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]
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
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[29]
Pathwidth[30]
Set cover (also called minimum cover problem) This is equivalent, by transposing the incidence matrix, to the hitting set problem.[2][31]
Set splitting problem[32]
Shortest total path length spanning tree[33]
Slope number two testing[34]
Treewidth[30]
Vertex cover[2][35]
Mathematical programming
3-partition problem[36]
Bin packing problem[37]
Knapsack problem, quadratic knapsack problem, and several variants[2][38]
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.[39]
Bottleneck traveling salesman[40]
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][41]
Latin squares (The problem of determining if a partially filled square can be completed to form one)
Numerical 3-dimensional matching[42]
Partition problem[2][43]
Quadratic assignment problem[44]
Solving two-variable quadratic polynomials over the integers.[45] 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[46]
Formal languages and string processing
Closest string[47]
Longest common subsequence problem over multiple sequences[48]
The bounded variant of the Post correspondence problem[49]
Shortest common supersequence[50]
String-to-string correction problem[51]
Games and puzzles
Battleship
Bulls and Cows, marketed as Master Mind: certain optimisation problems but not the game itself.
Eternity II
(Generalized) FreeCell[52]
Fillomino[53]
Hashiwokakero[54]
Heyawake[55]
(Generalized) Instant Insanity[56]
Kakuro (Cross Sums)
Kingdomino[57]
Kuromasu (also known as Kurodoko)[58]
LaserTank [59]
Lemmings (with a polynomial time limit)[60]
Light Up[61]
Masyu[62]
Minesweeper Consistency Problem[63] (but see Scott, Stege, & van Rooij[64])
Nimber (or Grundy number) of a directed graph.[65]
Numberlink
Nonograms
Nurikabe
(Generalized) Pandemic[66]
Optimal solution for the N×N×N Rubik's Cube[67]
SameGame
(Generalized) Set[68]
Slither Link on a variety of grids[69][70][71]
(Generalized) Sudoku[69][72]
Problems related to Tetris[73]
Verbal arithmetic
Other
Berth allocation problem[74]
Betweenness
Assembling an optimal Bitcoin block.[75]
Boolean satisfiability problem (SAT).[2][76] 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.[77]
Conjunctive Boolean query[78]
Cyclic ordering
Circuit satisfiability problem
Uncapacitated facility location problem
Flow Shop Scheduling Problem
Generalized assignment problem
Upward planarity testing[34]
Hospitals-and-residents problem with couples
Some problems related to Job-shop scheduling
Monochromatic triangle[79]
Minimum maximal independent set a.k.a. minimum independent dominating set[80]
NP-complete special cases include the minimum maximal matching problem,[81] which is essentially equal to the edge dominating set problem (see above).
Maximum common subgraph isomorphism problem[82]
Minimum degree spanning tree
Minimum k-spanning tree
Metric k-center
Maximum 2-Satisfiability[83]
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.[84]
Minimal addition chains for sequences.[85] The complexity of minimal addition chains for individual numbers is unknown.[86]
Non-linear univariate polynomials over GF[2n], n the length of the input. Indeed, over any GF[qn].
Open-shop scheduling
Pathwidth,[30] or, equivalently, interval thickness, and vertex separation number[87]
Pancake sorting distance problem for strings[88]
k-Chinese postman
Subgraph isomorphism problem[89]
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.[90]
Set packing[2][91]
Serializability of database histories[92]
Scheduling to minimize weighted completion time
Sparse approximation
Block Sorting[93] (Sorting by Block Moves)
Second order instantiation
Treewidth[30]
Testing whether a tree may be represented as Euclidean minimum spanning tree
Three-dimensional Ising model[94]
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
Grigoriev & Bodlaender (2007).
Karp (1972)
Garey & Johnson (1979): SP1
Garey & Johnson (1979): GT18
Garey & Johnson (1979): ND5
Garey & Johnson (1979): ND25, ND27
Garey & Johnson (1979): GT19
Garey & Johnson (1979): GT5
Garey & Johnson (1979): GT3
Garey & Johnson (1979): GT2
Garey & Johnson (1979): ND2
Garey & Johnson (1979): GT40
Garey & Johnson (1979): GT17
Garey & Johnson (1979): ND1
Garey & Johnson (1979): SP2
Garey & Johnson (1979): GT7
Garey & Johnson (1979): GT8
Garey & Johnson (1979): GT52
Garey & Johnson (1979): GT4
Garey & Johnson (1979): GT11, GT12, GT13, GT14, GT15, GT16, ND14
Garey & Johnson (1979): GT34
Garey & Johnson (1979): GT37, GT38, GT39
Garey & Johnson (1979): ND29
Garey & Johnson (1979): GT25, ND16
Garey & Johnson (1979): GT20
Garey & Johnson (1979): GT23
Garey & Johnson (1979): GT59
Garey & Johnson (1979): GT61
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
Arnborg, Corneil & Proskurowski (1987)
Garey & Johnson (1979): SP5, SP8
Garey & Johnson (1979): SP4
Garey & Johnson (1979): ND3
Garg, Ashim; Tamassia, Roberto (1995). "On the computational complexity of upward and rectilinear planarity testing". Lecture Notes in Computer Science. 894/1995. pp. 286–297. doi:10.1007/3-540-58950-3_384. ISBN 978-3-540-58950-1.
Garey & Johnson (1979): GT1
Garey & Johnson (1979): SP15
Garey & Johnson (1979): SR1
Garey & Johnson (1979): MP9
Garey & Johnson (1979): ND22, ND23
Garey & Johnson (1979): ND24
Garey & Johnson (1979): MP1
Garey & Johnson (1979): SP16
Garey & Johnson (1979): SP12
Garey & Johnson (1979): ND43
NP-complete decision problems for Quadratic Polynomials
Garey & Johnson (1979): SP13
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
Garey & Johnson (1979): SR10
Garey & Johnson (1979): SR11
Garey & Johnson (1979): SR8
Garey & Johnson (1979): SR20
Malte Helmert, Complexity results for standard benchmark domains in planning, Artificial Intelligence 143(2):219-262, 2003.
Yato, Takauki (2003). Complexity and Completeness of Finding Another Solution and its Application to Puzzles. CiteSeerX 10.1.1.103.8380.
"HASHIWOKAKERO Is NP-Complete".
Holzer & Ruepp (2007)
Garey & Johnson (1979): GP15
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.
Kölker, Jonas (2012). "Kurodoko is NP-complete" (PDF). Journal of Information Processing. 20 (3): 694–706. doi:10.2197/ipsjjip.20.694. S2CID 46486962.
Alexandersson, Per; Restadh, Petter (2020). "LaserTank is NP-Complete". Mathematical Aspects of Computer and Information Sciences. Lecture Notes in Computer Science. Springer International Publishing. 11989: 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".
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.
Garey & Johnson (1979): GT56
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.
http://pbg.cs.illinois.edu/papers/set.pdf
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
Garey & Johnson (1979): LO1
Garey & Johnson (1979): p. 48
Garey & Johnson (1979): SR31
Garey & Johnson (1979): GT6
Minimum Independent Dominating Set
Garey & Johnson (1979): GT10
Garey & Johnson (1979): GT49
Garey & Johnson (1979): LO5
https://web.archive.org/web/20150203193923/http://www.meliksah.edu.tr/acivril/max-vol-original.pdf
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)
Kashiwabara & Fujisawa (1979); Ohtsuki et al. (1979); Lengauer (1981).
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.
Garey & Johnson (1979): GT48
Garey & Johnson (1979): ND13
Garey & Johnson (1979): SP3
Garey & Johnson (1979): SR33
Bein, W. W.; Larmore, L. L.; Latifi, S.; Sudborough, I. H. (1 January 2002). Block sorting is hard. International Symposium on Parallel Architectures, Algorithms and Networks, 2002. I-SPAN '02. Proceedings. 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, 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. (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. 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). How the World Computes. Lecture Notes in Computer Science. 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.
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