Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHardness of Modern Games
76
0
0.0
(
0
)
Added by
Lu\\'is M. S. Russo
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We consider the complexity properties of modern puzzle games, Hexiom, Cut the Rope and Back to Bed. The complexity of games plays an important role in the type of experience they provide to players. Back to Bed is shown to be PSPACE-Hard and the first two are shown to be NP-Hard. These results give further insight into the structure of these games and the resulting constructions may be useful in further complexity studies.
rate research
Read More
We study the problem of approximating the value of a Unique Game instance in the streaming model. A simple count of the number of constraints divided by $p$, the alphabet size of the Unique Game, gives a trivial $p$-approximation that can be computed in $O(log n)$ space. Meanwhile, with high probability, a sample of $tilde{O}(n)$ constraints suffices to estimate the optimal value to $(1+epsilon)$ accuracy. We prove that any single-pass streaming algorithm that achieves a $(p-epsilon)$-approximation requires $Omega_epsilon(sqrt{n})$ space. Our proof is via a reduction from lower bounds for a communication problem that is a $p$-ary variant of the Boolean Hidden Matching problem studied in the literature. Given the utility of Unique Games as a starting point for reduction to other optimization problems, our strong hardness for approximating Unique Games could lead to downemph{stream} hardness results for streaming approximability for other CSP-like problems.
Games on graphs provide a natural and powerful model for reactive systems. In this paper, we consider generalized reachability objectives, defined as conjunctions of reachability objectives. We first prove that deciding the winner in such games is $PSPACE$-complete, although it is fixed-parameter tractable with the number of reachability objectives as parameter. Moreover, we consider the memory requirements for both players and give matching upper and lower bounds on the size of winning strategies. In order to allow more efficient algorithms, we consider subclasses of generalized reachability games. We show that bounding the size of the reachability sets gives two natural subclasses where deciding the winner can be done efficiently.
The use of monotonicity and Tarskis theorem in existence proofs of equilibria is very widespread in economics, while Tarskis theorem is also often used for similar purposes in the context of verification. However, there has been relatively little in the way of analysis of the complexity of finding the fixed points and equilibria guaranteed by this result. We study a computational formalism based on monotone functions on the $d$-dimensional grid with sides of length $N$, and their fixed points, as well as the closely connected subject of supermodular games and their equilibria. It is known that finding some (any) fixed point of a monotone function can be done in time $log^d N$, and we show it requires at least $log^2 N$ function evaluations already on the 2-dimensional grid, even for randomized algorithms. We show that the general Tarski problem of finding some fixed point, when the monotone function is given succinctly (by a boolean circuit), is in the class PLS of problems solvable by local search and, rather surprisingly, also in the class PPAD. Finding the greatest or least fixed point guaranteed by Tarskis theorem, however, requires $dcdot N$ steps, and is NP-hard in the white box model. For supermodular games, we show that finding an equilibrium in such games is essentially computationally equivalent to the Tarski problem, and finding the maximum or minimum equilibrium is similarly harder. Interestingly, two-player supermodular games where the strategy space of one player is one-dimensional can be solved in $O(log N)$ steps. We also observe that computing (approximating) the value of Condons (Shapleys) stochastic games reduces to the Tarski problem. An important open problem highlighted by this work is proving a $Omega(log^d N)$ lower bound for small fixed dimension $d geq 3$.
Stochastic games combine controllable and adversarial non-determinism with stochastic behavior and are a common tool in control, verification and synthesis of reactive systems facing uncertainty. Multi-objective stochastic games are natural in situations where several - possibly conflicting - performance criteria like time and energy consumption are relevant. Such conjunctive combinations are the most studied multi-objective setting in the literature. In this paper, we consider the dual disjunctive problem. More concretely, we study turn-based stochastic two-player games on graphs where the winning condition is to guarantee at least one reachability or safety objective from a given set of alternatives. We present a fine-grained overview of strategy and computational complexity of such emph{disjunctive queries} (DQs) and provide new lower and upper bounds for several variants of the problem, significantly extending previous works. We also propose a novel value iteration-style algorithm for approximating the set of Pareto optimal thresholds for a given DQ.
The analysis of several algorithms and data structures can be framed as a peeling process on a random hypergraph: vertices with degree less than k and their adjacent edges are removed until no vertices of degree less than k are left. Often the question is whether the remaining hypergraph, the k-core, is empty or not. In some settings, it may be possible to remove either vertices or edges from the hypergraph before peeling, at some cost. For example, in hashing applications where keys correspond to edges and buckets to vertices, one might use an additional side data structure, commonly referred to as a stash, to separately handle some keys in order to avoid collisions. The natural question in such cases is to find the minimum number of edges (or vertices) that need to be stashed in order to realize an empty k-core. We show that both these problems are NP-complete for all $k geq 2$ on graphs and regular hypergraphs, with the sole exception being that the edge variant of stashing is solvable in polynomial time for $k = 2$ on standard (2-uniform) graphs.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
