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

Some Simple Distributed Algorithms for Sparse Networks

We give simple, deterministic, distributed algorithms for computing maximal matchings, maximal independent sets and colourings. We show that edge colourings with at most 2D-1 colours, and maximal matchings can be computed within O(log^* n + D) deterministic rounds, where D is the maximum degree of the network. We also show how to find maximal independent sets and (D+1)-vertex colourings within O(log^* n + D^2) deterministic rounds. All hidden constants are very small and the algorithms are very simple

Motif counting beyond five nodes

Counting graphlets is a well-studied problem in graph mining and social network analysis. Recently, several papers explored very simple and natural algorithms based on Monte Carlo sampling of Markov Chains (MC), and reported encouraging results. We show, perhaps surprisingly, that such algorithms are outperformed by color coding (CC) [2], a sophisticated algorithmic technique that we extend to the case of graphlet sampling and for which we prove strong statistical guarantees. Our computational experiments on graphs with millions of nodes show CC to be more accurate than MC; furthermore, we formally show that the mixing time of the MC approach is too high in general, even when the input graph has high conductance. All this comes at a price however. While MC is very efficient in terms of space, CC’s memory requirements become demanding when the size of the input graph and that of the graphlets grow. And yet, our experiments show that CC can push the limits of the state-of-the-art, both in terms of the size of the input graph and of that of the graphlets

The Limits of Popularity-Based Recommendations, and the Role of Social Ties

In this paper we introduce a mathematical model that captures some of the salient features of recommender systems that are based on popularity and that try to exploit social ties among the users. We show that, under very general conditions, the market always converges to a steady state, for which we are able to give an explicit form. Thanks to this we can tell rather precisely how much a market is altered by a recommendation system, and determine the power of users to influence others. Our theoretical results are complemented by experiments with real world social networks showing that social graphs prevent large market distortions in spite of the presence of highly influential users.Comment: 10 pages, 9 figures, KDD 201

The Distortion of Locality Sensitive Hashing

Given a pairwise similarity notion between objects, locality sensitive hashing (LSH) aims to construct a hash function family over the universe of objects such that the probability two objects hash to the same value is their similarity. LSH is a powerful algorithmic tool for large-scale applications and much work has been done to understand LSHable similarities, i.e., similarities that admit an LSH. In this paper we focus on similarities that are provably non-LSHable and propose a notion of distortion to capture the approximation of such a similarity by a similarity that is LSHable. We consider several well-known non-LSHable similarities and show tight upper and lower bounds on their distortion. We also experimentally show that our upper bounds translate to

About latent roles in forecasting players in team sports

Forecasting players in sports has grown in popularity due to the potential for a tactical advantage and the applicability of such research to multi-agent interaction systems. Team sports contain a significant social component that influences interactions between teammates and opponents. However, it still needs to be fully exploited. In this work, we hypothesize that each participant has a specific function in each action and that role-based interaction is critical for predicting players' future moves. We create RolFor, a novel end-to-end model for Role-based Forecasting. RolFor uses a new module we developed called Ordering Neural Networks (OrderNN) to permute the order of the players such that each player is assigned to a latent role. The latent role is then modeled with a RoleGCN. Thanks to its graph representation, it provides a fully learnable adjacency matrix that captures the relationships between roles and is subsequently used to forecast the players' future trajectories. Extensive experiments on a challenging NBA basketball dataset back up the importance of roles and justify our goal of modeling them using optimizable models. When an oracle provides roles, the proposed RolFor compares favorably to the current state-of-the-art (it ranks first in terms of ADE and second in terms of FDE errors). However, training the end-to-end RolFor incurs the issues of differentiability of permutation methods, which we experimentally review. Finally, this work restates differentiable ranking as a difficult open problem and its great potential in conjunction with graph-based interaction models. Project is available at: https://www.pinlab.org/aboutlatentrolesComment: AI4ABM@ICLR2023 Worksho