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

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

Preliminary Notes on Open Data Licensing

Open data is fuel for the future economy. Opening and sharing data owned by public bodies, communities and companies has an incredible economic value. This will potentially lead our society to a new data-driven thinking paradigm. They also enable a smarter urban space where companies can provide better and innovative services. In particular, accessing to government data held by public bodies generates accountability, transparency and fosters economic growth. Two main aspects define data as open: data formats and licenses. This paper aims at listing some preliminary notes on the copyright framework in which open data are released and presenting the idea of considering licenses as metadata. Many tools, according to the semantic web paradigm, aim at enforcing this aspect, managing data exchange in a more compliance way and reducing costs for reuse of data. Finally, a research path for a new approach to digital ownership will be presented

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

Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees

Let $G$ be an $n$-node and $m$-edge positively real-weighted undirected graph. For any given integer $f \ge 1$, we study the problem of designing a sparse \emph{f-edge-fault-tolerant} ($f$-EFT) $\sigma${\em -approximate single-source shortest-path tree} ($\sigma$-ASPT), namely a subgraph of $G$ having as few edges as possible and which, following the failure of a set $F$ of at most $f$ edges in $G$, contains paths from a fixed source that are stretched at most by a factor of $\sigma$. To this respect, we provide an algorithm that efficiently computes an $f$-EFT $(2|F|+1)$-ASPT of size $O(f n)$. Our structure improves on a previous related construction designed for \emph{unweighted} graphs, having the same size but guaranteeing a larger stretch factor of $3(f+1)$, plus an additive term of $(f+1) \log n$. Then, we show how to convert our structure into an efficient $f$-EFT \emph{single-source distance oracle} (SSDO), that can be built in $\widetilde{O}(f m)$ time, has size $O(fn \log^2 n)$, and is able to report, after the failure of the edge set $F$, in $O(|F|^2 \log^2 n)$ time a $(2|F|+1)$-approximate distance from the source to any node, and a corresponding approximate path in the same amount of time plus the path's size. Such an oracle is obtained by handling another fundamental problem, namely that of updating a \emph{minimum spanning forest} (MSF) of $G$ after that a \emph{batch} of $k$ simultaneous edge modifications (i.e., edge insertions, deletions and weight changes) is performed. For this problem, we build in $O(m \log^3 n)$ time a \emph{sensitivity} oracle of size $O(m \log^2 n)$, that reports in $O(k^2 \log^2 n)$ time the (at most $2k$) edges either exiting from or entering into the MSF. [...]Comment: 16 pages, 4 figure

Specializations and Generalizations of the Stackelberg Minimum Spanning Tree Game

Let be given a graph $G=(V,E)$ whose edge set is partitioned into a set $R$ of \emph{red} edges and a set $B$ of \emph{blue} edges, and assume that red edges are weighted and form a spanning tree of $G$. Then, the \emph{Stackelberg Minimum Spanning Tree} (\stack) problem is that of pricing (i.e., weighting) the blue edges in such a way that the total weight of the blue edges selected in a minimum spanning tree of the resulting graph is maximized. \stack \ is known to be \apx-hard already when the number of distinct red weights is 2. In this paper we analyze some meaningful specializations and generalizations of \stack, which shed some more light on the computational complexity of the problem. More precisely, we first show that if $G$ is restricted to be \emph{complete}, then the following holds: (i) if there are only 2 distinct red weights, then the problem can be solved optimally (this contrasts with the corresponding \apx-hardness of the general problem); (ii) otherwise, the problem can be approximated within $7/4 + \epsilon$, for any $\epsilon > 0$. Afterwards, we define a natural extension of \stack, namely that in which blue edges have a non-negative \emph{activation cost} associated, and it is given a global \emph{activation budget} that must not be exceeded when pricing blue edges. Here, after showing that the very same approximation ratio as that of the original problem can be achieved, we prove that if the spanning tree of red edges can be rooted so as that any root-leaf path contains at most $h$ edges, then the problem admits a $(2h+\epsilon)$-approximation algorithm, for any $\epsilon > 0$.Comment: 22 pages, 7 figure

The Max-Distance Network Creation Game on General Host Graphs

In this paper we study a generalization of the classic \emph{network creation game} in the scenario in which the $n$ players sit on a given arbitrary \emph{host graph}, which constrains the set of edges a player can activate at a cost of $\alpha \geq 0$ each. This finds its motivations in the physical limitations one can have in constructing links in practice, and it has been studied in the past only when the routing cost component of a player is given by the sum of distances to all the other nodes. Here, we focus on another popular routing cost, namely that which takes into account for each player its \emph{maximum} distance to any other player. For this version of the game, we first analyze some of its computational and dynamic aspects, and then we address the problem of understanding the structure of associated pure Nash equilibria. In this respect, we show that the corresponding price of anarchy (PoA) is fairly bad, even for several basic classes of host graphs. More precisely, we first exhibit a lower bound of $\Omega (\sqrt{ n / (1+\alpha)})$ for any $\alpha = o(n)$. Notice that this implies a counter-intuitive lower bound of $\Omega(\sqrt{n})$ for very small values of $\alpha$ (i.e., edges can be activated almost for free). Then, we show that when the host graph is restricted to be either $k$-regular (for any constant $k \geq 3$), or a 2-dimensional grid, the PoA is still $\Omega(1+\min\{\alpha, \frac{n}{\alpha}\})$, which is proven to be tight for $\alpha=\Omega(\sqrt{n})$. On the positive side, if $\alpha \geq n$, we show the PoA is $O(1)$. Finally, in the case in which the host graph is very sparse (i.e., $|E(H)|=n-1+k$, with $k=O(1)$), we prove that the PoA is $O(1)$, for any $\alpha$.Comment: 17 pages, 4 figure