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...ess''' is a property of computational problems that is a special case of [[NP-complete]]ness. A general computational problem may have numerical parameters. For ...ss=free}}</ref> A problem is said to be strongly [[NP-hard]] if a strongly NP-complete problem has a pseudo-polynomial reduction to it. This pseudo-polynomial red ...

...]] problem with known pseudo-polynomial time algorithms is called [[weakly NP-complete]]. An [[NP-complete]] problem is called [[strongly NP-complete]] if it is proven that it cannot be solved by a pseudo-polynomial time alg ...

{{Short description|List of unsolved computational problems}} This article is a [[Lists of unsolved problems|list of notable unsolved problems]] in [[computer science]]. A problem in computer science is considered unso ...

{{Short description|Strongly NP-complete problem in computer science}} The '''3-partition problem''' is a [[Strong NP-completeness|strongly NP-complete]] problem in [[computer science]]. The problem is to decide whether a give ...

The '''[[knapsack problem]]''' is one of the most studied problems in [[combinatorial optimization]], with many real-life applications. For th | title = Knapsack Problems: Algorithms and Computer Implementations ...

...first2 = Michel | last3 = Laporte | first3 = Gilbert | title = Arc Routing Problems, Part 1: The Chinese Postman Problem | journal = [[Operations Research (jou Various combinatorial problems have been reduced to the Chinese Postman Problem, including finding a [[max ...

...ry of computation|theoretical computer science]] (e.g. some [[P-complete]] problems), where it is used to "artificially" decrease the run-time or space require ...t includes some numerical parameters is strongly NP-complete if it remains NP-complete even when the size of the input is made artificially larger by representing ...

...ling]], which is itself a special case of [[optimal job scheduling]]. Many problems, which are NP-hard in general, can be solved in polynomial time in the sing ...d [[Optimal job scheduling|three-field notation for optimal job scheduling problems]], the single-machine variant is denoted by '''1''' in the first field. For ...

==Complete problems and logical characterization== ...'NL''' is contained in the complexity class '''[[P (complexity)|P]]''' of problems solvable in deterministic polynomial time.<ref>{{harvtxt|Sipser|1997}}, Cor ...

{{Short description|NP-complete problem in computer science}} ...m of the numbers in ''S''₂. Although the partition problem is [[NP-complete]], there is a [[pseudo-polynomial time]] [[dynamic programming]] solution, ...

...achines. '''BPP''' also contains '''[[P (complexity)|P]]''', the class of problems solvable in polynomial time with a deterministic machine, since a determini ...The error probability does not even have to be constant: the same class of problems is defined by allowing error as high as 1/2 − ''n''^−''c'' on the ...

{{Short description|Set of problems solved by small circuits}} In the perspective of circuit complexity, '''P/poly''' is the class of problems that can be solved by small circuits. More precisely, it is the set of [[fo ...

...be characterized by a property generalizing the defining property of the [[strongly perfect graph]]s: in every induced subgraph of a Meyniel graph, every verte }}.</ref> and several graph optimization problems including [[graph coloring]] that are [[NP-hard]] for arbitrary graphs can ...

...roblems of determining whether such paths and cycles exist in graphs are [[NP-complete]]; see [[Hamiltonian path problem]] for details. ...er]].<ref>{{citation|title=Across the Board: The Mathematics of Chessboard Problems | first=John J.|last=Watkins|publisher=Princeton University Press|year=2004 ...

...'}} for a given graph {{mvar|G}} and input {{mvar|K}}; it is a classical [[NP-complete]] [[decision problem]] in [[computational complexity theory]].{{sfnp|Garey| ...'}} is defined as: <math>\gamma^{strong}(G) := \min \{|D| : D \text{ is a strongly-dominating set of } G \}</math>; obviously, <math>\gamma^{strong}(G) \geq \ ...

...m]]), but also impossible to get a good polynomial-time approximation. The problems for which such an inapproximability result would hold include [[constraint Such instances are strongly constrained in the sense that the colour of a vertex uniquely defines the c ...

...ision versions of these problems are classical examples of [[NP-complete]] problems.<ref>{{citation ...g (decision version) is the problem GT10 in Appendix&nbsp;A1.1.</ref> Both problems can be [[approximation algorithm|approximated]] within factor 2 in polynomi ...

...is in the [[low hierarchy]] of [[class NP]], which implies that it is not NP-complete unless the [[polynomial time hierarchy]] collapses to its second level.{{sf ...at is isomorphic to another given graph ''H''; this problem is known to be NP-complete. It is also known to be a special case of the [[Abelian group|non-abelian]] ...

...he '''maximum independent set problem.''' It is a [[Strong NP-completeness|strongly NP-hard]] problem.<ref>{{Cite journal|last1=Garey|first1=M. R.|last2=Johnso In [[computer science]], several [[computational problems]] related to independent sets have been studied. ...

In the context of optimization problems, the correct value is understood to be the value of the optimal solution, a ...and M. Protasi. ''Complexity and Approximation: Combinatorial optimization problems and their approximability properties'', Springer-Verlag, 1999.</ref> ...