Minesweeper May Not Be NP-Complete but Is Hard Nonetheless

Scott, Allan; Stege, Ulrike; Rooij, Iris van

doi:10.1007/s00283-011-9256-x

Cited by 44 publications

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“…However, small-scale instances can make a good puzzle for humans to solve. Other popular puzzle games that are computationally hard are Sudoku (Ercsey-Ravasz and Toroczkai 2012) and Minesweeper (Scott et al 2011). …”

Section: Introductionmentioning

confidence: 99%

Planning Your Route: Where to Start?

2018

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Tour planning is an important part of location-based applications. A tour planner provides an optimized path through places of interests (targets) by minimizing the tour length or by applying some other constraints. It is usually formulated as a travelling salesman problem (TSP) or vehicle routing problem (VRP). In the present study, we focus on how to choose the best starting location in case of an open-loop TSP. We consider three different strategies for selecting the starting location and compare their effectiveness with regard to optimizing tour length. If all targets are visible, most humans tend to start on the convex hull or from the furthest point. However, there are also cases where not all targets are visible beforehand, and the only information given is the bounding box. An optimum tour then typically starts from the corner or the shorter side of the box. Humans also have a strong preference to start from a corner. A good strategy can result in the shortest tour, while a bad strategy can even add 20% to the total tour length.

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Section: Introductionmentioning

confidence: 99%

Planning Your Route: Where to Start?

2018

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“…The problem of uncovering the (hidden) information of a cell on a given minesweeper board is also computationally hard (in fact, this problem is co-NP-complete). In this latter version, we assume the existence of a solution to the minesweeper board (Scott, Stege, & van Rooij, 2011). We will pick up on this topic in the last section of this article.…”

Section: Known Hard Problems We Play For Funmentioning

confidence: 99%

“…This, in effect, prevents players from exploring moves that may lead down incorrect paths through the problem space. Indeed, even the problem to decide whether the content of another cell on a given minesweeper board can be inferred is computationally hard (Scott et al, 2011). Even if the correct answer is that an inference is possible, backtracking may be a useful aid for humans playing minesweeper.…”

Section: Presentationmentioning

confidence: 99%

“…The complexity of these different problems can be different: for example, in the case of Minesweeper, the Minesweeper Consistency problem that asks whether or not a given Minesweeper board does have a solution, is shown to be NP-complete (Kaye, 2000), whereas Minesweeper Inference, the promise problem that assumes as input a consistent Minesweeper board that can be solved asks, whether or not with the information available the content of a hidden cell can be inferred, is co-NP-complete (Scott et al, 2011). It is theoretically possible that the promise problem of its intractable non-promise version is a tractable one.…”

Section: Future Research: Research Questionsmentioning

confidence: 99%

“…Consider the game Minesweeper, which has been shown to be computationally hard in its general case (Scott et al, 2011), and which human performance has been studied, at least in a limited sense (Walker, 2010; also see above). A typical implementation of the Minesweeper game-as played for example on a PC-prevents backtracking.…”

Section: Presentationmentioning

confidence: 99%

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On Evaluating Human Problem Solving of Computationally Hard Problems

Carruthers

Stege²

2013

The Journal of Problem Solving

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This article is concerned with how computer science, and more exactly computational complexity theory, can inform cognitive science. In particular, we suggest factors to be taken into account when investigating how people deal with computational hardness. This discussion will address the two upper levels of Marr's Level Theory: the computational level and the algorithmic level. Our reasons for believing that humans indeed deal with hard cognitive functions are threefold: (1) Several computationally hard functions are suggested in the literature, e.g., in the areas of visual search, visual perception and analogical reasoning, linguistic processing, and decision making. This article gives a brief introduction to some theories and foundations of complexity theory and motivates the use of computationally hard problems in human problem solving with a short survey of known results of human performance, a review of some computationally hard games and puzzles, and the connection between complexity theory and models of cognitive functions. We aim to illuminate the role that computer science, in particular complexity theory, can play in the study of human problem solving. Theoretical computer science can provide a wealth of interesting problems for human study, but it can also help to provide deep insight into these problems. In particular, we discuss the role that computer science can play when choosing computational problems for study and designing experiments to investigate human performance. Finally, we enumerate issues and pitfalls that can arise when choosing computationally hard problems as the subject of study, in turn motivating some interesting potential future lines of study. The pitfalls addressed include: choice of presentation and representation of problem instances, evaluation of problem comprehension, and the role of cognitive support in experiments. Our goal is not to exhaustively list all the ways in which these choices may impact experimental studies, but rather to provide a few simple examples in order to highlight possible pitfalls. Keywords problem solving, computational complexity, intractability, algorithmic level theories, cognitive functions 1 University of Victoria. Please direct correspondence to scarruth@uvic.ca.

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LaserTank is NP-Complete

Alexandersson

Restadh

2020

Mathematical Aspects of Computer and Information Sciences

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Minesweeper May Not Be NP-Complete but Is Hard Nonetheless

Cited by 44 publications

References 0 publications

Planning Your Route: Where to Start?

Planning Your Route: Where to Start?

On Evaluating Human Problem Solving of Computationally Hard Problems

LaserTank is NP-Complete

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