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Register a new userGames and Computational Complexity
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Added by
Alapan Chaudhuri
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Computers are known to solve a wide spectrum of problems, however not all problems are computationally solvable. Further, the solvable problems themselves vary on the amount of computational resources they require for being solved. The rigorous analysis of problems and assigning them to complexity classes what makes up the immense field of complexity theory. Do protein folding and sudoku have something in common? It might not seem so but complexity theory tells us that if we had an algorithm that could solve sudoku efficiently then we could adapt it to predict for protein folding. This same property is held by classic platformer games such as Super Mario Bros, which was proven to be NP-complete by Erik Demaine et. al. This article attempts to review the analysis of classical platformer games. Here, we explore the field of complexity theory through a broad survey of literature and then use it to prove that that solving a generalized level in the game Celeste is NP-complete. Later, we also show how a small change in it makes the game presumably harder to compute. Various abstractions and formalisms related to modelling of games in general (namely game theory and constraint logic) and 2D platformer video games, including the generalized meta-theorems originally formulated by Giovanni Viglietta are also presented.
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Recently, a standardized framework was proposed for introducing quantum-inspired moves in mathematical games with perfect information and no chance. The beauty of quantum games-succinct in representation, rich in structures, explosive in complexity, dazzling for visualization, and sophisticated for strategic reasoning-has drawn us to play concrete games full of subtleties and to characterize abstract properties pertinent to complexity consequence. Going beyond individual games, we explore the tractability of quantum combinatorial games as whole, and address fundamental questions including: Quantum Leap in Complexity: Are there polynomial-time solvable games whose quantum extensions are intractable? Quantum Collapses in Complexity: Are there PSPACE-complete games whose quantum extensions fall to the lower levels of the polynomial-time hierarchy? Quantumness Matters: How do outcome classes and strategies change under quantum moves? Under what conditions doesnt quantumness matter? PSPACE Barrier for Quantum Leap: Can quantum moves launch PSPACE games into outer polynomial space We show that quantum moves not only enrich the game structure, but also impact their computational complexity. In settling some of these basic questions, we characterize both the powers and limitations of quantum moves as well as the superposition of game configurations that they create. Our constructive proofs-both on the leap of complexity in concrete Quantum Nim and Quantum Undirected Geography and on the continuous collapses, in the quantum setting, of complexity in abstract PSPACE-complete games to each level of the polynomial-time hierarchy-illustrate the striking computational landscape over quantum games and highlight surprising turns with unexpected quantum impact. Our studies also enable us to identify several elegant open questions fundamental to quantum combinatorial game theory (QCGT).
We describe the Turing Machine, list some of its many influences on the theory of computation and complexity of computations, and illustrate its importance.
Covering spaces of graphs have long been useful for studying expanders (as graph lifts) and unique games (as the label-extended graph). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linials 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot-Kindler-Mossel-ODonnell, FOCS 2004; SICOMP, 2007) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new Unique Games-complete problem in over a decade. Our results partially settle an open question of Chen and Freedman (SODA 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproximability over $mathbb{Z}/2mathbb{Z}$, and we show Unique Games-completeness over $mathbb{Z}/kmathbb{Z}$ for large $k$.) This equivalence comes from the fact that when the structure group $G$ of the covering space is Abelian - or more generally for principal $G$-bundles - Maximum Section of a $G$-Covering Space is the same as the well-studied problem of 1-Homology Localization. Although our most technically demanding result is an application of Unique Games to computational topology, we hope that our observations on the topological nature of the Unique Games Conjecture will lead to applications of algebraic topology to the Unique Games Conjecture in the future.
In two papers, Burgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity. In this paper we show that deciding positivity of plethysm coefficients is NP-hard, and that computing plethysm coefficients is #P-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.
We analyze the computational complexity of optimally playing the two-player board game Push Fight, generalized to an arbitrary board and number of pieces. We prove that the game is PSPACE-hard to decide who will win from a given position, even for simple (almost rectangular) hole-free boards. We also analyze the mate-in-1 problem: can the player win in a single turn? One turn in Push Fight consists of up to two moves followed by a mandatory push. With these rules, or generalizing the number of allowed moves to any constant, we show mate-in-1 can be solved in polynomial time. If, however, the number of moves per turn is part of the input, the problem becomes NP-complete. On the other hand, without any limit on the number of moves per turn, the problem becomes polynomially solvable again.
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