An Efficient Local Search for Partial Latin Square Extension Problem

A partial Latin square (PLS) is a partial assignment of n symbols to an nxn grid such that, in each row and in each column, each symbol appears at most once. The partial Latin square extension problem is an NP-hard problem that asks for a largest extension of a given PLS. In this paper we propose an efficient local search for this problem. We focus on the local search such that the neighborhood is defined by (p,q)-swap, i.e., removing exactly p symbols and then assigning symbols to at most q empty cells. For p in {1,2,3}, our neighborhood search algorithm finds an improved solution or concludes that no such solution exists in O(n^{p+1}) time. We also propose a novel swap operation, Trellis-swap, which is a generalization of (1,q)-swap and (2,q)-swap. Our Trellis-neighborhood search algorithm takes O(n^{3.5}) time to do the same thing. Using these neighborhood search algorithms, we design a prototype iterated local search algorithm and show its effectiveness in comparison with state-of-the-art optimization solvers such as IBM ILOG CPLEX and LocalSolver.Comment: 17 pages, 2 figure

Using Small MUSes to Explain How to Solve Pen and Paper Puzzles

In this paper, we present Demystify, a general tool for creating human-interpretable step-by-step explanations of how to solve a wide range of pen and paper puzzles from a high-level logical description. Demystify is based on Minimal Unsatisfiable Subsets (MUSes), which allow Demystify to solve puzzles as a series of logical deductions by identifying which parts of the puzzle are required to progress. This paper makes three contributions over previous work. First, we provide a generic input language, based on the Essence constraint language, which allows us to easily use MUSes to solve a much wider range of pen and paper puzzles. Second, we demonstrate that the explanations that Demystify produces match those provided by humans by comparing our results with those provided independently by puzzle experts on a range of puzzles. We compare Demystify to published guides for solving a range of different pen and paper puzzles and show that by using MUSes, Demystify produces solving strategies which closely match human-produced guides to solving those same puzzles (on average 89% of the time). Finally, we introduce a new randomised algorithm to find MUSes for more difficult puzzles. This algorithm is focused on optimised search for individual small MUSes

Uma escala para medir a dificuldade dos quebra-cabeças de Sudoku

In the last few years, Sudoku has become a popular game, deserving the attention of many researchers. Most Sudoku puzzles are classified as easy, average and hard according to their degree of difficulty. However when asked about the criteria used to classify them, there is no clear answer. This paper presents a PnP (paper-and-pencil) method to solve and measure the degree of difficulty of Sudoku puzzles. Contrary to other methods, this method can classify the puzzles automatically without human intervention. The scale used here was inspired on the type of scale used in the snow trails: green, blue, red and black. The Sudoku puzzle is transformed to SAT, and its level is set depending on the SAT problem being easy or hard to solve in SAT, which is equivalent to the puzzle being solved using an easy or hard PnP method. A classifier and a set of classified Sudoku problems are made available on the web.Resumo: Nos últimos anos, o Sudoku tornou-se um jogo muito popular, merecendo a atenção de muitos investigadores. A maior parte dos puzzles de Sudoku é classificada como simples, média ou difícil, consoante o seu grau de dificuldade. Contudo, quando questionados acerca do critério de classificação, não existe uma resposta clara. Este artigo apresenta um método manual para medir o grau de dificuldade dos puzzles de Sudoku. Ao contrário de outros métodos, este método classifica os puzzles automaticamente, sem intervenção humana. A escala utilizada é inspirada na escala utilizada nas pistas de neve: verde, azul, vermelho e negro. O puzzle de Sudoku é transformado num problema de SAT e o nível de dificuldade é dado pela resolução do problema de SAT, o qual é equivalente ao da resolução manual. O classificador de dificuldade e um conjunto de puzzles de Sudoku classificados estão disponíveis na Web.peerreviewe

Using small MUSes to explain how to solve pen and paper puzzles

Pen and paper puzzles like Sudoku, Futoshiki and Skyscrapers are hugely popular. Solving such puzzles can be a trivial task for modern AI systems. However, most AI systems solve problems using a form of backtracking, while people try to avoid backtracking as much as possible. This means that existing AI systems do not output explanations about their reasoning that are meaningful to people. We present Demystify, a tool which allows puzzles to be expressed in a high-level constraint programming language and uses MUSes to allow us to produce descriptions of steps in the puzzle solving. We give several improvements to the existing techniques for solving puzzles with MUSes, which allow us to solve a range of significantly more complex puzzles and give higher quality explanations. We demonstrate the effectiveness and generality of Demystify by comparing its results to documented strategies for solving a range of pen and paper puzzles by hand, showing that our technique can find many of the same explanations.Publisher PD

A Cellular Sudoku Solver

Sudoku is a very popular puzzle which consists on placing several numbers in a squared grid according to some simple rules. In this paper we present an efficient family of P systems which solve sudoku puzzles of any order verifying a specific property. The solution is searched by using a simple human-style method. If the sudoku cannot be solved by using this strategy, the P system detects this drawback and then the computations stops and returns No. Otherwise, the P system encodes the solution and returns Yes in the last computation step.Ministerio de Ciencia e Innovación TIN2008-04487-EMinisterio de Ciencia e Innovación TIN-2009-13192Junta de Andalucía P08-TIC-0420