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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 thousands of such problems known, this list is in no way comprehensive. Many problems of this type can be found in {{harvtxt|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).

::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.{{rp|ND2}}

::NP-complete special cases include the minimum maximal matching problem,{{rp|GT10}} which is essentially equal to the edge dominating set problem (see above).

  • Metric dimension of a graph{{rp|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. (The minimum spanning tree for an entire graph is solvable in polynomial time.)
  • Modularity maximization{{citation | title = Maximizing Modularity is hard | first1 = Ulrik | last1 = Brandes | author1-link = Ulrik Brandes | first2 = Daniel | last2 = Delling | first3 = Marco | last3 = Gaertler | first4 = Robert | last4 = Görke | first5 = Martin | last5 = Hoefer | first6 = Zoran | last6 = Nikoloski | first7 = Dorothea | last7 = Wagner | year = 2006 | arxiv = physics/0608255| bibcode = 2006physics...8255B }}
  • Monochromatic triangle{{rp|GT6}}
  • Pathwidth, or, equivalently, interval thickness, and vertex separation number{{harvtxt|Kashiwabara|Fujisawa|1979}}; {{harvtxt|Ohtsuki|Mori|Kuh|Kashiwabara|1979}}; {{harvtxt|Lengauer|1981}}.
  • Rank coloring
  • k-Chinese postman
  • Shortest total path length spanning tree{{rp|ND3}}
  • Slope number two testing{{cite book|title=Lecture Notes in Computer Science| year=1995| volume=894/1995 | pages=286–297| doi=10.1007/3-540-58950-3_384 | chapter=On the computational complexity of upward and rectilinear planarity testing| last1=Garg| first1=Ashim| last2=Tamassia| first2=Roberto| isbn=978-3-540-58950-1}}
  • Recognizing string graphs{{cite journal |last1=Schaefer |first1=Marcus |last2=Sedgwick |first2=Eric |last3=Štefankovič |first3=Daniel |title=Recognizing string graphs in NP |journal=Journal of Computer and System Sciences |date=September 2003 |volume=67 |issue=2 |pages=365–380 |doi=10.1016/S0022-0000(03)00045-X|doi-access=free }}
  • Subgraph isomorphism problem{{rp|GT48}}
  • Treewidth{{harvtxt|Arnborg|Corneil|Proskurowski|1987}}
  • Testing whether a tree may be represented as Euclidean minimum spanning tree
  • Vertex cover{{harvtxt|Karp|1972}}{{rp|GT1}}

Mathematical programming

Formal languages and string processing

| last1 = Lanctot | first1 = J. Kevin

| last2 = Li | first2 = Ming

| last3 = Ma | first3 = Bin

| last4 = Wang | first4 = Shaojiu

| last5 = Zhang | first5 = Louxin

| doi = 10.1016/S0890-5401(03)00057-9

| issue = 1

| journal = Information and Computation

| mr = 1994748

| pages = 41–55

| title = Distinguishing string selection problems

| volume = 185

| year = 2003| doi-access = free

}}

  • Longest common subsequence problem over multiple sequences{{rp|SR10}}
  • The bounded variant of the Post correspondence problem{{rp|SR11}}
  • Shortest common supersequence over multiple sequences{{rp|SR8}}
  • Extension of the string-to-string correction problem{{cite book |last1=Wagner |first1=Robert A. |title=Proceedings of seventh annual ACM symposium on Theory of computing - STOC '75 |chapter=On the complexity of the Extended String-to-String Correction Problem |date=May 1975 |pages=218–223 |doi=10.1145/800116.803771 |isbn=9781450374194 |s2cid=18705107 |chapter-url=https://dl.acm.org/doi/10.1145/800116.803771}}{{rp|SR8}}

Games and puzzles

  • Bag (Corral){{cite web |last1=Friedman |first1=Erich |title=Corral Puzzles are NP-complete |url=https://erich-friedman.github.io/papers/corral.pdf |access-date=17 August 2021}}
  • Battleship
  • Bulls and Cows, marketed as Master Mind: certain optimisation problems but not the game itself.
  • Edge-matching puzzles
  • Fillomino{{cite book | last = Yato | first = Takauki | title = Complexity and Completeness of Finding Another Solution and its Application to Puzzles | year = 2003 | citeseerx = 10.1.1.103.8380 }}
  • (Generalized) FreeCellMalte Helmert, Complexity results for standard benchmark domains in planning, Artificial Intelligence 143(2):219-262, 2003.
  • Goishi Hiroi
  • Hashiwokakero{{cite web | title = HASHIWOKAKERO Is NP-Complete | url = http://www.cs.au.dk/~koda/hashiwokakero }}
  • Heyawake{{harvtxt|Holzer|Ruepp|2007}}
  • (Generalized) Instant Insanity{{rp|GP15}}
  • Kakuro (Cross Sums){{cite web |url=http://www-imai.is.s.u-tokyo.ac.jp/~seta/paper/senior_thesis/seniorthesis.pdf |last=Takahiro |first=Seta |date=February 5, 2002 |title=The complexities of puzzles, cross sum and their another solution problems (ASP) |access-date=November 18, 2018}}
  • Kingdomino{{Cite journal|last1=Nguyen|first1=Viet-Ha|last2=Perrot|first2=Kévin|last3=Vallet|first3=Mathieu|date=2020-06-24|title=NP-completeness of the game KingdominoTM|journal=Theoretical Computer Science|language=en|volume=822|pages=23–35|doi=10.1016/j.tcs.2020.04.007|s2cid=218552723 |issn=0304-3975|doi-access=free}}
  • Kuromasu (also known as Kurodoko){{cite journal | last = Kölker | first = Jonas | title = Kurodoko is NP-complete | journal = Journal of Information Processing | volume = 20 | issue = 3 | pages = 694–706 | year = 2012 | doi = 10.2197/ipsjjip.20.694 | s2cid = 46486962 | url = https://pdfs.semanticscholar.org/4971/aa310e07aaa07c1ba7299f519b39ba5c5195.pdf | archive-url = https://web.archive.org/web/20200212021124/https://pdfs.semanticscholar.org/4971/aa310e07aaa07c1ba7299f519b39ba5c5195.pdf | url-status = dead | archive-date = 2020-02-12 }}
  • LaserTank{{cite book |last1=Alexandersson |first1=Per |last2=Restadh |first2=Petter |title=Mathematical Aspects of Computer and Information Sciences |chapter=LaserTank is NP-Complete |series=Lecture Notes in Computer Science |date=2020 |volume=11989 |pages=333–338 |doi=10.1007/978-3-030-43120-4_26 |publisher=Springer International Publishing |language=en|arxiv=1908.05966 |isbn=978-3-030-43119-8 |s2cid=201058355 }}
  • Lemmings (with a polynomial time limit){{cite conference | author = Cormode, Graham | title = The hardness of the lemmings game, or Oh no, more NP-completeness proofs | url= http://dimacs.rutgers.edu/~graham/pubs/papers/cormodelemmings.pdf | year = 2004}}
  • Light Up[https://web.archive.org/web/20080308132136/http://www.cs.umass.edu/~mcphailb/papers/2005lightup.pdf Light Up is NP-Complete]
  • Mahjong solitaire (with looking below tiles)
  • Masyu{{cite web | author = Friedman, Erich | title = Pearl Puzzles are NP-complete | url = http://www2.stetson.edu/~efriedma/papers/pearl/pearl.html | date = 2012-03-27 | archive-url = https://web.archive.org/web/20120204150006/http://www2.stetson.edu/~efriedma/papers/pearl/pearl.html | archive-date = 2012-02-04 | url-status = dead}}
  • Minesweeper Consistency Problem{{harvtxt|Kaye|2000}} (but see Scott, Stege, & van RooijAllan 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.)
  • Nonograms
  • Numberlink
  • Nurikabe{{cite journal

| last1 = Holzer | first1 = Markus

| last2 = Klein | first2 = Andreas

| last3 = Kutrib | first3 = Martin

| last4 = Ruepp | first4 = Oliver

| doi = 10.3233/FI-2011-534

| issue = 1–4

| journal = Fundamenta Informaticae

| pages = 159–174

| title = Computational Complexity of NURIKABE

| volume = 110

| year = 2011}}

  • (Generalized) Pandemic{{Cite journal|last1=Nakai|first1=Kenichiro|last2=Takenaga|first2=Yasuhiko|date=2012|title=NP-Completeness of Pandemic|journal=Journal of Information Processing|volume=20|issue=3|pages=723–726|doi=10.2197/ipsjjip.20.723|issn=1882-6652|doi-access=free}}
  • Peg solitaire
  • n-Queens completion
  • Optimal solution for the {{math|N×N×N}} Rubik's Cube{{cite conference|title=Solving the Rubik's Cube Optimally is NP-complete|first1=Erik|last1=Demaine|first2=Sarah|last2=Eisenstat|first3=Mikhail|last3=Rudoy|conference=35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)|year=2018|doi=10.4230/LIPIcs.STACS.2018.24|doi-access=free }}
  • SameGame
  • Shakashaka
  • Slither Link on a variety of grids{{cite journal | last1 = Nukui | last2 = Uejima | title = ASP-Completeness of the Slither Link Puzzle on Several Grids | journal = Ipsj Sig Notes | volume = 2007 | issue = 23 | pages = 129–136 | url = http://ci.nii.ac.jp/naid/110006249219/en/| date = March 2007 }}{{cite journal | last = Kölker | first = Jonas | title = Selected Slither Link Variants are NP-complete | journal = Journal of Information Processing | volume = 20 | issue = 3 | pages = 709–712 | year = 2012| doi = 10.2197/ipsjjip.20.709 | doi-access = free }}
  • (Generalized) Sudoku{{cite conference|url=http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/SIGAL87-2.pdf |title=Complexity and Completeness of Finding Another Solution and Its Application to Puzzles|first1=Takayuki|last1=Sato|first2=Takahiro|last2=Seta|conference= International Symposium on Algorithms (SIGAL 1987)|year=1987}}A SURVEY OF NP-COMPLETE PUZZLES, Section 23; Graham Kendall, Andrew Parkes, Kristian Spoerer; March 2008. [https://www.cs.wmich.edu/elise/courses/cs431/icga2008.pdf (icga2008.pdf)]
  • Tatamibari
  • Tentai Show
  • Problems related to Tetris{{cite conference|url=http://erikdemaine.org/papers/Tetris_COCOON2003/paper.pdf|title=Tetris is Hard, Even to Approximate|last1=Demaine|first1=Eric D.|last2=Hohenberger |first2=Susan|last3=Liben-Nowell|first3=David|date=25–28 July 2003|conference=Proceedings of the 9th International Computing and Combinatorics Conference (COCOON 2003)|conference-url= https://web.archive.org/web/20120105042734/http://www.cs.montana.edu/bhz/cocoon03.html|location=Big Sky, Montana}}
  • Verbal arithmetic

Other

| last = Lim | first = Andrew

| doi = 10.1016/S0167-6377(98)00010-8

| issue = 2–3

| journal = Operations Research Letters

| mr = 1653377

| pages = 105–110

| title = The berth planning problem

| volume = 22

| year = 1998}}

  • Betweenness
  • Assembling an optimal Bitcoin block.[https://freedom-to-tinker.com/blog/jbonneau/bitcoin-mining-is-np-hard/ J. Bonneau, "Bitcoin mining is NP-hard"]
  • Boolean satisfiability problem (SAT).{{rp|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.{{rp|p. 48}}
  • Circuit satisfiability problem
  • Conjunctive Boolean query{{rp|SR31}}
  • Cyclic ordering{{cite journal |last1=Galil |first1=Zvi |last2=Megiddo |first2=Nimrod |title=Cyclic ordering is NP-complete |journal=Theoretical Computer Science |date=October 1977 |volume=5 |issue=2 |pages=179–182 |doi=10.1016/0304-3975(77)90005-6 |doi-access=free }}
  • Exact cover problem. Remains NP-complete for 3-sets. Solvable in polynomial time for 2-sets (this is a matching).{{rp|SP2}}
  • Finding the global minimum solution of a Hartree-Fock problem{{cite journal | url=https://doi.org/10.1039/C2CP42695A | doi=10.1039/C2CP42695A | title=Computational complexity in electronic structure | year=2013 | last1=Whitfield | first1=James Daniel | last2=Love | first2=Peter John | last3=Aspuru-Guzik | first3=Alán | journal=Phys. Chem. Chem. Phys. | volume=15 | issue=2 | pages=397–411 | pmid=23172634 | arxiv=1208.3334 | bibcode=2013PCCP...15..397W | s2cid=12351374 }}
  • Upward planarity testing
  • Hospitals-and-residents problem with couples
  • Knot genus{{Cite book|last1=Agol|first1=Ian|author-link=Ian Agol|last2=Hass|first2=Joel|author-link2=Joel Hass|last3=Thurston|first3=William|title=Proceedings of the thiry-fourth annual ACM symposium on Theory of computing |chapter=3-manifold knot genus is NP-complete |author-link3=William Thurston|date=2002-05-19|chapter-url=https://doi.org/10.1145/509907.510016|series=STOC '02|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=761–766|arxiv=math/0205057|doi=10.1145/509907.510016|isbn=978-1-58113-495-7|s2cid=10401375}}
  • Latin square completion (the problem of determining if a partially filled square can be completed)
  • Maximum 2-satisfiability{{rp|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.{{citation

| last1 = Çivril | first1 = Ali

| last2 = Magdon-Ismail | first2 = Malik

| doi = 10.1016/j.tcs.2009.06.018

| issue = 47–49

| journal = Theoretical Computer Science

| mr = 2583677

| pages = 4801–4811

| title = On selecting a maximum volume sub-matrix of a matrix and related problems

| volume = 410

| archive-url = https://web.archive.org/web/20150203193923/http://www.meliksah.edu.tr/acivril/max-vol-original.pdf

| url = http://www.meliksah.edu.tr/acivril/max-vol-original.pdf

| archive-date = 2015-02-03

| year = 2009}}

  • Minimal addition chains for sequences.Peter Downey, Benton Leong, and Ravi Sethi. "Computing Sequences with Addition Chains" SIAM J. Comput., 10(3), 638–646, 1981 The complexity of minimal addition chains for individual numbers is unknown.[http://cr.yp.to/papers/pippenger.pdf D. J. Bernstein, "Pippinger's exponentiation algorithm" (draft)]
  • Modal logic S5-Satisfiability
  • Pancake sorting distance problem for strings{{cite journal | last1 = Hurkens | first1 = C. | last2 = Iersel | first2 = L. V. | last3 = Keijsper | first3 = J. | last4 = Kelk | first4 = S. | last5 = Stougie | first5 = L. | last6 = Tromp | first6 = J. | year = 2007 | title = Prefix reversals on binary and ternary strings | journal = SIAM J. Discrete Math. | volume = 21 | issue = 3| pages = 592–611 | doi=10.1137/060664252| arxiv = math/0602456 }}
  • Solubility of two-variable quadratic polynomials over the integers.{{cite book | chapter-url=https://dl.acm.org/doi/10.1145/800113.803627 | doi=10.1145/800113.803627 | chapter=NP-complete decision problems for quadratic polynomials | title=Proceedings of the eighth annual ACM symposium on Theory of computing - STOC '76 | year=1976 | last1=Manders | first1=Kenneth | last2=Adleman | first2=Leonard | pages=23–29 | isbn=9781450374149 | s2cid=18885088 }} 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 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{{rp|SR33}}
  • Set cover (also called "minimum cover" problem). This is equivalent, by transposing the incidence matrix, to the hitting set problem.{{rp|SP5, SP8}}
  • Set packing{{rp|SP3}}
  • Set splitting problem{{rp|SP4}}
  • Scheduling to minimize weighted completion time
  • Block Sorting{{Cite book|last1=Bein|first1=W. W.|last2=Larmore|first2=L. L.|last3=Latifi|first3=S.|last4=Sudborough|first4=I. H.|title=Proceedings International Symposium on Parallel Architectures, Algorithms and Networks. I-SPAN'02 |chapter=Block sorting is hard |date=2002-01-01|pages=307–312|doi=10.1109/ISPAN.2002.1004305|isbn=978-0-7695-1579-3|s2cid=32222403}} (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.{{rp|ND13}}
  • Three-dimensional Ising modelBarry Arthur Cipra, "The Ising Model Is NP-Complete", SIAM News, Vol 33, No 6.

See also

Notes

{{Reflist|30em}}

References

General

  • {{Garey-Johnson}}. This book is a classic, developing the theory, then cataloguing many NP-Complete problems.
  • {{cite conference

| last = Cook

| first = S.A.

| author-link = Stephen A. Cook

| title = The complexity of theorem proving procedures

| book-title = Proceedings, Third Annual ACM Symposium on the Theory of Computing, ACM, New York

| year = 1971

| pages = 151–158

| doi = 10.1145/800157.805047

| doi-access = free

}}

  • {{Cite book

| first = Richard M.

| last = Karp

| author-link = Richard Karp

| chapter = Reducibility among combinatorial problems

| title = Complexity of Computer Computations

| editor1-first = Raymond E.

| editor1-last = Miller

| editor2-first = James W.

| editor2-last = Thatcher

| publisher = Plenum

| pages = 85–103

| year = 1972

}}

  • {{cite web

| last = Dunne

| first = P.E.

| title = An annotated list of selected NP-complete problems

| publisher = COMP202, Dept. of Computer Science, University of Liverpool

| url = http://www.csc.liv.ac.uk/~ped/teachadmin/COMP202/annotated_np.html

| access-date = 2008-06-21

}}

  • {{cite web

| last = Crescenzi

| first = P.

|author2=Kann, V. |author3=Halldórsson, M. | author4-link = Marek Karpinski

|author4=Karpinski, M. | author5-link = Gerhard J. Woeginger

|author5=Woeginger, G

| title = A compendium of NP optimization problems

| publisher = KTH NADA, Stockholm

| url = https://www.csc.kth.se/~viggo/problemlist/compendium.html

| access-date = 2008-06-21

}}

  • {{cite web

| last = Dahlke

| first = K

| title = NP-complete problems

| work = Math Reference Project

| url = http://www.mathreference.com/lan-cx-np,intro.html

| access-date = 2008-06-21

}}

Specific problems

  • {{cite web

| first = E

| last = Friedman

| title = Pearl puzzles are NP-complete

| year = 2002

| publisher = Stetson University, DeLand, Florida

| url = http://www.stetson.edu/~efriedma/papers/pearl/pearl.html

| access-date = 2008-06-21

| archive-date = 4 September 2006

| archive-url = https://web.archive.org/web/20060904225147/http://www.stetson.edu/~efriedma/papers/pearl/pearl.html

| url-status = dead

}}

  • {{cite journal

| last1 = Grigoriev | first1 = A

| last2 = Bodlaender | first2 = H L | author2-link = Hans L. Bodlaender

| doi = 10.1007/s00453-007-0010-x

| issue = 1

| journal = Algorithmica

| mr = 2344391

| pages = 1–11

| title = Algorithms for graphs embeddable with few crossings per edge

| volume = 49

| year = 2007

| citeseerx = 10.1.1.61.3576

| s2cid = 8174422

}}

  • {{cite book

| first1 = S

| title = How the World Computes

| volume = 7318

| pages = 283–292

| last1 = Hartung

| first2 = A

| last2 = Nichterlein

| chapter = NP-Hardness and Fixed-Parameter Tractability of Realizing Degree Sequences with Directed Acyclic Graphs

| year = 2012

| publisher = Springer, Berlin, Heidelberg

| doi = 10.1007/978-3-642-30870-3_29

| series = Lecture Notes in Computer Science

| isbn = 978-3-642-30869-7

| citeseerx = 10.1.1.377.2077

| s2cid = 6112925

}}

  • {{cite conference

| last1 = Holzer

| first1 = Markus

|last2=Ruepp |first2=Oliver

| title = The Troubles of Interior Design–A Complexity Analysis of the Game Heyawake

| book-title = Proceedings, 4th International Conference on Fun with Algorithms, LNCS 4475

| publisher = Springer, Berlin/Heidelberg

| year = 2007

| isbn = 978-3-540-72913-6

| pages = 198–212

| doi = 10.1007/978-3-540-72914-3_18

| url = http://mediatum.ub.tum.de/doc/1289953/document.pdf

}}

  • {{cite journal

| doi = 10.1007/BF03025367

| last = Kaye

| first = Richard

| year = 2000

| title = Minesweeper is NP-complete

| journal = Mathematical Intelligencer

| volume = 22

| issue = 2

| pages = 9–15

| s2cid = 122435790

}} Further information available online at [http://web.mat.bham.ac.uk/R.W.Kaye/minesw/ Richard Kaye's Minesweeper pages].

  • {{Cite conference

| last1 = Kashiwabara | first1 = T.

| last2 = Fujisawa | first2 = T.

|title=NP-completeness of the problem of finding a minimum-clique-number interval graph containing a given graph as a subgraph

|book-title=Proceedings

| pages = 657–660

|conference=International Symposium on Circuits and Systems

| year = 1979

}}

  • {{Cite journal

| last1 = Ohtsuki | first1 = Tatsuo

| last2 = Mori | first2 = Hajimu

| last3 = Kuh | first3 = Ernest S.

| last4 = Kashiwabara | first4 = Toshinobu

| last5 = Fujisawa | first5 = Toshio

| doi = 10.1109/TCS.1979.1084695

| issue = 9

| journal = IEEE Transactions on Circuits and Systems

| pages = 675–684

| title = One-dimensional logic gate assignment and interval graphs

| volume = 26

| year = 1979

}}

  • {{Cite journal

| last = Lengauer | first = Thomas

| doi = 10.1007/BF00264496

| issue = 4

| journal = Acta Informatica

| pages = 465–475

| title = Black-white pebbles and graph separation

| volume = 16

| year = 1981

| s2cid = 19415148

}}

  • {{Cite journal

| last1 = Arnborg | first1 = Stefan

| last2 = Corneil | first2 = Derek G. | author2-link = Derek Corneil

| last3 = Proskurowski | first3 = Andrzej

| doi = 10.1137/0608024

| issue = 2

| journal = SIAM Journal on Algebraic and Discrete Methods

| pages = 277–284

| title = Complexity of finding embeddings in a k-tree

| volume = 8

| year = 1987

}}

  • {{cite conference

| last = Cormode

| first = Graham

| title = The hardness of the lemmings game, or Oh no, more NP-completeness proofs

| book-title = Proceedings of Third International Conference on Fun with Algorithms (FUN 2004)

| year = 2004

| pages = 65–76

}}

External links

  • [http://www.csc.kth.se/~viggo/problemlist/ A compendium of NP optimization problems]
  • [https://adriann.github.io/npc/npc.html Graph of NP-complete Problems]

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{{DEFAULTSORT:Np-Complete Problems}}

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