• Source: 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 thousands of such problems known, this list is in no way comprehensive. Many problems of this type can be found in Garey & Johnson (1979).




Graphs and hypergraphs


Graphs occur frequently in everyday applications. Examples include biological or social networks, which contain hundreds, thousands and even billions of nodes in some cases (e.g. Facebook or LinkedIn).

1-planarity
3-dimensional matching: SP1 
Bandwidth problem: GT40 
Bipartite dimension: GT18 
Capacitated minimum spanning tree: ND5 
Route inspection problem (also called Chinese postman problem) for mixed graphs (having both directed and undirected edges). The program is solvable in polynomial time if the graph has all undirected or all directed edges. Variants include the rural postman problem.: ND25, ND27 
Clique cover problem: GT17 
Clique problem: GT19 
Complete coloring, a.k.a. achromatic number: GT5 
Cycle rank
Degree-constrained spanning tree: ND1 
Domatic number: GT3 
Dominating set, a.k.a. domination number: GT2 
NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem.: ND2 
Feedback vertex set: GT7 
Feedback arc set: GT8 
Graph coloring: GT4 
Graph homomorphism problem: GT52 
Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph. A related problem is to find a partition that is optimal terms of the number of edges between parts.: GT11, GT12, GT13, GT14, GT15, GT16, ND14 
Grundy number of a directed graph.: GT56 
Hamiltonian completion: GT34 
Hamiltonian path problem, directed and undirected.: GT37, GT38, GT39 
Induced subgraph isomorphism problem
Graph intersection number: GT59 
Longest path problem: ND29 
Maximum bipartite subgraph or (especially with weighted edges) maximum cut.: GT25, ND16 
Maximum common subgraph isomorphism problem: GT49 
Maximum independent set: GT20 
Maximum Induced path: GT23 
Minimum maximal independent set a.k.a. minimum independent dominating set
NP-complete special cases include the minimum maximal matching problem,: GT10  which is essentially equal to the edge dominating set problem (see above).
Metric dimension of a graph: GT61 
Metric k-center
Minimum degree spanning tree
Minimum k-cut
Minimum k-spanning tree
Minor testing (checking whether an input graph



G


{\displaystyle G}

contains an input graph



H


{\displaystyle H}

as a minor); the same holds with topological minors
Steiner tree, or Minimum spanning tree for a subset of the vertices of a graph. (The minimum spanning tree for an entire graph is solvable in polynomial time.)
Modularity maximization
Monochromatic triangle: GT6 
Pathwidth, or, equivalently, interval thickness, and vertex separation number
Rank coloring
k-Chinese postman
Shortest total path length spanning tree: ND3 
Slope number two testing
Recognizing string graphs
Subgraph isomorphism problem: GT48 
Treewidth
Testing whether a tree may be represented as Euclidean minimum spanning tree
Vertex cover: GT1 




Mathematical programming


3-partition problem: SP15 
Bin packing problem: SR1 
Bottleneck traveling salesman: ND24 
Uncapacitated facility location problem
Flow Shop Scheduling Problem
Generalized assignment problem
Integer programming. The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete: MP1 
Some problems related to Job-shop scheduling
Knapsack problem, quadratic knapsack problem, and several variants: MP9 
Some problems related to Multiprocessor scheduling
Numerical 3-dimensional matching: SP16 
Open-shop scheduling
Partition problem: SP12 
Quadratic assignment problem: ND43 
Quadratic programming (NP-hard in some cases, P if convex)
Subset sum problem: SP13 
Variations on the Traveling salesman problem. The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric.: ND22, ND23 




Formal languages and string processing


Closest string
Longest common subsequence problem over multiple sequences: SR10 
The bounded variant of the Post correspondence problem: SR11 
Shortest common supersequence over multiple sequences: SR8 
Extension of the string-to-string correction problem: SR8 




Games and puzzles


Bag (Corral)
Battleship
Bulls and Cows, marketed as Master Mind: certain optimisation problems but not the game itself.
Edge-matching puzzles
Fillomino
(Generalized) FreeCell
Goishi Hiroi
Hashiwokakero
Heyawake
(Generalized) Instant Insanity: GP15 
Kakuro (Cross Sums)
Kingdomino
Kuromasu (also known as Kurodoko)
LaserTank
Lemmings (with a polynomial time limit)
Light Up
Mahjong solitaire (with looking below tiles)
Masyu
Minesweeper Consistency Problem (but see Scott, Stege, & van Rooij)
Nonograms
Numberlink
Nurikabe
(Generalized) Pandemic
Peg solitaire
n-Queens completion
Optimal solution for the N×N×N Rubik's Cube
SameGame
Shakashaka
Slither Link on a variety of grids
(Generalized) Sudoku
Tatamibari
Tentai Show
Problems related to Tetris
Verbal arithmetic




Other


Berth allocation problem
Betweenness
Assembling an optimal Bitcoin block.
Boolean satisfiability problem (SAT).: LO1  There are many variations that are also NP-complete. An important variant is where each clause has exactly three literals (3SAT), since it is used in the proof of many other NP-completeness results.: p. 48 
Circuit satisfiability problem
Conjunctive Boolean query: SR31 
Cyclic ordering
Exact cover problem. Remains NP-complete for 3-sets. Solvable in polynomial time for 2-sets (this is a matching).: SP2 
Finding the global minimum solution of a Hartree-Fock problem
Upward planarity testing
Hospitals-and-residents problem with couples
Knot genus
Latin square completion (the problem of determining if a partially filled square can be completed)
Maximum 2-satisfiability: LO5 
Maximum volume submatrix – Problem of selecting the best conditioned subset of a larger



m
×
n


{\displaystyle m\times n}

matrix. This class of problem is associated with Rank revealing QR factorizations and D optimal experimental design.
Minimal addition chains for sequences. The complexity of minimal addition chains for individual numbers is unknown.
Modal logic S5-Satisfiability
Pancake sorting distance problem for strings
Solubility of two-variable quadratic polynomials over the integers. Given positive integers




A
,
B
,
C



{\displaystyle \textstyle A,B,C}

, decide existence of positive integers



x
,
y


{\displaystyle x,y}

such that



A

x

2


+
B
y
−
C
=
0


{\displaystyle Ax^{2}+By-C=0}


By the same article existence of bounded modular square roots with arbitrarily composite modulus. Given positive integers




A
,
B
,
C
≥
0



{\displaystyle \textstyle A,B,C\geq 0}

, decide existence of an integer



x
∈
[
0
,
C
]


{\displaystyle x\in [0,C]}

such that




x

2


≡
A

mod

B




{\displaystyle x^{2}\equiv A{\bmod {B}}}

. The problem remains NP-complete even if a prime factorization of



B


{\displaystyle B}

is provided.
Serializability of database histories: SR33 
Set cover (also called "minimum cover" problem). This is equivalent, by transposing the incidence matrix, to the hitting set problem.: SP5, SP8 
Set packing: SP3 
Set splitting problem: SP4 
Scheduling to minimize weighted completion time
Block Sorting (Sorting by Block Moves)
Sparse approximation
Variations of the Steiner tree problem. Specifically, with the discretized Euclidean metric, rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric.: ND13 
Three-dimensional Ising model




See also


Existential theory of the reals § Complete problems
Karp's 21 NP-complete problems
List of PSPACE-complete problems
Reduction (complexity)




Notes






References


General

Garey, Michael R.; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences (1st ed.). New York: W. H. Freeman and Company. ISBN 9780716710455. MR 0519066. OCLC 247570676.. This book is a classic, developing the theory, then cataloguing many NP-Complete problems.
Cook, S.A. (1971). "The complexity of theorem proving procedures". Proceedings, Third Annual ACM Symposium on the Theory of Computing, ACM, New York. pp. 151–158. doi:10.1145/800157.805047.
Karp, Richard M. (1972). "Reducibility among combinatorial problems". In Miller, Raymond E.; Thatcher, James W. (eds.). Complexity of Computer Computations. Plenum. pp. 85–103.
Dunne, P.E. "An annotated list of selected NP-complete problems". COMP202, Dept. of Computer Science, University of Liverpool. Retrieved 21 June 2008.
Crescenzi, P.; Kann, V.; Halldórsson, M.; Karpinski, M.; Woeginger, G. "A compendium of NP optimization problems". KTH NADA, Stockholm. Retrieved 21 June 2008.
Dahlke, K. "NP-complete problems". Math Reference Project. Retrieved 21 June 2008.
Specific problems

Friedman, E (2002). "Pearl puzzles are NP-complete". Stetson University, DeLand, Florida. Archived from the original on 4 September 2006. Retrieved 21 June 2008.
Grigoriev, A; Bodlaender, H L (2007). "Algorithms for graphs embeddable with few crossings per edge". Algorithmica. 49 (1): 1–11. CiteSeerX 10.1.1.61.3576. doi:10.1007/s00453-007-0010-x. MR 2344391. S2CID 8174422.
Hartung, S; Nichterlein, A (2012). "NP-Hardness and Fixed-Parameter Tractability of Realizing Degree Sequences with Directed Acyclic Graphs". How the World Computes. Lecture Notes in Computer Science. Vol. 7318. Springer, Berlin, Heidelberg. pp. 283–292. CiteSeerX 10.1.1.377.2077. doi:10.1007/978-3-642-30870-3_29. ISBN 978-3-642-30869-7. S2CID 6112925.
Holzer, Markus; Ruepp, Oliver (2007). "The Troubles of Interior Design–A Complexity Analysis of the Game Heyawake" (PDF). Proceedings, 4th International Conference on Fun with Algorithms, LNCS 4475. Springer, Berlin/Heidelberg. pp. 198–212. doi:10.1007/978-3-540-72914-3_18. ISBN 978-3-540-72913-6.
Kaye, Richard (2000). "Minesweeper is NP-complete". Mathematical Intelligencer. 22 (2): 9–15. doi:10.1007/BF03025367. S2CID 122435790. Further information available online at Richard Kaye's Minesweeper pages.
Kashiwabara, T.; Fujisawa, T. (1979). "NP-completeness of the problem of finding a minimum-clique-number interval graph containing a given graph as a subgraph". Proceedings. International Symposium on Circuits and Systems. pp. 657–660.
Ohtsuki, Tatsuo; Mori, Hajimu; Kuh, Ernest S.; Kashiwabara, Toshinobu; Fujisawa, Toshio (1979). "One-dimensional logic gate assignment and interval graphs". IEEE Transactions on Circuits and Systems. 26 (9): 675–684. doi:10.1109/TCS.1979.1084695.
Lengauer, Thomas (1981). "Black-white pebbles and graph separation". Acta Informatica. 16 (4): 465–475. doi:10.1007/BF00264496. S2CID 19415148.
Arnborg, Stefan; Corneil, Derek G.; Proskurowski, Andrzej (1987). "Complexity of finding embeddings in a k-tree". SIAM Journal on Algebraic and Discrete Methods. 8 (2): 277–284. doi:10.1137/0608024.
Cormode, Graham (2004). "The hardness of the lemmings game, or Oh no, more NP-completeness proofs". Proceedings of Third International Conference on Fun with Algorithms (FUN 2004). pp. 65–76.




External links


A compendium of NP optimization problems
Graph of NP-complete Problems

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• Daftar masalah matematika yang belum terpecahkan
• Optimisasi multiobjektif
• Penghargaan Turing
• List of NP-complete problems
• Karp's 21 NP-complete problems
• NP-completeness
• NP-hardness
• P versus NP problem
• Lists of unsolved problems
• NP (complexity)
• Lists of problems
• Strong NP-completeness
• List of PSPACE-complete problems