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. 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^[1]
3-dimensional matching^[2]^[3]^: SP1
Bandwidth problem^[3]^: GT40
Bipartite dimension^[3]^: GT18
Capacitated minimum spanning tree^[3]^: 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.^[3]^{: ND25, ND27}
Clique cover problem^[2]^[3]^: GT17
Clique problem^[2]^[3]^: GT19
Complete coloring, a.k.a. achromatic number^[3]^: GT5
Cycle rank
Degree-constrained spanning tree^[3]^: ND1
Domatic number^[3]^: GT3
Dominating set, a.k.a. domination number^[3]^: 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.^[3]^: ND2

Feedback vertex set^[2]^[3]^: GT7
Feedback arc set^[2]^[3]^: GT8
Graph coloring^[2]^[3]^: GT4
Graph homomorphism problem^[3]^: 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.^[3]^{: GT11, GT12, GT13, GT14, GT15, GT16, ND14}
Grundy number of a directed graph.^[3]^: GT56
Hamiltonian completion^[3]^: GT34
Hamiltonian path problem, directed and undirected.^[2]^[3]^{: GT37, GT38, GT39}
Induced subgraph isomorphism problem
Graph intersection number^[3]^: GT59
Longest path problem^[3]^: ND29
Maximum bipartite subgraph or (especially with weighted edges) maximum cut.^[2]^[3]^{: GT25, ND16}
Maximum common subgraph isomorphism problem^[3]^: GT49
Maximum independent set^[3]^: GT20
Maximum Induced path^[3]^: GT23
Minimum maximal independent set a.k.a. minimum independent dominating set^[4]

NP-complete special cases include the minimum maximal matching problem,^[3]^: GT10 which is essentially equal to the edge dominating set problem (see above).

Metric dimension of a graph^[3]^: GT61
Metric k-center
Minimum degree spanning tree
Minimum k-cut
Minimum k-spanning tree
Minor testing (checking whether an input graph $G$ contains an input graph $H$ as a minor); the same holds with topological minors
Steiner tree, or Minimum spanning tree for a subset of the vertices of a graph.^[2] (The minimum spanning tree for an entire graph is solvable in polynomial time.)
Modularity maximization^[5]
Monochromatic triangle^[3]^: GT6
Pathwidth,^[6] or, equivalently, interval thickness, and vertex separation number^[7]
Rank coloring
k-Chinese postman
Shortest total path length spanning tree^[3]^: ND3
Slope number two testing^[8]
Recognizing string graphs^[9]
Subgraph isomorphism problem^[3]^: GT48
Treewidth^[6]
Testing whether a tree may be represented as Euclidean minimum spanning tree
Vertex cover^[2]^[3]^: GT1

Mathematical programming

3-partition problem^[3]^: SP15
Bin packing problem^[3]^: SR1
Bottleneck traveling salesman^[3]^: 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^[2]^[3]^: MP1
Some problems related to Job-shop scheduling
Knapsack problem, quadratic knapsack problem, and several variants^[2]^[3]^: MP9
Some problems related to Multiprocessor scheduling
Numerical 3-dimensional matching^[3]^: SP16
Open-shop scheduling
Partition problem^[2]^[3]^: SP12
Quadratic assignment problem^[3]^: ND43
Quadratic programming (NP-hard in some cases, P if convex)
Subset sum problem^[3]^: 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.^[3]^{: ND22, ND23}
Vehicle routing problem

Formal languages and string processing

Closest string^[10]
Longest common subsequence problem over multiple sequences^[3]^: SR10
The bounded variant of the Post correspondence problem^[3]^: SR11
Shortest common supersequence over multiple sequences^[3]^: SR8
Extension of the string-to-string correction problem^[11]^[3]^: SR8

Games and puzzles

Bag (Corral)^[12]
Battleship
Bulls and Cows, marketed as Master Mind: certain optimisation problems but not the game itself.
Edge-matching puzzles
Fillomino^[13]
(Generalized) FreeCell^[14]
Goishi Hiroi
Hashiwokakero^[15]
Heyawake^[16]
(Generalized) Instant Insanity^[3]^: GP15
Kakuro (Cross Sums)^[17]
Kingdomino^[18]
Kuromasu (also known as Kurodoko)^[19]
LaserTank^[20]
Lemmings (with a polynomial time limit)^[21]
Light Up^[22]
Mahjong solitaire (with looking below tiles)
Masyu^[23]
Minesweeper Consistency Problem^[24] (but see Scott, Stege, & van Rooij^[25])
Nonograms
Numberlink
Nurikabe^[26]
(Generalized) Pandemic^[27]
Peg solitaire
n-Queens completion
Optimal solution for the N×N×N Rubik's Cube^[28]
SameGame
Shakashaka
Slither Link on a variety of grids^[29]^[30]^[31]
(Generalized) Sudoku^[29]^[32]
Tatamibari
Tentai Show
Problems related to Tetris^[33]
Verbal arithmetic

Other

Berth allocation problem^[34]
Betweenness
Assembling an optimal Bitcoin block.^[35]
Boolean satisfiability problem (SAT).^[2]^[3]^: 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.^[3]^{: p. 48}
Circuit satisfiability problem
Conjunctive Boolean query^[3]^: SR31
Cyclic ordering^[36]
Exact cover problem. Remains NP-complete for 3-sets. Solvable in polynomial time for 2-sets (this is a matching).^[2]^[3]^: SP2
Finding the global minimum solution of a Hartree-Fock problem^[37]
Upward planarity testing^[8]
Hospitals-and-residents problem with couples
Knot genus^[38]
Latin square completion (the problem of determining if a partially filled square can be completed)
Maximum 2-satisfiability^[3]^: LO5
Maximum volume submatrix – Problem of selecting the best conditioned subset of a larger $m\times n$ matrix. This class of problem is associated with Rank revealing QR factorizations and D optimal experimental design.^[39]
Minimal addition chains for sequences.^[40] The complexity of minimal addition chains for individual numbers is unknown.^[41]
Modal logic S5-Satisfiability
Pancake sorting distance problem for strings^[42]
Solubility of two-variable quadratic polynomials over the integers.^[43] Given positive integers $\textstyle A,B,C$ , decide existence of positive integers $x,y$ such that $Ax^{2}+By-C=0$
By the same article^[43] existence of bounded modular square roots with arbitrarily composite modulus. Given positive integers $\textstyle A,B,C\geq 0$ , decide existence of an integer $x\in [0,C]$ such that $x^{2}\equiv A{\bmod {B}}$ . The problem remains NP-complete even if a prime factorization of $B$ is provided.
Serializability of database histories^[3]^: SR33
Set cover (also called minimum cover problem) This is equivalent, by transposing the incidence matrix, to the hitting set problem.^[2]^[3]^{: SP5, SP8}
Set packing^[2]^[3]^: SP3
Set splitting problem^[3]^: SP4
Scheduling to minimize weighted completion time
Block Sorting^[44] (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.^[3]^: ND13
Three-dimensional Ising model^[45]

Notes

^ Grigoriev & Bodlaender (2007).
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q Karp (1972)
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q ^r ^s ^t ^u ^v ^w ^x ^y ^z ^aa ^ab ^ac ^ad ^ae ^af ^ag ^ah ^ai ^aj ^ak ^al ^am ^an ^ao ^ap ^aq ^ar ^as ^at ^au ^av ^aw ^ax ^ay ^az ^ba ^bb ^bc ^bd ^be Garey & Johnson (1979)
^ Minimum Independent Dominating Set
^ Brandes, Ulrik; Delling, Daniel; Gaertler, Marco; Görke, Robert; Hoefer, Martin; Nikoloski, Zoran; Wagner, Dorothea (2006), Maximizing Modularity is hard, arXiv:physics/0608255, Bibcode:2006physics...8255B
^ ^a ^b Arnborg, Corneil & Proskurowski (1987)
^ Kashiwabara & Fujisawa (1979); Ohtsuki et al. (1979); Lengauer (1981).
^ ^a ^b Garg, Ashim; Tamassia, Roberto (1995). "On the computational complexity of upward and rectilinear planarity testing". Lecture Notes in Computer Science. Vol. 894/1995. pp. 286–297. doi:10.1007/3-540-58950-3_384. ISBN 978-3-540-58950-1.
^ Schaefer, Marcus; Sedgwick, Eric; Štefankovič, Daniel (September 2003). "Recognizing string graphs in NP". Journal of Computer and System Sciences. 67 (2): 365–380. doi:10.1016/S0022-0000(03)00045-X.
^ Lanctot, J. Kevin; Li, Ming; Ma, Bin; Wang, Shaojiu; Zhang, Louxin (2003), "Distinguishing string selection problems", Information and Computation, 185 (1): 41–55, doi:10.1016/S0890-5401(03)00057-9, MR 1994748
^ Wagner, Robert A. (May 1975). "On the complexity of the Extended String-to-String Correction Problem". Proceedings of seventh annual ACM symposium on Theory of computing - STOC '75. pp. 218–223. doi:10.1145/800116.803771. ISBN 9781450374194. S2CID 18705107.
^ Friedman, Erich. "Corral Puzzles are NP-complete" (PDF). Retrieved 17 August 2021.
^ Yato, Takauki (2003). Complexity and Completeness of Finding Another Solution and its Application to Puzzles. CiteSeerX 10.1.1.103.8380.
^ Malte Helmert, Complexity results for standard benchmark domains in planning, Artificial Intelligence 143(2):219-262, 2003.
^ "HASHIWOKAKERO Is NP-Complete".
^ Holzer & Ruepp (2007)
^ Takahiro, Seta (5 February 2002). "The complexities of puzzles, cross sum and their another solution problems (ASP)" (PDF). Retrieved 18 November 2018.
^ Nguyen, Viet-Ha; Perrot, Kévin; Vallet, Mathieu (24 June 2020). "NP-completeness of the game KingdominoTM". Theoretical Computer Science. 822: 23–35. doi:10.1016/j.tcs.2020.04.007. ISSN 0304-3975. S2CID 218552723.
^ Kölker, Jonas (2012). "Kurodoko is NP-complete" (PDF). Journal of Information Processing. 20 (3): 694–706. doi:10.2197/ipsjjip.20.694. S2CID 46486962. Archived from the original (PDF) on 12 February 2020.
^ Alexandersson, Per; Restadh, Petter (2020). "LaserTank is NP-Complete". Mathematical Aspects of Computer and Information Sciences. Lecture Notes in Computer Science. Vol. 11989. Springer International Publishing. pp. 333–338. arXiv:1908.05966. doi:10.1007/978-3-030-43120-4_26. ISBN 978-3-030-43119-8. S2CID 201058355.
^ Cormode, Graham (2004). The hardness of the lemmings game, or Oh no, more NP-completeness proofs (PDF).
^ Light Up is NP-Complete
^ Friedman, Erich (27 March 2012). "Pearl Puzzles are NP-complete". Archived from the original on 4 February 2012.
^ Kaye (2000)
^ Allan Scott, Ulrike Stege, Iris van Rooij, Minesweeper may not be NP-complete but is hard nonetheless, The Mathematical Intelligencer 33:4 (2011), pp. 5–17.
^ Holzer, Markus; Klein, Andreas; Kutrib, Martin; Ruepp, Oliver (2011). "Computational Complexity of NURIKABE". Fundamenta Informaticae. 110 (1–4): 159–174. doi:10.3233/FI-2011-534.
^ Nakai, Kenichiro; Takenaga, Yasuhiko (2012). "NP-Completeness of Pandemic". Journal of Information Processing. 20 (3): 723–726. doi:10.2197/ipsjjip.20.723. ISSN 1882-6652.
^ Demaine, Erik; Eisenstat, Sarah; Rudoy, Mikhail (2018). Solving the Rubik's Cube Optimally is NP-complete. 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). doi:10.4230/LIPIcs.STACS.2018.24.
^ ^a ^b Sato, Takayuki; Seta, Takahiro (1987). Complexity and Completeness of Finding Another Solution and Its Application to Puzzles (PDF). International Symposium on Algorithms (SIGAL 1987).
^ Nukui; Uejima (March 2007). "ASP-Completeness of the Slither Link Puzzle on Several Grids". Ipsj Sig Notes. 2007 (23): 129–136.
^ Kölker, Jonas (2012). "Selected Slither Link Variants are NP-complete". Journal of Information Processing. 20 (3): 709–712. doi:10.2197/ipsjjip.20.709.
^ A SURVEY OF NP-COMPLETE PUZZLES, Section 23; Graham Kendall, Andrew Parkes, Kristian Spoerer; March 2008. (icga2008.pdf)
^ Demaine, Eric D.; Hohenberger, Susan; Liben-Nowell, David (25–28 July 2003). Tetris is Hard, Even to Approximate (PDF). Proceedings of the 9th International Computing and Combinatorics Conference (COCOON 2003). Big Sky, Montana.
^ Lim, Andrew (1998), "The berth planning problem", Operations Research Letters, 22 (2–3): 105–110, doi:10.1016/S0167-6377(98)00010-8, MR 1653377
^ J. Bonneau, "Bitcoin mining is NP-hard"
^ Galil, Zvi; Megiddo, Nimrod (October 1977). "Cyclic ordering is NP-complete". Theoretical Computer Science. 5 (2): 179–182. doi:10.1016/0304-3975(77)90005-6.
^ Whitfield, James Daniel; Love, Peter John; Aspuru-Guzik, Alán (2013). "Computational complexity in electronic structure". Phys. Chem. Chem. Phys. 15 (2): 397–411. arXiv:1208.3334. Bibcode:2013PCCP...15..397W. doi:10.1039/C2CP42695A. PMID 23172634. S2CID 12351374.
^ Agol, Ian; Hass, Joel; Thurston, William (19 May 2002). "3-manifold knot genus is NP-complete". Proceedings of the thiry-fourth annual ACM symposium on Theory of computing. STOC '02. New York, NY, USA: Association for Computing Machinery. pp. 761–766. arXiv:math/0205057. doi:10.1145/509907.510016. ISBN 978-1-58113-495-7. S2CID 10401375.
^ Çivril, Ali; Magdon-Ismail, Malik (2009), "On selecting a maximum volume sub-matrix of a matrix and related problems" (PDF), Theoretical Computer Science, 410 (47–49): 4801–4811, doi:10.1016/j.tcs.2009.06.018, MR 2583677, archived from the original (PDF) on 3 February 2015
^ Peter Downey, Benton Leong, and Ravi Sethi. "Computing Sequences with Addition Chains" SIAM J. Comput., 10(3), 638–646, 1981
^ D. J. Bernstein, "Pippinger's exponentiation algorithm" (draft)
^ Hurkens, C.; Iersel, L. V.; Keijsper, J.; Kelk, S.; Stougie, L.; Tromp, J. (2007). "Prefix reversals on binary and ternary strings". SIAM J. Discrete Math. 21 (3): 592–611. arXiv:math/0602456. doi:10.1137/060664252.
^ ^a ^b Manders, Kenneth; Adleman, Leonard (1976). "NP-complete decision problems for quadratic polynomials". Proceedings of the eighth annual ACM symposium on Theory of computing - STOC '76. pp. 23–29. doi:10.1145/800113.803627. ISBN 9781450374149. S2CID 18885088.
^ Bein, W. W.; Larmore, L. L.; Latifi, S.; Sudborough, I. H. (1 January 2002). "Block sorting is hard". Proceedings International Symposium on Parallel Architectures, Algorithms and Networks. I-SPAN'02. pp. 307–312. doi:10.1109/ISPAN.2002.1004305. ISBN 978-0-7695-1579-3. S2CID 32222403.
^ Barry Arthur Cipra, "The Ising Model Is NP-Complete", SIAM News, Vol 33, No 6.

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.