The Alcuin number of a graph and its connections to the vertex cover number

We consider a planning problem that generalizes Alcuin's river crossing problem to scenarios with arbitrary conflict graphs. This generalization leads to the so-called Alcuin number of the underlying conflict graph. We derive a variety of combinatorial, structural, algorithmical, and complexity theoretical results around the Alcuin number. Our technical main result is an NP-certificate for the Alcuin number. It turns out that the Alcuin number of a graph is closely related to the size of a minimum vertex cover in the graph, and we unravel several surprising connections between these two graph parameters. We provide hardness results and a fixed parameter tractability result for computing the Alcuin number. Furthermore we demonstrate that the Alcuin number of chordal graphs, bipartite graphs, and planar graphs is substantially easier to analyze than the Alcuin number of general graphs

The Computational Complexity of Some Games and Puzzles With Theoretical Applications

The subject of this thesis is the algorithmic properties of one- and two-player games people enjoy playing, such as Sudoku or Chess. Questions asked about puzzles and games in this context are of the following type: can we design efficient computer programs that play optimally given any opponent (for a two-player game), or solve any instance of the puzzle in question? We examine four games and puzzles and show algorithmic as well as intractability results. First, we study the wolf-goat-cabbage puzzle, where a man wants to transport a wolf, a goat, and a cabbage across a river by using a boat that can carry only one item at a time, making sure that no incompatible items are left alone together. We study generalizations of this puzzle, showing a close connection with the Vertex Cover problem that implies NP-hardness as well as inapproximability results. Second, we study the SET game, a card game where the objective is to form sets of cards that match in a certain sense using cards from a special deck. We study single- and multi-round variations of this game and establish interesting con- nections with other classical computational problems, such as Perfect Multi- Dimensional Matching, Set Packing, Independent Edge Dominating Set, and Arc Kayles. We prove algorithmic and hardness results in the classical and the parameterized sense. Third, we study the UNO game, a game of colored numbered cards where players take turns discarding cards that match either in color or in number. We extend results by Demaine et. al. (2010 and 2014) that connected one- and two-player generaliza- tions of the game to Edge Hamiltonian Path and Generalized Geography, proving that a solitaire version parameterized by the number of colors is fixed param- eter tractable and that a k-player generalization for k greater or equal to 3 is PSPACE-hard. Finally, we study the Scrabble game, a word game where players are trying to form words in a crossword fashion by placing letter tiles on a grid board. We prove that a generalized version of Scrabble is PSPACE-hard, answering a question posed by Demaine and Hearn in 2008

The Alcuin number of a graph

We consider a planning problem that generalizes Alcuin’s river crossing problem (also known as: The wolf, goat, and cabbage puzzle) to scenarios with arbitrary conflict graphs. We derive a variety of combinatorial, structural, algorithmical, and complexity theoretical results around this problem

The Alcuin number of a graph

We consider a planning problem that generalizes Alcuin’s river crossing problem (also known as: The wolf, goat, and cabbage puzzle) to scenarios with arbitrary conflict graphs. We derive a variety of combinatorial, structural, algorithmical, and complexity theoretical results around this problem

MIP models for problems based on the Alcuin Number of a graph

openIl presente lavoro di tesi si prefigge lo studio di una generalizzazione dell’Alcuin river crossing problem, noto anche come il problema dell’attraversamento del fiume di Alcuin, con l’obiettivo di analizzare i tempi necessari per determinare la soluzione del problema e di valutare il legame tra il numero di Alcuin di un grafo e la sua densita`. Questo elaborato e` stato articolato in 4 capitoli: inizialmente si presentano al- cune nozioni teoriche fondamentali, necessarie per la comprensione dello studio, successivamente si presentano le modalita` usate per affrontare le prove sperimen- tali e si analizzano i risultati ottenuti. Infine si elaborano le conclusioni del lavoro svolto. Per affrontare questo studio sono stati scritti diversi script Python[1] che svolgo- no le varie funzioni necessarie partendo dalla generazione casuale di grafi fatta usando il pacchetto python Netwokx[2], fino all’elaborazione dei risultati ottenu- ti. Tutto il codice e` consultabile liberamente online tramite la repository usata per il controllo versione del progetto[14]. Per la risoluzione dei modelli matematici e` stato usato il software CPLEX 22.1.1[9] e la sua interfaccia su Python[1] trami- te il pacchetto Pyomo[3]. Per la presentazione dei risultati in maniera grafica e per l’elaborazione del file .csv sono stati utilizzati i seguenti pacchetti Python[1]: Matplotlib[4], Seaborn[5], Numpy[6], Pandas[7]. I tempi di calcolo presenti nei risultati di questo elaborato fanno riferimento al cluster del Dipartmento di Ingegneria dell’Informazione dell’Universita` degli Stu- di di Padova. Per poter usare il cluster e` stato necessario l’utilizzo dello scheduler generico SLURM[8]. Molte delle definizioni presenti in questo elaborato sono ispirate alle dispense del corso di Modelli e Software per l’Ottimizzazione Discreta, tenuto dal professore Domenico Salvagnin[11][12][13] presso l’Universita` degli Studi di Padova.This thesis work aims to study a generalization of the river crossing problem with the aim to analyzing the necessary time to determine the problem solution and to evaluate the link between the Alcuin number of a graph and its density. This work is divided in four chapters: initially some fundamental theoretical notions are presented, necessary to understand the study, subsequently the methods used to deal with the experimental tests are presented and the results obtainded are analyzed. Finally the conclusions of the work carried out are drawn up. To tackle this study several Python[1] script were written which perform all the necessary functions starting from the graphs random generation done using Python package Networkx[2], up to the processing of the obtained results. All of the code can be freely consulted online via the repository used for project version control[14]. To solve mathematical models was used Cplex22.1.1[9] software and its Python[1] interface via the Pyomo[3] package. The following Python[1] packages were used to present the results graphically and to process the .csv result file: Matplotlib[4], Seaborn[5], Numpy[6], Pandas[7]. The calculation times presented in this paper refer to the cluster of Department of Information Engineering of the University of Padova. To be able to use the cluster it was necessary to use the generic scheduler SLURM[8]. Most of the definition presented in this paper are inspired by the lecture notes of Models and Software for Discrete Optimization course, held by professor Domenico Salvagnin[11][12][13] at University of Padua

The Alcuin number of a graph and its connections to the vertex cover number

We consider a planning problem that generalizes Alcuin's river crossing problem to scenarios with arbitrary conflict graphs. This generalization leads to the so-called Alcuin number of the underlying conflict graph. We derive a variety of combinatorial, structural, algorithmical, and complexity theoretical results around the Alcuin number. Our technical main result is an NP-certificate for the Alcuin number. It turns out that the Alcuin number of a graph is closely related to the size of a minimum vertex cover in the graph, and we unravel several surprising connections between these two graph parameters. We provide hardness results and a fixed parameter tractability result for computing the Alcuin number. Furthermore we demonstrate that the Alcuin number of chordal graphs, bipartite graphs, and planar graphs is substantially easier to analyze than the Alcuin number of general graphs. Key words: transportation problem, scheduling and planning, graph theory, vertex cover This paper originally appeared in SIAM Journal on Discrete Mathematics, Volume 24, Number 3, 2010, pages 757–769