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 player's 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 on 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 also explicitly or succinctly. When the games are given in general form, we asume a 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.Postprint (published version

The Complexity of angel-daemons and game isomorphism

The analysis of the computational aspects of strategic situations is a basic field in Computer Sciences. Two main topics related to strategic games have been developed. First, introduction and analysis of a class of games (so called angel/daemon games) designed to asses web applications, have been considered. Second, the problem of isomorphism between strategic games has been analysed. Both parts have been separately considered. Angel-Daemon Games A service is a computational method that is made available for general use through a wide area network. The performance of web-services may fluctuate; at times of stress the performance of some services may be degraded (in extreme cases, to the point of failure). In this thesis uncertainty profiles and Angel-Daemon games are used to analyse servicebased behaviours in situations where probabilistic reasoning may not be appropriate. In such a game, an angel player acts on a bounded number of ¿angelic¿ services in a beneficial way while a daemon player acts on a bounded number of ¿daemonic¿ services in a negative way. Examples are used to illustrate how game theory can be used to analyse service-based scenarios in a realistic way that lies between over-optimism and over-pessimism. The resilience of an orchestration to service failure has been analysed - here angels and daemons are used to model services which can fail when placed under stress. The Nash equilibria of a corresponding Angel-Daemon game may be used to assign a ¿robustness¿ value to an orchestration. Finally, the complexity of equilibria problems for Angel-Daemon games has been analysed. It turns out that Angel-Daemon games are, at the best of our knowledge, the first natural example of zero-sum succinct games. The fact that deciding the existence of a pure Nash equilibrium or a dominant strategy for a given player is Sp 2-complete has been proven. Furthermore, computing the value of an Angel-Daemon game is EXP-complete. Thus, matching the already known complexity results of the corresponding problems for the generic families of succinctly represented games with exponential number of actions. Game Isomorphism The question of whether two multi-player strategic games are equivalent and the computational complexity of deciding such a property has been addressed. Three notions of isomorphisms, strong, weak and local have been considered. Each one of these isomorphisms preserves a different structure of the game. Strong isomorphism is defined to preserve the utility functions and Nash equilibria. Weak isomorphism preserves only the player preference relations and thus pure Nash equilibria. Local isomorphism preserves preferences defined only on ¿close¿ neighbourhood of strategy profiles. The problem of the computational complexity of game isomorphism, which depends on the level of succinctness of the description of the input games but it is independent of the isomorphism to consider, has been shown. Utilities in games can be given succinctly by Turing machines, boolean circuits or boolean formulas, or explicitly by tables. Actions can be given also explicitly or succinctly. When the games are given in general form, an explicit description of actions and a succinct description of utilities have been assumed. It is has been established that the game isomorphism problem for general form games is equivalent to the circuit isomorphism when utilities are described by Turing Machines; and to the boolean formula isomorphism problem when utilities are described by formulas. When the game is given in explicit form, it is has been proven that the game isomorphism problem is equivalent to the graph isomorphism problem. Finally, an equivalence classes of small games and their graphical representation have been also examined.Postprint (published version

Chain models, trees of singular cardinality and dynamic EF games

Let κ be a singular cardinal. Karp's notion of a chain model of size ? is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf(κ). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size κ with no κ-branches (even no cf(κ)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κ-branches under this notion and prove that when cf(κ) = ω this collection is rather regular; in particular it has universality number exactly κ+. Such trees are then used to develop a descriptive set theory of the space cf(κ)κ.The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models

Canonisation and Definability for Graphs of Bounded Rank Width

We prove that the combinatorial Weisfeiler-Leman algorithm of dimension $(3k+4)$ is a complete isomorphism test for the class of all graphs of rank width at most $k$. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width. It was known that isomorphism of graphs of rank width $k$ is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time $n^{f(k)}$ for a non-elementary function $f$. Our result yields an isomorphism test for graphs of rank width $k$ running in time $n^{O(k)}$. Another consequence of our result is the first polynomial time canonisation algorithm for graphs of bounded rank width. Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.Comment: 32 page

Limitations of Algebraic Approaches to Graph Isomorphism Testing

We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Gr\"obner basis computation. The idea of these algorithms is to encode two graphs into a system of equations that are satisfiable if and only if if the graphs are isomorphic, and then to (try to) decide satisfiability of the system using, for example, the Gr\"obner basis algorithm. In some cases this can be done in polynomial time, in particular, if the equations admit a bounded degree refutation in an algebraic proof systems such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on the polynomial calculus degree over all fields of characteristic different from 2 and also linear lower bounds for the degree of Positivstellensatz calculus derivations. We compare this approach to recently studied linear and semidefinite programming approaches to isomorphism testing, which are known to be related to the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system that lies between degree-k Nullstellensatz and degree-k polynomial calculus

Games orbits play and obstructions to Borel reducibility

We introduce a new, game-theoretic approach to anti-classification results for orbit equivalence relations. Within this framework, we give a short conceptual proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classification by orbits of CLI groups. We apply this criterion to the relation of equality of countable sets of reals, and the relations of unitary conjugacy of unitary and selfadjoint operators on the separable infinite-dimensional Hilbert space.Comment: 13 pages. Final version, to appear in Groups, Geometry, and Dynamic

An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism

The individualization-refinement paradigm provides a strong toolbox for testing isomorphism of two graphs and indeed, the currently fastest implementations of isomorphism solvers all follow this approach. While these solvers are fast in practice, from a theoretical point of view, no general lower bounds concerning the worst case complexity of these tools are known. In fact, it is an open question whether individualization-refinement algorithms can achieve upper bounds on the running time similar to the more theoretical techniques based on a group theoretic approach. In this work we give a negative answer to this question and construct a family of graphs on which algorithms based on the individualization-refinement paradigm require exponential time. Contrary to a previous construction of Miyazaki, that only applies to a specific implementation within the individualization-refinement framework, our construction is immune to changing the cell selector, or adding various heuristic invariants to the algorithm. Furthermore, our graphs also provide exponential lower bounds in the case when the $k$-dimensional Weisfeiler-Leman algorithm is used to replace the standard color refinement operator and the arguments even work when the entire automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page