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Complexity class
as hard as NP, but not necessarily in NP. NP-equivalent Decision problems that are both NP-hard and NP-easy, but not necessarily in NP. NP-intermediate
NP-hardness
Concept in complexity theory
solved by a pseudo-polynomial time algorithm unless P = NP. The strong/weak kinds of NP-hardness are defined analogously. Consider solving the problem of
Pseudo-polynomial_time
Computational property
reduction is more restrictive than the usual poly-time reduction used for NP-hardness proofs. In particular, the pseudo-polynomial reduction cannot output
Strong_NP-completeness
Unsolved problem in computer science
to P versus NP problem. Fortnow, L.; Gasarch, W. "Computational complexity". Aviad Rubinstein's Hardness of Approximation Between P and NP, winner of the
P_versus_NP_problem
problem. Both weak NP-hardness and weak polynomial-time correspond to encoding the input integers in binary coding. If a problem is strongly NP-hard, then it
Weak_NP-completeness
Optimization problem
problems when k is small (say k < 5). Farthest-point clustering For the hardness of the problem, it's impractical to get an exact solution or precise approximation
Optimal_facility_location
The hardness of the lemmings game, or Oh no, more NP-completeness proofs (PDF). Light Up is NP-Complete Friedman, Erich. "Pearl Puzzles are NP-complete"
List_of_NP-complete_problems
Cycles in a graph that generate all cycles
average of the pairwise distances; both are NP-hard. Finding a minimum weight weakly fundamental basis is also NP-hard, and approximating it is MAXSNP-hard
Cycle_basis
Complexity class used to classify decision problems
results about the hardness of approximation algorithms to be proven. All problems in P, denoted P ⊆ N P {\displaystyle {\mathsf {P\subseteq NP}}} . Given a
NP_(complexity)
problem becomes NP-hard, implying that finding a polynomial time approximation for the problem is impossible unless NP=P. Some hardness of approximation
Hardness_of_approximation
Hypothesis in computational complexity theory
computational hardness assumption such as P ≠ NP. Computer scientists have different ways of assessing which hardness assumptions are more reliable. We say that
Computational hardness assumption
Computational_hardness_assumption
Problem of finding the longest simple path for a given graph
cannot be solved in polynomial time for arbitrary graphs unless P = NP. Stronger hardness results are also known showing that it is difficult to approximate
Longest_path_problem
Unsolved problem in computational complexity theory
game, known as a unique game, has NP-hard computational complexity. It has broad applications in the theory of hardness of approximation. If the unique
Unique_games_conjecture
Optimization problem in computer science
in the Euclidean norm is NP-hard under randomized polynomial-time reductions. Subsequent work established randomized hardness of approximation for every
Lattice_problem
Vector quantization algorithm minimizing the sum of squared deviations
with up to 4,177 entities and 20,531 features. As expected, due to the NP-hardness of the subjacent optimization problem, the computational time of optimal
K-means_clustering
Optimization problem
Since the traveling salesman problem is NP-hard, the job-shop problem with sequence-dependent setup is also NP-hard since the TSP is a special case of
Job-shop_scheduling
should be used. Explicit proofs of NP-hardness are given in later papers: Yu, Hoogeveen and Lenstra prove NP-hardness of a very restricted version of Numerical
Numerical 3-dimensional matching
Numerical_3-dimensional_matching
Mathematical optimization problem restricted to integers
vertex cover to integer programming that will serve as the proof of NP-hardness. Let G = ( V , E ) {\displaystyle G=(V,E)} be an undirected graph. Define
Integer_programming
Type of algorithm, produces approximately correct solutions
(e.g., they may be used to generate good seed values). Results about NP-hardness in theoretical computer science make heuristics the only viable option
Heuristic_(computer_science)
Field in logic and theoretical computer science
Koroth, Mertz and Pitassi (2020) to NP-hardness of automating Cutting Planes; and by Garlík (2020) to NP-hardness of automating k-DNF Resolution. It is
Proof_complexity
Complexity class
or even impossible to prove NP-hardness for TFNP problems. For example, if any TFNP problem is NP-complete, then NP = coNP, which is generally conjectured
TFNP
Complexity class of problems
problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such problems
NP-intermediate
Problem in computational complexity theory
satisfiability problem and the quantified boolean formula problem. Because of its NP-hardness, large-size MAX-SAT instances cannot in general be solved exactly, and
Maximum satisfiability problem
Maximum_satisfiability_problem
Node ordering for directed acyclic graphs
ordering exists, and whether a Hamiltonian path exists, despite the NP-hardness of the Hamiltonian path problem for more general directed graphs (i.e
Topological_sorting
Computer hardware technology that uses quantum mechanics
the class of NP-complete problems (if an NP-complete problem were in BQP, then it would follow from NP-hardness that all problems in NP are in BQP). Hanyuan-1
Quantum_computing
Solving an optimization problem with a quadratic objective function
well-known NP-hard problem; hence, solving the quadratic program is NP-hard too. Some important special cases are also NP-hard: Sahni proved NP-hardness for
Quadratic_programming
Physics phenomenon
1145/780542.780545. ISBN 1-58113-674-9. Gharibian, Sevag (2010). "Strong NP-Hardness of the Quantum Separability Problem". Quantum Information and Computation
Quantum_entanglement
2012 American film
Selection. Complexity theory List of films about mathematicians NP-hardness P-hardness Regan, K. W. (April 22, 2012). "The Travelling Salesman's Power"
Travelling Salesman (2012 film)
Travelling_Salesman_(2012_film)
co-NP is a subset of NP/poly, which (by the Karp–Lipton theorem) would cause the collapse of the polynomial hierarchy. The same computational hardness assumption
NP/poly
Function used in computer cryptography
worst-case sense. This is different from much of complexity theory (e.g., NP-hardness), where the term "hard" is meant in the worst-case. That is why even
One-way_function
Matrix with non-zero elements only in a diagonal band
Mathematics, ISBN 978-0-898716-13-9. Feige, Uriel (2000), "Coping with the NP-Hardness of the Graph Bandwidth Problem", Algorithm Theory - SWAT 2000, Lecture
Band_matrix
Branch of chemistry
evaluating and transforming the two-electron integrals. This proof of NP-hardness or NP-completeness comes from embedding problems like the Ising model into
Computational_chemistry
Mathematical proof at least partially generated by computer
computed several van der Waerden numbers using FPGA-based SAT-solver. NP-hardness of minimum-weight triangulation, 2008 Ahmed (between 2009 and 2014) computed
Computer-assisted_proof
Problem in combinatorial optimization
that the hardness of the knapsack problem depends on the form of the input. If the weights and profits are given as integers, it is weakly NP-complete
Knapsack_problem
NP-hard problem in combinatorial optimization
showed in 1972 that the Hamiltonian cycle problem was NP-complete, which implies the NP-hardness of TSP. This supplied a mathematical explanation for the
Travelling_salesman_problem
Strongly NP-complete problem in computer science
we construct a 4-set in A, with sum T. The NP-hardness of 3-partition was used to prove the NP-hardness rectangle packing, as well as of Tetris and some
3-partition_problem
Classic NP-complete problem in computer science
polynomial time. Thus Circuit SAT belongs to complexity class NP. To show NP-hardness, it is possible to construct a reduction from 3SAT to Circuit SAT
Circuit satisfiability problem
Circuit_satisfiability_problem
Toy problem in artificial intelligence research
doi:10.1016/s0004-3702(00)00079-5. Chenoweth, Stephen V (1991). On the NP-Hardness of Blocks World. AAAI Proceedings of the ninth national conference on
Blocks_world
Japanese manga series
depressed. Ikeda suggests Kosuke find science in his everyday life, such as NP-hardness in complex datings sim, or Chaos theory in the random trajectory of a
Science Fell in Love, So I Tried to Prove It
Science_Fell_in_Love,_So_I_Tried_to_Prove_It
Edges that hit all cycles in a graph
of its hardness proof, unless P = NP, it has no polynomial time approximation ratio better than 1.3606. This is the same threshold for hardness of approximation
Feedback_arc_set
Subfield of mathematical optimization
optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to
Combinatorial_optimization
Independent set which is not a subset of any other independent set
Rinnooy Kan, A. H. G. (1980), "Generating all maximal independent sets: NP-hardness and polynomial time algorithms" (PDF), SIAM Journal on Computing, 9 (3):
Maximal_independent_set
Single-winner ranked-choice voting system
S2CID 154114517. The article only directly proves NP-hardness, but it is clear that the decision problem is in NP since given a candidate and a list of k swaps
Dodgson's_method
Theorem in computational complexity theory
the PCP characterization theorem) states that every decision problem in the NP complexity class has probabilistically checkable proofs (proofs that can be
PCP_theorem
Problem of counting solutions to logic formulas
This is the counting version of Planar Monotone Rectilinear 3SAT. The NP-hardness reduction given by de Berg & Khosravi is parsimonious. Therefore, this
♯SAT
British businessman (born 1980)
exponentially hard problems found in academia and industry known as NP-hardness. In 2016, Satalia was the only UK company to appear in the Gartner Cool
Daniel_J._Hulme
Graph without four-vertex star subgraphs
that the NP-hardness of cliques in claw-free graphs follows from the NP-hardness of the independent set problem in triangle-free graphs, proven NP-hard by
Claw-free_graph
Construction analogous to that of a dual vector space
S2CID 13921988. Cai, Jin-Yi; Nerurkar, Ajay (2000). "A note on the non-NP-hardness of approximate lattice problems under general Cook reductions". Information
Dual_lattice
we should minimize. It can be proved that the talent scheduling problem is NP-hard by a reduction to the optimal linear arrangement (OLA) problem. Even
Talent_scheduling
Graph of short distances in another graph
S2CID 2706926. Agnarsson & Halldórsson (2000) list publications proving NP-hardness for general graphs by McCormick (1983) and Lin and Skiena (1995), and
Graph_power
Optimization problem in computer science
m\geq 4} , the problem is also strongly NP-hard (this result improved a previous result showing strong NP-hardness for m ≥ 5 {\displaystyle m\geq 5} ). If
Parallel_task_scheduling
Undirected unit-distance graph requiring four colors
spindle were used by Horvat, Kratochvíl & Pisanski (2011) to show the NP-hardness of testing whether a given graph has a two-dimensional unit distance
Moser_spindle
Type of database query
AC0, which is contained in LOGSPACE and thus in polynomial time. The NP-hardness of conjunctive queries may appear surprising, since relational algebra
Conjunctive_query
NP-complete problem in computer science
same sum). This problem is strongly NP-hard. Kovalyov and Pesch discuss a generic approach to proving NP-hardness of partition-type problems. One application
Partition_problem
Application of computational algorithms, methods and programs to phylogenetic analyses
is believed to be computationally intractable to compute due to its NP-hardness. The pruning algorithm, a variant of dynamic programming, is often used
Computational_phylogenetics
Quantum computational theorem on problem complexity
Chinmay Nirkhe, and was presented at STOC 2023. The classical theory of NP-hardness is well-suited for characterizing problems which are unlikely to be solvable
NLTS_conjecture
Overview of and topical guide to algorithms
element method Fast multipole method P (complexity) NP (complexity) NP-completeness NP-hardness EXPTIME PSPACE BPP (complexity) BQP Undecidable problem
Outline_of_algorithms
Optimal job scheduling with some jobs done in parts
{\displaystyle 2n-2} preemptions. Shachnai, Tamir, and Woeginger proved NP-hardness for the case where the number of preemption is strictly less than 2 n
Fractional_job_scheduling
Task of computing complete subgraphs
the theory of NP-complete problems. Because of the hardness of the decision problem, the problem of finding a maximum clique is also NP-hard. If one could
Clique_problem
Minimum-cost tree with exactly k vertices
Because the NP-hardness reduction for the k-minimum spanning tree problem preserves the weight of all solutions, it also preserves the hardness of approximation
K-minimum_spanning_tree
Node labeling problem in graph theory
pathwidth, different NP-complete optimization problems involving linear layouts of graphs. Feige, Uriel (2000). "Coping with the NP-hardness of the graph bandwidth
Graph_bandwidth
Quantum states that are not entangled
Theory of Computing, ACM Press, New York, 2003. Sevag Gharibian, Strong NP-Hardness of the Quantum Separability Problem, Quantum Information and Computation
Separable_state
Computational complexity of quantum algorithms
from the class of NP-complete problems (if any NP-complete problem were in BQP, then it follows from NP-hardness that all problems in NP are in BQP). The
Quantum_complexity_theory
Decision problem in computer science
{\displaystyle T} . The problem is known to be NP-complete. Moreover, some restricted variants of it are NP-complete too, for example: The variant in which
Subset_sum_problem
Difficulty measures for computer science problems
{BH}}_{k-2}} Hardness for classes of the Boolean hierarchy can be proved by showing a reduction from a number of instances of an arbitrary NP-complete problem
Boolean_hierarchy
Economical computational problem
problem 6 is still NP-hard even for zero-sum games (NP-complete for any number of players). Conitzer and Sandholm prove the following hardness results, even
Nash_equilibrium_computation
Property in graph theory
communication. In database theory, the NP-hardness of the cutwidth problem has been used to show that it is also NP-hard to schedule the transfer of blocks
Cutwidth
List of concepts in artificial intelligence
input is "yes" if the solution set is non-empty and "no" if it is empty. NP-hardness In computational complexity theory, the defining property of a class
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
Subset of a graph's vertices, including at least one endpoint of every edge
classical optimization problem. It is NP-hard, so it cannot be solved by a polynomial-time algorithm if P ≠ NP. Moreover, it is hard to approximate –
Vertex_cover
3-satisfiability (NAE3SAT) is an NP-complete variant of the Boolean satisfiability problem, often used in proofs of NP-completeness. Like 3-satisfiability
Not-all-equal 3-satisfiability
Not-all-equal_3-satisfiability
1762–1782. arXiv:1908.01669. doi:10.1287/opre.2022.2279. S2CID 247922344. "np hardness - A partition problem in which some numbers may be cut". Theoretical
List of unsolved problems in fair division
List_of_unsolved_problems_in_fair_division
objects that have that repeat time, pinwheel scheduling is NP-hard. Despite the NP-hardness of the pinwheel scheduling problem for general inputs, some
Pinwheel_scheduling
Set of hyperedges where every pair is disjoint
in polynomial time, so it cannot be used as a reduction for proving NP-hardness. But it has some theoretical uses. 3-dimensional matching – a special
Matching_in_hypergraphs
Graph layout on multiple half-planes
between secondary structures and book embeddings as part of a proof of the NP-hardness of certain problems in RNA secondary structure comparison. And if an
Book_embedding
Computational hardness assumption
the NP-hardness of several other computational problems. Because it is only a hypothesis, this does not prove that these problems actually are NP-hard
Small set expansion hypothesis
Small_set_expansion_hypothesis
Relation in theoretical computer science
word equations is NP-complete—remains open. (NP-hardness follows immediately from the fact that solving word equations generalises the NP-complete problem
Word_equation
Mathematical and computational problem
proved that BP-SPF is NP-hard. Menakerman and Rom showed that BP-SIF and BP-SPF are both strongly NP-hard. Despite the hardness, they present several
Bin_packing_problem
Set of problems in computational complexity theory
space complexity classes relate to each other in the following way: L⊆NL⊆P⊆NP⊆PSPACE⊆EXPTIME⊆NEXPTIME⊆EXPSPACE Where ⊆ denotes the subset relation. However
Complexity_class
Point set triangulation minimizing total length
logarithmic factor of the minimum weight. The hardness result of Mulzer and Rote also implies the NP-hardness of finding an approximate solution with relative
Minimum-weight_triangulation
Decision problem in graph theory
Franklin Institute, 103: 406–422 Nichterlein, André; Hartung, Sepp (2012), "NP-Hardness and Fixed-Parameter Tractability of Realizing Degree Sequences with Directed
Digraph_realization_problem
Single-player game played with mahjong tiles
the probability of removing all tiles is PSPACE-complete, and the game is NP-complete if looking below tiles is allowed. It has been proven that it is
Mahjong_solitaire
Proof checkable by a randomized algorithm
The PCP theorem proved in 1992 states that PCP[O(log n),O(1)] = NP. The theory of hardness of approximation requires a detailed understanding of the role
Probabilistically checkable proof
Probabilistically_checkable_proof
deciding whether s ∈ L(p) for an arbitrary string s ∈ Σ+ and pattern p is NP-complete (see picture), and so is hence the problem of deciding p ≤ q for
Pattern language (formal languages)
Pattern_language_(formal_languages)
Estimate of time taken for running an algorithm
the unsolved P versus NP problem asks if all problems in NP have polynomial-time algorithms. All the best-known algorithms for NP-complete problems like
Time_complexity
Class of algorithms that find approximate solutions to optimization problems
algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution
Approximation_algorithm
Vertices whose removal breaks all cycles
computational hardness assumption, it is NP-hard to approximate the problem to within any constant factor in polynomial time. The same hardness result was
Feedback_vertex_set
Subunit of a computational problem
computational problem to another, as part of proofs of NP-completeness or other types of computational hardness. The component design technique is a method for
Gadget_(computer_science)
Type of approximation algorithm
problem is APX-hard, after which the existence of a PTAS would show P = NP. APX-hardness is commonly shown via PTAS reduction or AP-reduction. Parameterized
Polynomial-time approximation scheme
Polynomial-time_approximation_scheme
Problem in social choice
budget-allocation always exists. However, similar to FJR, it is NP-hard to find an EJR allocation. The NP-hardness holds even with approval ballots, for any satisfaction
Combinatorial participatory budgeting
Combinatorial_participatory_budgeting
Complexity class of approximable problems
Unless P = NP, there exist problems in APX that are neither in PTAS nor APX-complete. Such problems can be thought of as having a hardness between PTAS
APX
Ali; Olshevsky, Alex; Parrilo, Pablo A.; Tsitsiklis, John N. (2013). "NP-hardness of deciding convexity of quartic polynomials and related problems". Mathematical
SOS-convexity
Abstract machine that models computation
optimization versions of certain NP-complete problems unless P = NP. Such problems are now studied in the field known as hardness of approximation. Oracle machine
Interactive_proof_system
_{\text{cer}}(G)=\gamma (G)} for graphs with no weak support vertices and the known NP-hardness of the domination problem in such graph classes. Determining whether
Certified_dominating_set
(this does not contradict the above hardness result, since the hardness result uses only a value oracle; in the hardness examples, the demand oracle itself
Welfare_maximization
Soft calcium sulfate mineral
[citation needed] The Mohs scale of mineral hardness defines gypsum as hardness value 2 based on scratch hardness comparison. Fine-grained white or lightly
Gypsum
Partition of a graph's nodes into cliques
problem remains NP-complete on some other special classes of graphs, including the cubic planar graphs and unit disk graphs. The same hardness of approximation
Clique_cover
Unsolved problem in computational complexity theory
solvable in polynomial time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. It is known that the graph
Graph_isomorphism_problem
Provides lower bounds on the circuit complexity of boolean functions
Under the assumption that pseudorandom functions exist with "exponential hardness" as specified in their main theorem, Razborov and Rudich show that these
Natural_proof
American computer scientist
the P versus NP problem. Algorithmica: P = NP; Heuristica: P is not NP, but NP problems are tractable on average; Pessiland: there are NP problems that
Russell_Impagliazzo
Branch of computational complexity theory
of efficient, exact, and deterministic solving algorithms for NP-complete, or otherwise NP-hard, problems is considered unlikely, if input parameters are
Parameterized_complexity
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