Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a simple path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding $\Sigma_2$-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving "subset Hamiltonian path" for terminals on the boundary of an embedded planar graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of this paper appeared at the 9th International Conference on Fun with Algorithms (FUN 2018

Theory Learning with Symmetry Breaking

This paper investigates the use of a Prolog coded SMT solver in tackling a well known constraints problem, namely packing a given set of consecutive squares into a given rectangle, and details the developments in the solver that this motivates. The packing problem has a natural model in the theory of quantifier-free integer difference logic, a theory supported by many SMT solvers. The solver used in this work exploits a data structure consisting of an incremental Floyd-Warshall matrix paired with a watch matrix that monitors the entailment status of integer difference constraints. It is shown how this structure can be used to build unsatisfiable theory cores on the fly, which in turn allows theory learning to be incorporated into the solver. Further, it is shown that a problem-specific and non-standard approach to learning can be taken where symmetry breaking is incorporated into the learning stage, magnifying the effect of learning. It is argued that the declarative framework allows the solver to be used in this white box manner and is a strength of the solver. The approach is experimentally evaluated

A general genetic algorithm for one and two dimensional cutting and packing problems

Cutting and packing problems are combinatorial optimisation problems. The major interest in these problems is their practical significance, in manufacturing and other business sectors. In most manufacturing situations a raw material usually in some standard size has to be divided or be cut into smaller items to complete the production of some product. Since the cost of this raw material usually forms a significant portion of the input costs, it is therefore desirable that this resource be used efficiently. A hybrid general genetic algorithm is presented in this work to solve one and two dimensional problems of this nature. The novelties with this algorithm are: A novel placement heuristic hybridised with a Genetic Algorithm is introduced and a general solution encoding scheme which is used to encode one dimensional and two dimensional problems is also introduced

Optimization Modulo Theories with Linear Rational Costs

In the contexts of automated reasoning (AR) and formal verification (FV), important decision problems are effectively encoded into Satisfiability Modulo Theories (SMT). In the last decade efficient SMT solvers have been developed for several theories of practical interest (e.g., linear arithmetic, arrays, bit-vectors). Surprisingly, little work has been done to extend SMT to deal with optimization problems; in particular, we are not aware of any previous work on SMT solvers able to produce solutions which minimize cost functions over arithmetical variables. This is unfortunate, since some problems of interest require this functionality. In the work described in this paper we start filling this gap. We present and discuss two general procedures for leveraging SMT to handle the minimization of linear rational cost functions, combining SMT with standard minimization techniques. We have implemented the procedures within the MathSAT SMT solver. Due to the absence of competitors in the AR, FV and SMT domains, we have experimentally evaluated our implementation against state-of-the-art tools for the domain of linear generalized disjunctive programming (LGDP), which is closest in spirit to our domain, on sets of problems which have been previously proposed as benchmarks for the latter tools. The results show that our tool is very competitive with, and often outperforms, these tools on these problems, clearly demonstrating the potential of the approach.Comment: Submitted on january 2014 to ACM Transactions on Computational Logic, currently under revision. arXiv admin note: text overlap with arXiv:1202.140

Master Texture Space: An Efficient Encoding for Projectively Mapped Objects

Projectively textured models are used in an increasingly large number of applicationsthat dynamically combine images with a simple geometric surface in a viewpoint dependentway. These models can provide visual fidelity while retaining the effects affordedby geometric approximation such as shadow casting and accurate perspective distortion.However, the number of stored views can be quite large and novel views must be synthesizedduring the rendering process because no single view may correctly texture the entireobject surface. This work introduces the Master Texture encoding and demonstrates thatthe encoding increases the utility of projectively textured objects by reducing render-timeoperations. Encoding involves three steps; 1) all image regions that correspond to the samegeometric mesh element are extracted and warped to a facet of uniform size and shape,2) an efficient packing of these facets into a new Master Texture image is computed, and3) the visibility of each pixel in the new Master Texture data is guaranteed using a simplealgorithm to discard occluded pixels in each view. Because the encoding implicitly representsthe multi-view geometry of the multiple images, a single texture mesh is sufficientto render the view-dependent model. More importantly, every Master Texture image cancorrectly texture the entire surface of the object, removing expensive computations suchas visibility analysis from the rendering algorithm. A benefit of this encoding is the supportfor pixel-wise view synthesis. The utility of pixel-wise view synthesis is demonstratedwith a real-time Master Texture encoded VDTM application. Pixel-wise synthesis is alsodemonstrated with an algorithm that distills a set of Master Texture images to a singleview-independent Master Texture image

Two-dimensional placement compaction using an evolutionary approach: a study

The placement problem of two-dimensional objects over planar surfaces optimizing given utility functions is a combinatorial optimization problem. Our main drive is that of surveying genetic algorithms and hybrid metaheuristics in terms of final positioning area compaction of the solution. Furthermore, a new hybrid evolutionary approach, combining a genetic algorithm merged with a non-linear compaction method is introduced and compared with referenced literature heuristics using both randomly generated instances and benchmark problems. A wide variety of experiments is made, and the respective results and discussions are presented. Finally, conclusions are drawn, and future research is defined