NP-complete problems

NP-complete problems are a class of problems in computational complexity theory. To understand NP-complete problems, we need to break down the concepts of "problem classes" and the related terminology. 1. **P (Polynomial time)**: This class contains decision problems (problems with a yes/no answer) for which a solution can be found in polynomial time.

Nonograms, also known as "Picross," "Griddlers," or "Paint by Numbers," are logic puzzle games involving a grid. The objective is to fill the grid with black squares according to a set of numerical clues provided for each row and column. These clues indicate consecutive blocks of filled squares, and players use them to deduce which squares should be filled in and which should remain empty.

Number partitioning is a problem in combinatorial mathematics and computer science that involves dividing a set of integers into two or more subsets such that certain conditions are met. The most common form of the problem involves partitioning a set of integers into two subsets that have equal sums, known as the "Partition Problem." The Partition Problem can be formally defined as follows: Given a set of integers, can it be divided into two subsets such that the sum of the elements in each subset is equal?

SAT solvers are algorithms or software tools designed to solve the Boolean satisfiability problem (SAT). The SAT problem involves determining whether there exists an assignment of truth values (true or false) to a set of variables such that a given Boolean formula evaluates to true. The problem is typically represented in conjunctive normal form (CNF), which is a conjunction (AND) of clauses, where each clause is a disjunction (OR) of literals (variables or their negations).

SMT solvers, or Satisfiability Modulo Theories solvers, are tools designed to determine the satisfiability of logical formulas with respect to specific theories. They extend the capabilities of traditional SAT solvers, which deal only with propositional logic, by incorporating more complex theories such as: 1. **Arithmetic**: Integers, reals, etc. 2. **Bit-vectors**: Operations on sequences of bits.

Strongly NP-complete problems are a subset of NP-complete problems that remain NP-complete even when the numerical values in the input are bounded by a polynomial in the size of the input. This contrasts with "weakly NP-complete" problems, which can be solved in polynomial time when the numbers involved are small (i.e., their magnitude is polynomially bounded) but may be hard in the general case where numerical values can be arbitrary.

Weakly NP-complete problems are a subset of decision problems that are considered to be NP-complete, but with a specific characteristic: they can be solved in polynomial time given a solution in the form of a polynomial-size numerical representation. This means that the problem can be solved in polynomial time when the numerical parameters or inputs are bounded by a polynomial size.

Battleship is a logic-based puzzle game that is reminiscent of the classic naval combat board game. In the puzzle format, the player is presented with a grid where they must place a set number of ships of varying lengths without overlapping them, while also ensuring that they adhere to specific clues provided in the puzzle. The grid typically has numbers along the top and left sides, which indicate how many segments of ships are located in each row and column.

The Generalized Assignment Problem (GAP) is an optimization problem that involves assigning a set of tasks to a set of agents (or resources) in a way that minimizes costs or maximizes efficiency while respecting certain constraints. Unlike the classical assignment problem, where each task is assigned to exactly one agent, the Generalized Assignment Problem allows each agent to handle multiple tasks, but with additional limitations.

Hashiwokakero is a logic-based puzzle game that originated in Japan. The name translates to "Build Bridges" in English, which describes the game's objective. The game is played on a grid that contains various islands, represented by circles. The goal is to connect these islands with bridges, following specific rules: 1. Each island has a number that indicates how many bridges must connect to it. 2. Bridges can only be drawn horizontally or vertically between islands.

Hitori is a logic-based puzzle game that originated in Japan. The objective of Hitori is to fill in cells in a grid based on certain rules, which typically involve the numbers provided in the cells. The game is played on a rectangular grid filled with numbers. Here are the basic rules: 1. **Grid Structure**: The grid consists of a set of numbers, where each number represents how many times that number appears in its respective row and column.

Kakuro is a number puzzle that is sometimes referred to as a "cross-sum" puzzle. It is similar to a crossword puzzle but uses numbers instead of words. The objective of Kakuro is to fill in the blank cells in a grid with digits from 1 to 9, such that the numbers in each group of contiguous cells sum up to the specified clues provided in the grid.

Richard Karp’s seminal 1972 paper, "Reducibility Among Combinatorial Problems," identified 21 specific problems that are NP-complete, which means they are among the most challenging problems in computational complexity theory. Below is a list of these 21 problems: 1. **Clique Problem**: Given a graph \( G \) and an integer \( k \), is there a complete subgraph (clique) of size \( k \)?

Light Up is a type of logic puzzle that involves placing "lights" in a grid to illuminate the entire area according to specific rules. The goal is to position lights (often represented as circles or bulbs) in such a way that all cells in the grid are lit up, while adhering to certain constraints. Here are some key features of Light Up puzzles: 1. **Grid and cells**: The puzzle is typically played on a rectangular grid made up of individual cells.

NP-complete problems are a class of problems in computational complexity theory that are particularly important because they are both in NP (nondeterministic polynomial time) and as hard as any problem in NP. If any NP-complete problem can be solved in polynomial time, then every problem in NP can also be solved in polynomial time.

Mastermind is a classic board game that involves strategy and logic. Originally created in the 1970s, it is typically played by two players. The game consists of a code maker and a code breaker. ### Game Components: - **Board**: A long, horizontal board with rows of pegs. - **Pegs**: Available in several colors. - **Key pegs**: Used to give feedback to the code breaker.

Masyu is a type of logic puzzle that typically appears on puzzle websites and in puzzle books. The objective of Masyu is to draw a single loop through a grid, following certain rules indicated by special circles within the grid. Here are the basic rules of Masyu: 1. **Loop Structure**: The loop must be a continuous, non-intersecting path that forms a closed loop.

Minesweeper is a classic single-player puzzle video game that involves logic and strategy. It gained widespread popularity with its inclusion in Microsoft Windows in the early 1990s, although it has origins dating back to earlier games. ### Gameplay Mechanics: - **Objective**: The primary goal of Minesweeper is to clear a rectangular board containing hidden "mines" or bombs without detonating any of them. Players aim to uncover all safe squares while avoiding the mines.

NP-completeness is a concept from computational complexity theory that classifies certain decision problems based on their solvability and the difficulty of solving them. Let's break down the relevant terms: 1. **P (Polynomial Time)**: This class consists of decision problems (problems with a yes/no answer) that can be solved by a deterministic Turing machine in polynomial time. In simpler terms, these are problems for which an efficient (i.e., fast) algorithm exists.

Not-All-Equal 3-Satisfiability (NAE-3-SAT) is a variation of the standard 3-Satisfiability (3-SAT) problem in the field of computational complexity and logic. In general, the 3-SAT problem involves determining whether there is an assignment of truth values to variables in a boolean formula written in conjunctive normal form (CNF), such that the entire formula evaluates to true.

Nurikabe is a logic-based puzzle that originated in Japan. The name "Nurikabe" translates to "painted wall" in English. It involves filling a grid with "walls" represented by cells, following specific rules to create an interesting path or area. ### Rules of Nurikabe: 1. **Grid Structure**: The puzzle is played on a rectangular grid. Some cells contain numbers, which indicate how many contiguous cells are part of a "region" or island.

As of my last update in October 2023, "Quadrel" could refer to various contexts, including companies, products, or concepts, depending on the industry or focus area. However, without specific context, it's hard to pinpoint exactly what you're referring to. If "Quadrel" refers to a company, product name, or technology, it would be helpful to provide additional details or context for a more precise answer.

SameGame is a puzzle video game designed by the computer scientist David S. Wilcox in 1991. The objective of the game is to clear the board of colored blocks by removing groups of two or more adjacent blocks of the same color. When a group is removed, the remaining blocks drop down to fill the gaps, which can potentially create new groups that can also be removed.

The Set Splitting Problem is a classic problem in computational complexity and combinatorial optimization.

Shakashaka is a logic puzzle that is a type of grid-based game, often similar to Sudoku or other number-placement puzzles. The objective of Shakashaka is to fill a grid with certain shapes or numbers based on specific rules, where clues provided around the grid guide how to complete it. The game involves critical thinking and pattern recognition, engaging players in a fun and challenging way.

Slitherlink is a logic-based puzzle that consists of a grid of dots. The objective of the puzzle is to create a single, continuous loop that connects the dots and satisfies certain numerical clues given within the grid. The loop can only go horizontally or vertically between dots and must not cross itself or branch off. Here are some key elements of Slitherlink: 1. **Grid Structure**: The puzzle is typically laid out on a rectangular or square grid defined by dots.

"Tentai Show" does not appear to be a widely recognized term or specific entity in popular culture, media, or significant events as of my last update in October 2023. It could refer to a variety of things such as a show, a brand, or a concept within a particular niche that hasn't gained widespread attention. If you meant something specific or if it refers to a local or emerging trend, can you provide more context or clarification?

Tetris is a classic video game that was created in 1984 by Russian computer scientist Alexey Pajitnov. The game is a puzzle-based arcade game where players manipulate falling blocks called "tetrominoes," which are geometric shapes composed of four squares each. The objective is to rotate and move these tetrominoes as they fall from the top of the screen, fitting them together to create complete horizontal lines at the bottom of the play area.

The Traveling Purchaser Problem (TPP) is a combinatorial optimization problem that is a variant of the well-known Traveling Salesman Problem (TSP). In the TPP, the objective is to identify the most efficient route for a purchaser who needs to buy a set of items from different locations while minimizing the total cost. This cost includes not only the travel costs between locations but also the prices of the items being purchased.

A unit disk graph is a type of geometric graph that is defined as follows: 1. **Vertices**: Each vertex in the graph corresponds to a point in a two-dimensional space (often represented as \(\mathbb{R}^2\)) with a specific location determined by its coordinates.

The Vehicle Routing Problem (VRP) is a combinatorial optimization problem that aims to determine the most efficient routes for a fleet of vehicles to deliver goods to a set of customers. It is a fundamental problem in logistics and transportation and has applications in various industries, including delivery services, food distribution, and waste collection. The primary objective of the VRP is to minimize certain costs, such as total distance traveled, time taken, or operational expenses while ensuring that all customer demands are met.