Bejeweled, Candy Crush and other Match-Three Games are (NP-)Hard

The twentieth century has seen the rise of a new type of video games targeted at a mass audience of "casual" gamers. Many of these games require the player to swap items in order to form matches of three and are collectively known as \emph{tile-matching match-three games}. Among these, the most influential one is arguably \emph{Bejeweled} in which the matched items (gems) pop and the above gems fall in their place. Bejeweled has been ported to many different platforms and influenced an incredible number of similar games. Very recently one of them, named \emph{Candy Crush Saga} enjoyed a huge popularity and quickly went viral on social networks. We generalize this kind of games by only parameterizing the size of the board, while all the other elements (such as the rules or the number of gems) remain unchanged. Then, we prove that answering many natural questions regarding such games is actually \NP-Hard. These questions include determining if the player can reach a certain score, play for a certain number of turns, and others. We also \href{http://candycrush.isnphard.com}{provide} a playable web-based implementation of our reduction.Comment: 21 pages, 12 figure

Trainyard is NP-Hard

Recently, due to the widespread diffusion of smart-phones, mobile puzzle games have experienced a huge increase in their popularity. A successful puzzle has to be both captivating and challenging, and it has been suggested that this features are somehow related to their computational complexity \cite{Eppstein}. Indeed, many puzzle games --such as Mah-Jongg, Sokoban, Candy Crush, and 2048, to name a few-- are known to be NP-hard \cite{CondonFLS97, culberson1999sokoban, GualaLN14, Mehta14a}. In this paper we consider Trainyard: a popular mobile puzzle game whose goal is to get colored trains from their initial stations to suitable destination stations. We prove that the problem of determining whether there exists a solution to a given Trainyard level is NP-hard. We also \href{http://trainyard.isnphard.com}{provide} an implementation of our hardness reduction

Large Peg-Army Maneuvers

Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Cs\'ak\'any and Juh\'asz (Mathematics Magazine, 2000). To appreciate the combinatorial beauty of our solutions, we recommend to visit the gallery of animations provided at http://solitairearmy.isnphard.com.Comment: Conference versio

Push-Pull Block Puzzles are Hard

This paper proves that push-pull block puzzles in 3D are PSPACE-complete to solve, and push-pull block puzzles in 2D with thin walls are NP-hard to solve, settling an open question by Zubaran and Ritt. Push-pull block puzzles are a type of recreational motion planning problem, similar to Sokoban, that involve moving a `robot' on a square grid with $1 \times 1$ obstacles. The obstacles cannot be traversed by the robot, but some can be pushed and pulled by the robot into adjacent squares. Thin walls prevent movement between two adjacent squares. This work follows in a long line of algorithms and complexity work on similar problems. The 2D push-pull block puzzle shows up in the video games Pukoban as well as The Legend of Zelda: A Link to the Past, giving another proof of hardness for the latter. This variant of block-pushing puzzles is of particular interest because of its connections to reversibility, since any action (e.g., push or pull) can be inverted by another valid action (e.g., pull or push).Comment: Full version of CIAC 2017 paper. 17 page

NP-completeness of the game Kingdomino

Kingdomino is a board game designed by Bruno Cathala and edited by Blue Orange since 2016. The goal is to place $2 \times 1$ dominoes on a grid layout, and get a better score than other players. Each $1 \times 1$ domino cell has a color that must match at least one adjacent cell, and an integer number of crowns (possibly none) used to compute the score. We prove that even with full knowledge of the future of the game, in order to maximize their score at Kingdomino, players are faced with an NP-complete optimization problem

Multi-Modal Data Analysis Based Game Player Experience Modeling Using LSTM-DNN

Game player modeling is a paradigm of computational models to exploit players’ behavior and experience using game and player analytics. Player modeling refers to descriptions of players based on frameworks of data derived from the interaction of a player’s behavior within the game as well as the player’s experience with the game. Player behavior focuses on dynamic and static information gathered at the time of gameplay. Player experience concerns the association of the human player during gameplay, which is based on cognitive and affective physiological measurements collected from sensors mounted on the player’s body or in the player’s surroundings. In this paper, player experience modeling is studied based on the board puzzle game “Candy Crush Saga” using cognitive data of players accessed by physiological and peripheral devices. Long Short-Term Memory-based Deep Neural Network (LSTM-DNN) is used to predict players’ effective states in terms of valence, arousal, dominance, and liking by employing the concept of transfer learning. Transfer learning focuses on gaining knowledge while solving one problem and using the same knowledge to solve different but related problems. The homogeneous transfer learning approach has not been implemented in the game domain before, and this novel study opens a new research area for the game industry where the main challenge is predicting the significance of innovative games for entertainment and players’ engagement. Relevant not only from a player’s point of view, it is also a benchmark study for game developers who have been facing problems of “cold start” for innovative games that strengthen the game industrial economy

The Computational Complexity of Angry Birds

The physics-based simulation game Angry Birds has been heavily researched by the AI community over the past five years, and has been the subject of a popular AI competition that is currently held annually as part of a leading AI conference. Developing intelligent agents that can play this game effectively has been an incredibly complex and challenging problem for traditional AI techniques to solve, even though the game is simple enough that any human player could learn and master it within a short time. In this paper we analyse how hard the problem really is, presenting several proofs for the computational complexity of Angry Birds. By using a combination of several gadgets within this game's environment, we are able to demonstrate that the decision problem of solving general levels for different versions of Angry Birds is either NP-hard, PSPACE-hard, PSPACE-complete or EXPTIME-hard. Proof of NP-hardness is by reduction from 3-SAT, whilst proof of PSPACE-hardness is by reduction from True Quantified Boolean Formula (TQBF). Proof of EXPTIME-hardness is by reduction from G2, a known EXPTIME-complete problem similar to that used for many previous games such as Chess, Go and Checkers. To the best of our knowledge, this is the first time that a single-player game has been proven EXPTIME-hard. This is achieved by using stochastic game engine dynamics to effectively model the real world, or in our case the physics simulator, as the opponent against which we are playing. These proofs can also be extended to other physics-based games with similar mechanics.Comment: 55 Pages, 39 Figure

Assessing Influential Users in Live Streaming Social Networks

abstract: Live streaming has risen to significant popularity in the recent past and largely this live streaming is a feature of existing social networks like Facebook, Instagram, and Snapchat. However, there does exist at least one social network entirely devoted to live streaming, and specifically the live streaming of video games, Twitch. This social network is unique for a number of reasons, not least because of its hyper-focus on live content and this uniqueness has challenges for social media researchers. Despite this uniqueness, almost no scientific work has been performed on this public social network. Thus, it is unclear what user interaction features present on other social networks exist on Twitch. Investigating the interactions between users and identifying which, if any, of the common user behaviors on social network exist on Twitch is an important step in understanding how Twitch fits in to the social media ecosystem. For example, there are users that have large followings on Twitch and amass a large number of viewers, but do those users exert influence over the behavior of other user the way that popular users on Twitter do? This task, however, will not be trivial. The same hyper-focus on live content that makes Twitch unique in the social network space invalidates many of the traditional approaches to social network analysis. Thus, new algorithms and techniques must be developed in order to tap this data source. In this thesis, a novel algorithm for finding games whose releases have made a significant impact on the network is described as well as a novel algorithm for detecting and identifying influential players of games. In addition, the Twitch network is described in detail along with the data that was collected in order to power the two previously described algorithms.Dissertation/ThesisDoctoral Dissertation Computer Science 201