Zero-Knowledge Proofs for Puzzles

This thesis will demonstrate zero-knowledge proofs for the puzzle games Kakuro and Rush- Hour. For Kakuro we will provide a classic zero-knowledge proof similar to the zero- knowledge proof for Sudoku. For Rush-Hour we will provide both a physical zero-knowledge proof and a classic zero-knowledge proof. These proofs share similarities with the zero- knowledge proof for the Rubik\u27s Cube

Domino Tatami Covering is NP-complete

A covering with dominoes of a rectilinear region is called \emph{tatami} if no four dominoes meet at any point. We describe a reduction from planar 3SAT to Domino Tatami Covering. As a consequence it is NP-complete to decide whether there is a perfect matching of a graph that meets every 4-cycle, even if the graph is restricted to be an induced subgraph of the grid-graph. The gadgets used in the reduction were discovered with the help of a SAT-solver.Comment: 10 pages, accepted at The International Workshop on Combinatorial Algorithms (IWOCA) 201

Card-Based ZKP Protocols for Takuzu and Juosan

International audienc

A Survey of Monte Carlo Tree Search Methods

Monte Carlo tree search (MCTS) is a recently proposed search method that combines the precision of tree search with the generality of random sampling. It has received considerable interest due to its spectacular success in the difficult problem of computer Go, but has also proved beneficial in a range of other domains. This paper is a survey of the literature to date, intended to provide a snapshot of the state of the art after the first five years of MCTS research. We outline the core algorithm's derivation, impart some structure on the many variations and enhancements that have been proposed, and summarize the results from the key game and nongame domains to which MCTS methods have been applied. A number of open research questions indicate that the field is ripe for future work

Mobile Application for Scanning Kakuro from Newspapers and Finishing It

Cílem této práce je vytvoření mobilní aplikace, která umožňuje naskenovat hrací plochu hry Kakuro z jakéhokoliv tištěného média a pomoci uživateli s jejím dohráním. Řešení využívá poznatky a metody z oblasti počítačového vidění a strojového učení, na základě kterých řeší problematiku detekce mřížky v obraze nebo rozpoznávání ručně psaných číslic a hracích polí za pomoci konvolučních neuronových sítí. Aplikace se celkově skládá ze serverové a klientské části. Klientské řešení zahrnuje mobilní aplikaci, vyvinutou pomocí technologie Flutter, která s pomocí serverové části aplikace, implementované v programovacím jazyce Python, zkonstruuje virtuální hrací plochu a poskytne uživateli pomoc s řešením hry. Aplikace je dostupná na zařízení s operačním systémem Android a iOS.The purpose of this thesis is to create a mobile application, which allows to scan a Kakuro game from any printed media and helps it's user solving it. The solution is composed of client and server sides, where client is the mobile application implemented in the Flutter framework, which collaborates with server side to construct a virtual playground and offer solution of the game to the user. This thesis also includes studies of computer vision and machine learning techniques, software engineering and algorithmic solving of the Kakuro game, which are necessary for constructing such  applications. The outcome of this work is a fully functional mobile application which allows it's user to scan the game and offers help with solution. The app is available for devices with Android and iOS operating systems.

All Paths Lead to Rome

All roads lead to Rome is the core idea of the puzzle game Roma. It is played on an $n \times n$ grid consisting of quadratic cells. Those cells are grouped into boxes of at most four neighboring cells and are either filled, or to be filled, with arrows pointing in cardinal directions. The goal of the game is to fill the empty cells with arrows such that each box contains at most one arrow of each direction and regardless where we start, if we follow the arrows in the cells, we will always end up in the special Roma-cell. In this work, we study the computational complexity of the puzzle game Roma and show that completing a Roma board according to the rules is an \NP-complete task, counting the number of valid completions is #Ptime-complete, and determining the number of preset arrows needed to make the instance \emph{uniquely} solvable is $\Sigma_2^P$-complete. We further show that the problem of completing a given Roma instance on an $n\times n$ board cannot be solved in time $\mathcal{O}\left(2^{{o}(n)}\right)$ under ETH and give a matching dynamic programming algorithm based on the idea of Catalan structures

Incremental inference on higher-order probabilistic graphical models applied to constraint satisfaction problems

Thesis (PhD)--Stellenbosch University, 2022.ENGLISH ABSTRACT: Probabilistic graphical models (PGMs) are used extensively in the probabilistic reasoning domain. They are powerful tools for solving systems of complex relationships over a variety of probability distributions, such as medical and fault diagnosis, predictive modelling, object recognition, localisation and mapping, speech recognition, and language processing [5, 6, 7, 8, 9, 10, 11]. Furthermore, constraint satisfaction problems (CSPs) can be formulated as PGMs and solved with PGM inference techniques. However, the prevalent literature on PGMs shows that suboptimal PGM structures are primarily used in practice and a suboptimal formulation for constraint satisfaction PGMs. This dissertation aimed to improve the PGM literature through accessible algorithms and tools for improved PGM structures and inference procedures, specifically focusing on constraint satisfaction. To this end, this dissertation presents three published contributions to the current literature: a comparative study to compare cluster graph topologies to the prevalent factor graphs [1], an application of cluster graphs in land cover classification in the field of cartography [2], and a comprehensive integration of various aspects required to formulate CSPs as PGMs and an algorithm to solve this formulation for problems too complex for traditional PGM tools [3]. First, we present a means of formulating and solving graph colouring problems with probabilistic graphical models. In contrast to the prevailing literature that mostly uses factor graph configurations, we approach it from a cluster graph perspective, using the general-purpose cluster graph construction algorithm, LTRIP. Our experiments indicate a significant advantage for preferring cluster graphs over factor graphs, both in terms of accuracy as well as computational efficiency. Secondly, we use these tools to solve a practical problem: land cover classification. This process is complex due to measuring errors, inefficient algorithms, and low-quality data. We proposed a PGM approach to boost geospatial classifications from different sources and consider the effects of spatial distribution and inter-class dependencies (similarly to graph colouring). Our PGM tools were shown to be robust and were able to produce a diverse, feasible, and spatially-consistent land cover classification even in areas of incomplete and conflicting evidence. Lastly, in our third publication, we investigated and improved the PGM structures used for constraint satisfaction. It is known that tree-structured PGMs always result in an exact solution [12, p355], but is usually impractical for interesting problems due to exponential blow-up. We, therefore, developed the “purge-and merge” algorithm to incrementally approximate a tree-structured PGM. This algorithm iteratively nudges a malleable graph structure towards a tree structure by selectively merging factors. The merging process is designed to avoid exponential blow-up through sparse data structures from which redundancy is purged as the algorithm progresses. This algorithm is tested on constraint satisfaction puzzles such as Sudoku, Fill-a-pix, and Kakuro and manages to outperform other PGM-based approaches reported in the literature [13, 14, 15]. Overall, the research reported in this dissertation contributed to developing a more optimised approach for higher order probabilistic graphical models. Further studies should concentrate on applying purge-and-merge on problems closer to probabilistic reasoning than constraint satisfaction and report its effectiveness in that domain.AFRIKAANSE OPSOMMING: Grafiese waarskynlikheidsmodelle (PGM) word wyd gebruik vir komplekse waarskynlikheidsprobleme. Dit is kragtige gereedskap om sisteme van komplekse verhoudings oor ‘n versameling waarskynlikheidsverspreidings op te los, soos die mediese en foutdiagnoses, voorspellingsmodelle, objekherkenning, lokalisering en kartering, spraakherkenning en taalprosessering [5, 6, 7, 8, 9, 10, 11]. Voorts kan beperkingvoldoeningsprobleme (CSP) as PGM’s geformuleer word en met PGM gevolgtrekkingtegnieke opgelos word. Die heersende literatuur oor PGM’s toon egter dat sub-optimale PGM-strukture hoofsaaklik in die praktyk gebruik word en ‘n sub-optimale PGM-formulering vir CSP’s. Die doel met die verhandeling is om die PGM-literatuur deur toeganklike algoritmes en gereedskap vir verbeterde PGM-strukture en gevolgtrekking-prosedures te verbeter deur op CSP toepassings te fokus. Na aanleiding hiervan voeg die verhandeling drie gepubliseerde bydraes by die huidige literatuur: ‘n vergelykende studie om bundelgrafieke tot die heersende faktorgrafieke te vergelyk [1], ‘n praktiese toepassing vir die gebruik van bundelgrafieke in “land-cover”- klassifikasie in die kartografieveld [2] en ‘n omvattende integrasie van verskeie aspekte om CSP’s as PGM’s te formuleer en ‘n algoritme vir die formulering van probleme te kompleks vir tradisionele PGM-gereedskap [3] Eerstens bied ons ‘n wyse van formulering en die oplos van grafiekkleurprobleme met PGM’s. In teenstelling met die huidige literatuur wat meestal faktorgrafieke gebruik, benader ons dit van ‘n bundelgrafiek-perspektief deur die gebruik van die automatiese bundelgrafiekkonstruksie-algoritme, LTRIP. Ons eksperimente toon ‘n beduidende voorkeur vir bundelgrafieke teenoor faktorgrafieke, wat akku raatheid asook berekende doeltreffendheid betref. Tweedens gebruik ons die gereedskap om ‘n praktiese probleem op te los: “landcover”-klassifikasie. Die proses is kompleks weens metingsfoute, ondoeltreffende algoritmes en lae-gehalte data. Ons stel ‘n PGM-benadering voor om die georuimtelike klassifikasies van verskillende bronne te versterk, asook die uitwerking van ruimtelike verspreiding en interklas-afhanklikhede (soortgelyk aan grafiekkleurprobleme). Ons PGM-gereedskap is robuus en kon ‘n diverse, uitvoerbare en ruimtelik-konsekwente “land-cover”-klassifikasie selfs in gebiede van onvoltooide en konflikterende inligting bewys. Ten slotte het ons in ons derde publikasie die PGM-strukture vir CSP’s ondersoek en verbeter. Dit is bekend dat boomstrukture altyd tot ‘n eksakte oplossing lei [12, p355], maar is weens eksponensiële uitbreiding gewoonlik onprakties vir interessante probleme. Ons het gevolglik die algoritme, purge-and-merge, ontwikkel om inkrementeel ‘n boomstruktuur na te doen. Die algoritme hervorm ‘n bundelgrafiek stapsgewys in ‘n boomstruktuur deur faktore selektief te “merge”. Die saamsmeltproses is ontwerp om eksponensiële uitbreiding te vermy deur van yl datastrukture gebruik te maak waarvan die waarskeinlikheidsruimte ge-“purge” word namate die algoritme vorder. Die algoritme is getoets op CSP-speletjies soos Sudoku, Fill-a-pix en Kakuro en oortref ander PGM-gegronde benaderings waaroor in die literatuur verslag gedoen word [13, 14, 15]. In die geheel gesien, het die navorsing bygedra tot die ontwikkeling van ‘n meer geoptimaliseerde benadering vir hoër-orde PGM’s. Verdere studies behoort te fokus op die toepassing van purge-and-merge op probleme nader aan waarskynlikheidsredenasie-probleme as aan CSP’s en moet sy effektiwiteit in daar die domein rapporteer.Doctora