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...r=519066|title-link=Computers and Intractability: A Guide to the Theory of NP-Completeness}}</ref> Some computational problems are parameterized by numbers whose magnitude exponentially exceed size of t ...

'''Numerical 3-dimensional matching''' is an [[NP-complete]] decision problem. It is given by three [[multisets]] of [[integer]]s <mat ...S. Johnson (1979), Computers and Intractability; A Guide to the Theory of NP-Completeness. {{ISBN|0-7167-1045-5}}</ref> ...

...variant of the [[Boolean satisfiability problem]], often used in proofs of NP-completeness.{{r|moqo}} [[File:Monotone NAE3SAT.svg|thumb|180px|'''Monotone''' NAE3SAT problems can be represented by [[vertex coloring]] a [[hypergraph]] so that every ed ...

...s with small total weight, so that the added edges make the graph into a [[strongly connected graph]]. ...|year=1976|txt}}. They showed that a weighted version of the problem is NP-complete, but the unweighted problem can be solved in [[linear time]].{{r|et76}} Sub ...

...vertices, the graph is said to be '''strongly rainbow-connected''' (or '''strongly rainbow colored''').{{sfnp|Chartrand|Johns|McKeon|Zhang|2008}} ...ber''' of a graph <math>G</math> is the minimum number of colors needed to strongly rainbow-connect <math>G</math>, and is denoted by <math>\text{src}(G)</math ...

...position.svg|thumb|upright=1.35|Walecki's Hamiltonian decomposition of the complete graph <math>K_9</math>]] A directed graph with such a decomposition must be [[strongly connected graph|strongly connected]] and all vertices must have the same in-degree and out-degree as ...

Like [[3SAT]], PLANAR-SAT is [[NP-complete]], and is commonly used in [[reduction (complexity)|reductions]]. ...ar graphs. Hence, Planar 3SAT provides a way to prove those problems to be NP-hard. ...

...ach easy-to-verify problem can be solved as fast as SAT, which is hence NP-complete.]] In [[computational complexity theory]], a problem is '''NP-complete''' when: ...

...utational complexity theory]] that studies the complexity of computational problems on "most inputs". ...ic case complexity is because for a wide variety of concrete computational problems, the most difficult instances seem to be rare. Typical instances are relati ...

...f any size by verifying if the graph is strongly connected. The problem is NP hard if we restrict the postman tour to traverse each arc exactly once or i Input: A strongly connected, mixed graph <math>G=(V,E,A)</math> with cost <math>c(e)\geq0</ma ...

...hat take an input and produce a list of solutions, similarly to [[function problems]]. For each input, the enumeration algorithm must produce the list of all s ...theory]], and several [[complexity class]]es have been introduced for such problems. ...

...the components. The fifth looks similar to the fourth, but since it is not strongly connected, and the maximum cycle rank of its components is 1, it has the sa cycle rank zero, while a [[complete graph|complete digraph]] of [[Graph (discrete mathematics)#Definitions|order]] ''n'' with ...

...mixed) Nash equilibrium in a graphical game is [[PPAD (complexity)|PPAD]]-complete.<ref name="Daskalakis2006"/> Finding a [[correlated equilibrium]] of a grap ...actions grows with the number of players (even linearly) the problem is NP-complete.<ref name="Brandt2009"/> In any symmetric game there exists a [[symmetric e ...

.... According to the Graham–Pollak theorem, a partition into fewer than five complete bipartite subgraphs is not possible.]] ...[complete graph]] cannot be partitioned into fewer than <math>n-1</math> [[complete bipartite graph]]s.{{r|az18}} It was first published by [[Ronald Graham]] a ...

...time algorithm then they all do, and that many known algorithms for these problems have optimal or near-optimal time {{nowrap|complexity.{{r|ip99}}}} ...T}} can be solved in {{nowrap|time <math>O(2^{\delta n})</math>.}} Because problems with larger <math>k</math> cannot be easier, these numbers are ordered as { ...

...basis for proofs that various games and puzzles are PSPACE-hard or PSPACE-complete. | title = PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation ...

...nders. This assumption implies the hardness of several other computational problems, and the optimality of certain known [[approximation algorithm]]s. ...e or small. It asserts that, for every <math>\varepsilon>0</math>, it is [[NP-hard]] to distinguish between {{nowrap|<math>d</math>-regular}} graphs with ...

...math> P|size_j|C_{\max} </math> in the [[Notation for theoretic scheduling problems|three-field notation]] introduced by Graham et al.<ref name="initial specif ...mation algorithm with a ratio smaller than <math>3/2</math> unless <math>P=NP</math>.{{Citation needed|date=June 2021}} ...

...assical complexity classes such as [[P (complexity)|P]], [[NP (complexity)|NP]], [[BPP (complexity)|BPP]], and [[PSPACE]]. ...\varnothing</math>. All of the previous complexity classes contain promise problems.<ref name=":27">{{cite arXiv| last=Watrous|first=John| date=2008-04-21| tit ...

...equivalence is the one used by {{harvtxt|Goldberg|Karzanov|1996}} to model problems of matching in terms of skew-symmetric graphs; in that application, the two ...between matched and unmatched edges are of importance in weighted matching problems. ...