, and parallel for Wide Area Networks. This paper introduces the architecture of the MALLBA library, details some of the implemented skeletons, and offers computational results for some classical optimization problems to show the viability of our library. Among other conclusions, we claim that the design used to develop the optimization techniques included in the library is generic and efficient at the same time.

The complexity of game isomorphism

We address the question of whether two multiplayer strategic games are equivalent and the computational complexity of deciding such a property. We introduce two notions of isomorphisms, strong and weak. Each one of those isomorphisms preserves a different structure of the game. Strong isomorphisms are defined to preserve the utility functions and Nash equilibria. Weak isomorphisms preserve only the players preference relations and thus pure Nash equilibria. We show that the computational complexity of the game isomorphism problem depends on the level of succinctness of the description of the input games but it is independent of which of the two types of isomorphisms is considered. Utilities in games can be given succinctly by Turing machines, boolean circuits or boolean formulas, or explicitly by tables. Actions can be given both explicitly or succinctly. When the games are given in general form, we assume an explicit description of actions and a succinct description of utilities. We show that the game isomorphism problem for general form games is equivalent to the circuit isomorphism when utilities are described by TMs and to the boolean formula isomorphism problem when utilities are described by formulas. When the game is given in explicit form, we show that the game isomorphism problem is equivalent to the graph isomorphism problem.

Prefixes of infinite words and ambiguous context-free languages

Abstract Two ‘gap’ theorems are shown for languages formed with words that fail to be prefixes of an infinite word: such languages can never be described by unambiguous context-free grammars.

Pure nash equilibria in games with a large number of actions

We study the computational complexity of deciding the existence of a Pure Nash Equilibrium in multi-player strategic games. We address two fundamental questions: how can we represent a game? and how can we represent a game with polynomial pay-off functions? Our results show that the computational complexity of deciding the existence of a pure Nash equilibrium in a strategic game depends on two parameters: the number of players and the size of the sets of strategies. In particular we show that deciding the existence of a Nash equilibrium in a strategic game is NP-complete when the number of players is large and the number of strategies for each player is constant, while the problem is Σ

A design of a parallel dictionary using skip lists

^{p}_{\rm 2}

On the complexity of game isomorphism

-complete when the number of players is a constant and the size of the sets of strategies is exponential (with respect to the length of the strategies).

An approach to correctness of data parallel algorithms

We present a top-down design of a parallel PRAM dictionary using skip lists. More precisely, we give detailed algorithms to search for, insert or delete k elements in a skip list of n elements in parallel. The algorithms are iterative and easy to implement on real machines. We discuss some implementation issues and give concrete examples in C∗. The algorithms run on an EREW PRAM in expected time O(log n + log k) using k processors. We also show an explicit protocol to avoid read conflicts thus obtaining an efficient EREW version of our algorithms. Although the asymptotic performance of the parallel skip list algorithms is not better compared to that of other parallel dictionaries, they are a practical alternative. Skip list algorithms are very simple and there is a small probability of large deviations from their expected performance.

On characterizations of the class pspace/poly

We consider the question of when two games are equivalent and the computational complexity of deciding such a property for strategic games. We introduce three types of isomorphisms depending on which structure of the game is preserved: strict, weak, and local. We show that the computational complexity of the game isomorphism problem depends on the level of succinctness of the description of the input games but it is independent of the way the isomorphism is defined. Utilities or preferences in games can be represented by Turing machines (general form) or tables (explicit form). When the games are given in general form, we show that the game isomorphism problem is equivalent to the circuit isomorphism problem. When the games are given in explicit form, we show that the game isomorphism problem is equivalent to the graph isomorphism problem.