List of NP-complete problems

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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 (see e.g. Facebook or LinkedIn).

• 1-planarity^[1]
• 3-dimensional matching^[2][3]
• Bipartite dimension^[4]
• Capacitated minimum spanning tree^[5]
• 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.^[6]
• Clique problem^[2][7]
• Complete coloring, a.k.a. achromatic number^[8]
• Domatic number^[9]
• Dominating set, a.k.a. domination number^[10]

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.^[11]

• Bandwidth problem^[12]
• Clique cover problem^[2][13]
• Rank coloring a.k.a. cycle rank
• Degree-constrained spanning tree^[14]
• Exact cover problem. Remains NP-complete for 3-sets. Solvable in polynomial time for 2-sets (this is a matching).^[2][15]
• Feedback vertex set^[2][16]
• Feedback arc set^[2][17]
• Graph homomorphism problem^[18]
• Graph coloring^[2][19]
• 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.^[20]
• Hamiltonian completion^[21]
• Hamiltonian path problem, directed and undirected.^[2][22]
• Longest path problem^[23]
• Maximum bipartite subgraph or (especially with weighted edges) maximum cut.^[2][24]
• Maximum independent set^[25]
• Maximum Induced path^[26]
• Graph intersection number^[27]
• Metric dimension of a graph^[28]
• Minimum k-cut
• Minimum spanning tree, or Steiner tree, for a subset of the vertices of a graph.^[2] (The minimum spanning tree for an entire graph is solvable in polynomial time.)
• Pathwidth^[29]
• Set cover (also called minimum cover problem) This is equivalent, by transposing the incidence matrix, to the hitting set problem.^[2][30]
• Set splitting problem ^[31]
• Shortest total path length spanning tree^[32]
• Slope number two testing^[33]
• Treewidth^[29]
• Vertex cover^[2][34]

Mathematical programming

• 3-partition problem^[35]
• Bin packing problem^[36]
• Knapsack problem and several variants^[2][37]
• 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.^[38]
• Bottleneck traveling salesman^[39]
• 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][40]
• Numerical 3-dimensional matching^[41]
• Partition problem^[2][42]
• Quadratic assignment problem^[43]
• Solving two-variable quadratic polynomials over the integers.^[44] Given positive integers $\textstyle A,B,C \geq 0$, find positive integers $x,y$ such that $Ax^2+By-C=0$
• Quadratic programming (NP-hard in some cases, P if convex)
• Subset sum problem^[45]

Formal languages and string processing

• Closest string^[46]
• Longest common subsequence problem^[47]
• The bounded variant of the Post correspondence problem^[48]
• Shortest common supersequence^[49]
• String-to-string correction problem^[50]

Games and puzzles

• Battleship
• Bejeweled^[51]
• Bulls and Cows, marketed as Master Mind: certain optimisation problems but not the game itself.
• Candy Crush Saga^[51][52]
• Donkey Kong^[53]
• Eternity II
• (Generalized) FreeCell^[54]
• Fillomino^[55]
• Hashiwokakero^[56]
• Heyawake^[57]
• (Generalized) Instant Insanity^[58]
• Kakuro (Cross Sums)
• Kuromasu (also known as Kurodoko)^[59]
• Legend of Zelda^[53]
• Lemmings^[60]
• Light Up^[61]
• Masyu^[62]
• Metroid^[53]
• Minesweeper Consistency Problem^[63] (but see Scott, Stege, & van Rooij^[64])
• Nimber (or Grundy number) of a directed graph.^[65]
• Nonograms
• Nurikabe
• Pokémon^[53]
• SameGame
• Slither Link on a variety of grids^{[66][67][68][69]}
• (Generalized) Sudoku
• Super Mario Bros^[53]
• Problems related to Tetris^[70]
• Verbal arithmetic

Other

• Art gallery problem and its variations.
• Berth allocation problem^[71]
• Betweenness
• Assembling an optimal Bitcoin block.^[72]
• Boolean satisfiability problem (SAT).^[2][73] 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.^[74]
• Conjunctive Boolean query^[75]
• Cyclic ordering
• Circuit satisfiability problem
• Uncapacitated Facility Location
• Flow Shop Scheduling Problem
• Generalized assignment problem
• Upward planarity testing^[33]
• Hospitals-and-residents problem with couples
• Some problems related to Job-shop scheduling
• Monochromatic triangle^[76]
• Minimum maximal independent set a.k.a. minimum independent dominating set^[77]

NP-complete special cases include the minimum maximal matching problem,^[78] which is essentially equal to the edge dominating set problem (see above).

• Maximum common subgraph isomorphism problem^[79]
• Minimum degree spanning tree
• Minimum k-spanning tree
• Metric k-center
• Maximum 2-Satisfiability^[80]
• Modal logic S5-Satisfiability
• Some problems related to Multiprocessor scheduling
• Maximum volume submatrix – Problem of selecting the best conditioned subset of a larger m x n matrix. This class of problem is associated with Rank revealing QR factorizations and D optimal experimental design.^[81]
• Minimal addition chains for sequences.^[82] The complexity of minimal addition chains for individual numbers is unknown.^[83]
• Non-linear univariate polynomials over GF[2ⁿ], n the length of the input. Indeed, over any GF[qⁿ].
• Open-shop scheduling
• Pathwidth,^[29] or, equivalently, interval thickness, and vertex separation number^[84]
• Pancake sorting distance problem for strings^[85]
• k-Chinese postman
• Subgraph isomorphism problem^[86]
• 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.^[87]
• Set packing^[2][88]
• Serializability of database histories^[89]
• Scheduling to minimize weighted completion time
• Sparse approximation
• Block Sorting (Sorting by Block Moves)
• Second order instantiation
• Treewidth^[29]
• Testing whether a tree may be represented as Euclidean minimum spanning tree
• Three-dimensional Ising model^[90]
• Vehicle routing problem

See also

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

Notes

1. Grigoriev & Bodlaender (2007).
2. Karp (1972)
3. Garey & Johnson (1979): SP1
4. Garey & Johnson (1979): GT18
5. Garey & Johnson (1979): ND5
6. Garey & Johnson (1979): ND25, ND27
7. Garey & Johnson (1979): GT19
8. Garey & Johnson (1979): GT5
9. Garey & Johnson (1979): GT3
10. Garey & Johnson (1979): GT2
11. Garey & Johnson (1979): ND2
12. Garey & Johnson (1979): GT40
13. Garey & Johnson (1979): GT17
14. Garey & Johnson (1979): ND1
15. Garey & Johnson (1979): SP2
16. Garey & Johnson (1979): GT7
17. Garey & Johnson (1979): GT8
18. Garey & Johnson (1979): GT52
19. Garey & Johnson (1979): GT4
20. Garey & Johnson (1979): GT11, GT12, GT13, GT14, GT15, GT16, ND14
21. Garey & Johnson (1979): GT34
22. Garey & Johnson (1979): GT37, GT38, GT39
23. Garey & Johnson (1979): ND29
24. Garey & Johnson (1979): GT25, ND16
25. Garey & Johnson (1979): GT20
26. Garey & Johnson (1979): GT23
27. Garey & Johnson (1979): GT59
28. Garey & Johnson (1979): GT61
29. Arnborg, Corneil & Proskurowski (1987)
30. Garey & Johnson (1979): SP5, SP8
31. Garey & Johnson (1979): SP4
32. Garey & Johnson (1979): ND3
33. "On the computational complexity of upward and rectilinear planarity testing". Lecture Notes in Computer Science. 894/1995. 1995. pp. 286–297. doi:10.1007/3-540-58950-3_384.
34. Garey & Johnson (1979): GT1
35. Garey & Johnson (1979): SP15
36. Garey & Johnson (1979): SR1
37. Garey & Johnson (1979): MP9
38. Garey & Johnson (1979): ND22, ND23
39. Garey & Johnson (1979): ND24
40. Garey & Johnson (1979): MP1
41. Garey & Johnson (1979): SP16
42. Garey & Johnson (1979): SP12
43. Garey & Johnson (1979): ND43
44. NP-complete decision problems for Quadratic Polynomials
45. Garey & Johnson (1979): SP13
46. 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
47. Garey & Johnson (1979): SR10
48. Garey & Johnson (1979): SR11
49. Garey & Johnson (1979): SR8
50. Garey & Johnson (1979): SR20
51. L. Gualà, S. Leucci, E. Natale (2014). "Bejeweled, Candy Crush and other Match-Three Games are (NP-)Hard".
52. Walsh, Toby (2014). "Candy Crush is NP-hard".
53. G. Aloupis, E. D. Demaine, A. Guo (2012-03-09). "Classic Nintendo Games are (NP-)Hard" (PDF).
54. Malte Helmert, Complexity results for standard benchmark domains in planning, Artificial Intelligence Journal 143(2):219-262, 2003.
55. Yato, Takauki (2003). "Complexity and Completeness of Finding Another Solution and its Application to Puzzles". CiteSeerX: 10.1.1.103.8380.
56. "HASHIWOKAKERO Is NP-Complete".
57. Holzer & Ruepp (2007)
58. Garey & Johnson (1979): GP15
59. Kölker, Jonas (2012). "Kurodoko is NP-complete".
60. Cormode, Graham (2004). The hardness of the lemmings game, or Oh no, more NP-completeness proofs (PDF).
61. Light Up is NP-Complete
62. Friedman, Erich (2012-03-27). "Pearl Puzzles are NP-complete".
63. Kaye (2000)
64. 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.
65. Garey & Johnson (1979): GT56
66. Yato; Seta. "Complexity and Completeness of Finding Another Solution and Its Application to Puzzles".
67. Yato. "Complexity and completeness of finding another solution and its application to puzzles" (PDF).
68. Nukui; Uejima. "ASP-Completeness of the Slither Link Puzzle on Several Grids".
69. Kölker, Jonas (2012). "Selected Slither Link Variants are NP-complete".
70. Demaine, Eric D.; Hohenberger, Susan; Liben-Nowell, David (July 25–28, 2003). Tetris is Hard, Even to Approximate (PDF). Proceedings of the 9th International Computing and Combinatorics Conference (COCOON 2003). Big Sky, Montana.
71. 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
72. J. Bonneau, "Bitcoin mining is NP-hard
73. Garey & Johnson (1979): LO1
74. Garey & Johnson (1979): p. 48
75. Garey & Johnson (1979): SR31
76. Garey & Johnson (1979): GT6
77. Minimum Independent Dominating Set
78. Garey & Johnson (1979): GT10
79. Garey & Johnson (1979): GT49
80. Garey & Johnson (1979): LO5
81. http://wayback.archive.org/web/20150203193923/http://www.meliksah.edu.tr/acivril/max-vol-original.pdf
82. Peter Downey, Benton Leong, and Ravi Sethi. "Computing Sequences with Addition Chains" SIAM J. Comput., 10(3), 638–646, 1981
83. D. J. Bernstein, "Pippinger's exponentiation algorithm (draft)
84. Kashiwabara & Fujisawa (1979); Ohtsuki et al. (1979); Lengauer (1981).
85. Hurkens, C., Iersel, L. V., Keijsper, J., Kelk, S., Stougie, L. and Tromp J. "Prefix reversals on binary and ternary strings". SIAM J. Discrete Math. 21(3)(2007) 592–611.
86. Garey & Johnson (1979): GT48
87. Garey & Johnson (1979): ND13
88. Garey & Johnson (1979): SP3
89. Garey & Johnson (1979): SR33
90. Barry A. 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, W. H. Freeman, ISBN 0-7167-1045-5. 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. 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 2008-06-21.
• Crescenzi, P.; Kann, V.; Halldórsson, M.; Karpinski, M.; Woeginger, G. "A compendium of NP optimization problems". KTH NADA, Stockholm. Retrieved 2008-06-21. Cite uses deprecated parameter |coauthors= (help)
• Dahlke, K. "NP-complete problems". Math Reference Project. Retrieved 2008-06-21.

Specific problems

• Friedman, E (2002). "Pearl puzzles are NP-complete". Stetson University, DeLand, Florida. Retrieved 2008-06-21.

• Grigoriev, A; Bodlaender, H L (2007). "Algorithms for graphs embeddable with few crossings per edge". Algorithmica 49 (1): 1–11. doi:10.1007/s00453-007-0010-x. MR 2344391.

• Hartung, S; Nichterlein, A (2012). "NP-Hardness and Fixed-Parameter Tractability of Realizing Degree Sequences with Directed Acyclic Graphs". Springer, Berlin, Heidelberg. Retrieved 2013-01-24.

• Holzer, Markus; Ruepp, Oliver (2007). "The Troubles of Interior Design–A Complexity Analysis of the Game Heyawake". 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. Further information available online at Richard Kaye's Minesweeper pages.
• Kashiwabara, T.; Fujisawa, T. (1979). "Proc. International Symposium on Circuits and Systems": 657–660. |contribution= ignored (help).
• 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..
• 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