# Category: NP-complete problems

Minesweeper (video game)
Minesweeper is a logic puzzle video game genre generally played on personal computers. The game features a grid of clickable squares, with hidden "mines" scattered throughout the board. The objective
Minimum k-cut
In mathematics, the minimum k-cut, is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges
Mastermind (board game)
Mastermind or Master Mind is a code-breaking game for two players. It resembles an earlier pencil and paper game called Bulls and Cows that may date back a century.
Satisfiability modulo theories
In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability
Shakashaka
Shakashaka (シャカシャカ) is a logic puzzle developed by publisher Nikoli.
15 puzzle
The 15 puzzle (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and many others) is a sliding puzzle having 15 square tiles numbered 1–15 in a frame that is 4 tiles high and 4 tiles
Hamiltonian path problem
In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path (a path in an undirected or directed gra
Minimum routing cost spanning tree
In computer science, the minimum routing cost spanning tree of a weighted graph is a spanning tree minimizing the sum of pairwise distances between vertices in the tree. It is also called the optimum
Tetris
Tetris (Russian: Тетрис) is a puzzle video game created by Soviet software engineer Alexey Pajitnov in 1984. It has been published by several companies for multiple platforms, most prominently during
Feedback vertex set
In the mathematical discipline of graph theory, a feedback vertex set (FVS) of a graph is a set of vertices whose removal leaves a graph without cycles ("removal" means deleting the vertex and all edg
Betweenness
Betweenness is an algorithmic problem in order theory about ordering a collection of items subject to constraints that some items must be placed between others. It has applications in bioinformatics a
List of NP-complete problems
This is a list of some of the more commonly known problems that are NP-complete when expressed as decision problems. As there are hundreds of such problems known, this list is in no way comprehensive.
Longest path problem
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any r
Steiner tree problem
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Ste
The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a
Hitori
Hitori (Japanese: "Alone" or "one person"; ひとりにしてくれ Hitori ni shite kure; literally "leave me alone") is a type of logic puzzle published by Nikoli. Hitori is NP complete.
Monochromatic triangle
In graph theory and theoretical computer science, the monochromatic triangle problem is an algorithmic problem on graphs,in which the goal is to partition the edges of a given graph into two triangle-
Battleship (puzzle)
The Battleship puzzle (sometimes called Bimaru, Yubotu, Solitaire Battleships or Battleship Solitaire) is a logic puzzle based on the Battleship guessing game. It and its variants have appeared in sev
Boolean satisfiability problem
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if
Edge dominating set
In graph theory, an edge dominating set for a graph G = (V, E) is a subset D ⊆ E such that every edge not in D is adjacent to at least one edge in D. An edge dominating set is also known as a line dom
Unit disk graph
In geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex for each disk in the family, and with an
Set splitting problem
In computational complexity theory, the set splitting problem is the following decision problem: given a family F of subsets of a finite set S, decide whether there exists a partition of S into two su
Masyu
Masyu (ましゅ, Mashu, IPA [maɕu͍]; translates as "evil influence") is a type of logic puzzle designed and published by Nikoli. The purpose of its creation was to present a puzzle that uses no numbers or
Hashiwokakero
Hashiwokakero (橋をかけろ Hashi o kakero; lit. "build bridges!") is a type of logic puzzle published by Nikoli. It has also been published in English under the name Bridges or Chopsticks (based on a mistra
Sudoku
Sudoku (/suːˈdoʊkuː, -ˈdɒk-, sə-/; Japanese: 数独, romanized: sūdoku, lit. 'digit-single'; originally called Number Place) is a logic-based, combinatorial number-placement puzzle. In classic Sudoku, the
Quadrel is a puzzle video game developed by Loriciels and released in June 1991. It was released for MS-DOS, Amiga, Atari ST, and Amstrad CPC
In graph theory, a branch of mathematics, a radio coloring of an undirected graph is a form of graph coloring in which one assigns positive integer labels to the graphssuch that the labels of adjacent
Dominating set
In graph theory, a dominating set for a graph G is a subset D of its vertices, such that any vertex of G is either in D, or has a neighbor in D. The domination number γ(G) is the number of vertices in
Circuit satisfiability problem
In theoretical computer science, the circuit satisfiability problem (also known as CIRCUIT-SAT, CircuitSAT, CSAT, etc.) is the decision problem of determining whether a given Boolean circuit has an as
Bipartite dimension
In the mathematical fields of graph theory and combinatorial optimization, the bipartite dimension or biclique cover number of a graph G = (V, E) is the minimum number of bicliques (that is complete b
Subgraph isomorphism problem
In theoretical computer science, the subgraph isomorphism problem is a computational task in which two graphs G and H are given as input, and one must determine whether G contains a subgraph that is i
Slitherlink (also known as Fences, Takegaki, Loop the Loop, Loopy, Ouroboros, Suriza and Dotty Dilemma) is a logic puzzle developed by publisher Nikoli.
Clique cover
In graph theory, a clique cover or partition into cliques of a given undirected graph is a partition of the vertices into cliques, subsets of vertices within which every two vertices are adjacent. A m
Token reconfiguration
In computational complexity theory and combinatorics, the token reconfiguration problem is a reconfiguration problem on a graph with both an initial and desired state for tokens. Given a graph , an in
Graph partition
In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the gro
Planar SAT
In computer science, the planar 3-satisfiability problem (abbreviated PLANAR 3SAT or PL3SAT) is an extension of the classical Boolean 3-satisfiability problem to a planar incidence graph. In other wor
Maximum common induced subgraph
In graph theory and theoretical computer science, a maximum common induced subgraph of two graphs G and H is a graph that is an induced subgraph of both G and H,and that has as many vertices as possib
Maximum coverage problem
The maximum coverage problem is a classical question in computer science, computational complexity theory, and operations research.It is a problem that is widely taught in approximation algorithms. As
Hamiltonian path
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian
In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic
Set packing
Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems.Suppose one has a finite set S and a list of subsets
Hamiltonian cycle polynomial
In mathematics, the Hamiltonian cycle polynomial of an n×n-matrix is a polynomial in its entries, defined as where is the set of n-permutations having exactly one cycle. This is an algebraic option us
Induced subgraph isomorphism problem
In complexity theory and graph theory, induced subgraph isomorphism is an NP-complete decision problem that involves finding a given graph as an induced subgraph of a larger graph.
Vertex cycle cover
In mathematics, a vertex cycle cover (commonly called simply cycle cover) of a graph G is a set of cycles which are subgraphs of G and contain all vertices of G. If the cycles of the cover have no ver
Clique problem
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several dif
Knapsack problem
The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total we
Not-all-equal 3-satisfiability
In computational complexity, not-all-equal 3-satisfiability (NAE3SAT) is an NP-complete variant of the Boolean satisfiability problem, often used in proofs of NP-completeness.
Hamiltonian completion
The Hamiltonian completion problem is to find the minimal number of edges to add to a graph to make it Hamiltonian. The problem is clearly NP-hard in general case (since its solution gives an answer t
Zero-weight cycle problem
In computer science and graph theory, the zero-weight cycle problem is the problem of deciding whether a directed graph with weights on the edges (which may be positive or negative or zero) has a cycl
Light Up (puzzle)
Light Up (Japanese: 美術館 bijutsukan, art gallery), also called Akari, is a binary-determination logic puzzle published by Nikoli. As of 2011, three books consisting entirely of Light Up puzzles have be
Vehicle routing problem
The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a fleet of vehicles to traverse in order to deliver
Independent set (graph theory)
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set of vertices such that for every two vertices
Nurikabe (puzzle)
Nurikabe (hiragana: ぬりかべ) is a binary determination puzzle named for Nurikabe, an invisible wall in Japanese folklore that blocks roads and delays foot travel. Nurikabe was apparently invented and nam
Instant Insanity
Instant Insanity is the name given by Parker Brothers to their 1967 version of a puzzle which has existed since antiquity, and which has been marketed by many toy and puzzle makers under a variety of
Feedback arc set
In graph theory and graph algorithms, a feedback arc set or feedback edge set in a directed graph is a subset of the edges of the graph that contains at least one edge out of every cycle in the graph.
NP-completeness
In computational complexity theory, a problem is NP-complete when: 1. * it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-for
Tentai Show
Tentai Show (Japanese: 天体ショー tentai shō), also known by the names Tentaisho, Galaxies, Spiral Galaxies, or Sym-a-Pix, is a binary-determination logic puzzle published by Nikoli.
Karp's 21 NP-complete problems
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete. In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Ka
Shortest common supersequence problem
In computer science, the shortest common supersequence of two sequences X and Y is the shortest sequence which has X and Y as subsequences. This is a problem closely related to the longest common subs
Travelling salesman problem
The travelling salesman problem (also called the travelling salesperson problem or TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the
Traveling purchaser problem
The traveling purchaser problem (TPP) is an NP-hard problem studied in theoretical computer science. Given a list of marketplaces, the cost of travelling between different marketplaces, and a list of
Domatic number
In graph theory, a domatic partition of a graph is a partition of into disjoint sets , ,..., such that each Vi is a dominating set for G. The figure on the right shows a domatic partition of a graph;
SameGame
SameGame (さめがめ) is a tile-matching puzzle originally released under the name Chain Shot! in 1985 by Kuniaki Moribe (Morisuke). It has since been ported to numerous computer platforms, handheld devices
Hitting set
No description available.
Degree-constrained spanning tree
In graph theory, a degree-constrained spanning tree is a spanning tree where the maximum vertex degree is limited to a certain constant k. The degree-constrained spanning tree problem is to determine
Vertex cover
In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In computer science, the problem of finding a min
Longest common subsequence problem
The longest common subsequence (LCS) problem is the problem of finding the longest subsequence common to all sequences in a set of sequences (often just two sequences). It differs from the longest com
Set cover problem
The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory. It is one of Karp's 21 NP-complete problems shown to be NP-complete in 197
(SAT, ε-UNSAT)
In computational complexity theory, (SAT, ε-UNSAT) is a language that is used in the proof of the PCP theorem, which relates the language NP to probabilistically checkable proof systems. For a given 3
Chinese postman problem
In graph theory, a branch of mathematics and computer science, Guan's route problem, the Chinese postman problem, postman tour or route inspection problem is to find a shortest closed path or circuit
Generalized assignment problem
In applied mathematics, the maximum generalized assignment problem is a problem in combinatorial optimization. This problem is a generalization of the assignment problem in which both tasks and agents
Kakuro
Kakuro or Kakkuro or Kakoro (Japanese: カックロ) is a kind of logic puzzle that is often referred to as a mathematical transliteration of the crossword. Kakuro puzzles are regular features in many math-an
Set TSP problem
In combinatorial optimization, the set TSP, also known as the generalized TSP, group TSP, One-of-a-Set TSP, Multiple Choice TSP or Covering Salesman Problem, is a generalization of the traveling sales
Graph coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest
Exact cover
In the mathematical field of combinatorics, given a collection S of subsets of a set X, an exact cover is a subcollection S* of S such that each element in X is contained in exactly one subset in S*.
Nonogram
Nonograms, also known as Hanjie, Paint by Numbers, Picross, Griddlers, and Pic-a-Pix, and by various other names, are picture logic puzzles in which cells in a grid must be colored or left blank accor
Maximum cut
For a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets S and T, such that the number of ed
3-dimensional matching
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyp