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Problem in computational complexity theory
computational complexity theory, the maximum satisfiability problem (MAX-SAT) is the problem of determining the maximum number of clauses, of a given Boolean
Maximum satisfiability problem
Maximum_satisfiability_problem
Problem of determining if a Boolean formula could be made true
science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) asks whether
Boolean satisfiability problem
Boolean_satisfiability_problem
Logic problem, AND of pairwise ORs
more general problems, which are NP-complete, 2-satisfiability can be solved in polynomial time. Instances of the 2-satisfiability problem are typically
2-satisfiability
Task of computing complete subgraphs
sequence of bits. An instance of the satisfiability problem should have a valid proof if and only if it is satisfiable. The proof is checked by an algorithm
Clique_problem
Problem in graph theory
NP-completeness of the problem can be shown, for example, by a reduction from maximum 2-satisfiability (a restriction of the maximum satisfiability problem). The weighted
Maximum_cut
Set of computational problems stated by Richard Karp (1973)
NP-complete by reducing Exact cover to Knapsack. Satisfiability: the Boolean satisfiability problem for formulas in conjunctive normal form (often referred
Karp's 21 NP-complete problems
Karp's_21_NP-complete_problems
Set of objects whose state must satisfy limits
are incomplete satisfiability algorithms. They may find a solution of a problem, but they may fail even if the problem is satisfiable. They work by iteratively
Constraint satisfaction problem
Constraint_satisfaction_problem
MAXEkSAT is a problem in computational complexity theory that is a maximization version of the Boolean satisfiability problem 3SAT. In MAXEkSAT, each
MAXEkSAT
Complexity class
the halting problem is NP-hard but not NP-complete. For example, the Boolean satisfiability problem can be reduced to the halting problem by transforming
NP-hardness
Study of mathematical algorithms for optimization problems
optimization and simulated annealing. The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at
Mathematical_optimization
Complexity class
between a problem in P and an NP-complete problem. For example, the 3-satisfiability problem, a restriction of the Boolean satisfiability problem, remains
NP-completeness
a partially filled square can be completed) Maximum 2-satisfiability Maximum volume submatrix – Problem of selecting the best conditioned subset of a
List_of_NP-complete_problems
Optimization by removing non-optimal solutions to subproblems
NP-hard problems: Integer programming Nonlinear programming Travelling salesman problem (TSP) Quadratic assignment problem (QAP) Maximum satisfiability problem
Branch_and_bound
Computational Formula that can be measured in terms of True or False
theory, the quantified Boolean formula problem (QBF) is a generalization of the Boolean satisfiability problem in which both existential quantifiers and
True quantified Boolean formula
True_quantified_Boolean_formula
Problem of finding a cycle through all vertices of a graph
NP-complete problems. They remain NP-complete even for special kinds of graphs, such as: bipartite graphs, undirected planar graphs of maximum degree three
Hamiltonian_path_problem
Spanish computer scientist
science, including proof complexity and algorithms for the maximum satisfiability problem. She is a professor of computer science at the Polytechnic University
María_Luisa_Bonet
Pythagorean Triples Problem via Cube-and-Conquer". In Creignou, N.; Le Berre, D. (eds.). Theory and Applications of Satisfiability Testing – SAT 2016.
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Method for automated planning
Satisfiability) is a method for automated planning. It converts the planning problem instance into an instance of the Boolean satisfiability problem (SAT)
Satplan
Subset of a graph's vertices, including at least one endpoint of every edge
NP-completeness can be proven by reduction from 3-satisfiability or, as Karp did, by reduction from the clique problem. Vertex cover remains NP-complete even in
Vertex_cover
Class of problems in computer science
shown by a reduction from the following version of the Boolean satisfiability problem, which was shown to be NP-complete likewise to the unrestricted
Interval_scheduling
Boolean satisfiability problem despite there being no known efficient algorithm in the general case. The Boolean satisfiability (or SAT) problem can be
Boolean satisfiability algorithm heuristics
Boolean_satisfiability_algorithm_heuristics
Area of discrete mathematics
which are strictly compositional, graph unification is the sufficient satisfiability and combination function. Well-known applications include automatic
Graph_theory
Boolean satisfiability problem restricted to a planar incidence graph
the planar 3-satisfiability problem (abbreviated PLANAR 3SAT or PL3SAT) is an extension of the classical Boolean 3-satisfiability problem to a planar incidence
Planar_SAT
Mathematics problem
namely design theory techniques, SAT formulations (propositional satisfiability problem), constraint-based approaches, metaheuristic methods, and radix
Social_golfer_problem
Code-breaking game
the hints in the previous guesses). The Mastermind satisfiability problem (MSP) is a decision problem that asks, "Given a set of guesses and the number
Mastermind_(board_game)
split subsets. Set splitting is special case of the not-all-equal satisfiability problem without negated variables. Additionally, Ek-set splitting equals
Set_splitting_problem
Complexity class of approximable problems
and Gerhard Woeginger. Maximum Satisfiability Archived 2007-04-13 at the Wayback Machine. A compendium of NP optimization problems Archived 2007-04-05 at
APX
Metaheuristic method for optimization problems
Parreira, A (2000). "Variable neighborhood search for weighted maximum satisfiability problem". Les Cahiers du GERAD G–2000–62, HEC Montréal, Canada. Hansen
Variable_neighborhood_search
Standard form of Boolean function
for satisfiability is to convert it into a DNF, the satisfiability of which can be checked in linear time 1 ≤ m ≤ {\displaystyle 1\leq m\leq } maximum number
Conjunctive_normal_form
Quantum physics-based metaheuristic for optimization problems
algorithms for solving instances of the max-SAT (maximum satisfiable problem) and Minimum Multicut problems, together with an overview of the quantum annealing
Quantum_annealing
Unproven computational hardness assumption
The k {\displaystyle k} -SAT problem is a version of the Boolean satisfiability problem in which the input to the problem is a Boolean expression in conjunctive
Exponential_time_hypothesis
The network flow algorithms are used to solve instances of weighted 2-satisfiability, and these in turn are used to compute utilitarian stable matchings
Optimal_stable_matching
Partition of a graph's nodes into 2 disjoint subsets
D. P. (1995), "Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming", Journal of the ACM, 42
Cut_(graph_theory)
Problem in computer science
MAX-3SAT is a problem in the computational complexity subfield of computer science. It generalises the Boolean satisfiability problem (SAT) which is a
MAX-3SAT
On linear-time algorithms for graph logic
formulas, this problem is undecidable. However, satisfiability of MSO2 formulas is decidable for the graphs of bounded treewidth, and satisfiability of MSO1
Courcelle's_theorem
Standard form of a boolean function
Boolean satisfiability problem on conjunctive normal form formulas is NP-complete. By the duality principle, so is the falsifiability problem on DNF formulas
Disjunctive_normal_form
Task to construct a program meeting a formal specification
possible to encode program synthesis problems in Boolean logic and use algorithms for the Boolean satisfiability problem to automatically find programs. In
Program_synthesis
Inherent difficulty of computational problems
many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the
Computational complexity theory
Computational_complexity_theory
Directed graph isomorphic to its own transpose graph
and in the implication graphs used to efficiently solve the 2-satisfiability problem. As defined, e.g., by Goldberg & Karzanov (1996), a skew-symmetric
Skew-symmetric_graph
Amount of resources to perform an algorithm
other NP problem. Many combinatorial problems, such as the Knapsack problem, the travelling salesman problem, and the Boolean satisfiability problem are NP-complete
Computational_complexity
Theorem in computational complexity theory
optimization problems including maximum boolean formula satisfiability, maximum independent set in graphs, and the shortest vector problem for lattices
PCP_theorem
Method for problem solving in optimization
target is to minimize the total length of the cycle The Boolean satisfiability problem, in which a candidate solution is a truth assignment, and the target
Local_search_(optimization)
whether the problem is satisfiable. Enforcing strong directional i {\displaystyle i} -consistency allows telling the satisfiability of problems that have
Local_consistency
placed, then it may be solved efficiently by using an instance of 2-satisfiability to find a placement avoiding any conflicting pairs of placements; several
Automatic_label_placement
Unsolved problem in computational complexity theory
David P. (1995), "Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming", Journal of the ACM, 42
Unique_games_conjecture
Probabilistic optimization technique and metaheuristic
traveling salesman problem, the boolean satisfiability problem, protein structure prediction, and job-shop scheduling). For problems where a fixed amount
Simulated_annealing
Proposition in mathematical logic
problems in set theory, and establishing its truth or falsehood was the first of Hilbert's 23 problems presented in 1900. The answer to this problem is
Continuum_hypothesis
Methodic assignment of colors to elements of a graph
related to the problem of symmetry breaking. The current state-of-the-art randomized algorithms are faster for sufficiently large maximum degree Δ than
Graph_coloring
Graph with a median for each three vertices
solution of 2-satisfiability instances, below. Median graphs have a close connection to the solution sets of 2-satisfiability problems that can be used
Median_graph
Independent set which is not a subset of any other independent set
either the maximum set packing or the maximal matching problem or by an N C 2 {\displaystyle NC^{2}} reduction from the 2-satisfiability problem. Typically
Maximal_independent_set
Set of hypergraph nodes to which every hyperedge is connected
computer science such as machine learning, indexing of databases, the satisfiability problem, data mining, and computer program optimization. Matching in hypergraphs
Vertex_cover_in_hypergraphs
Computational complexity class
minimum dominating set in a tournament could be used to solve Boolean satisfiability with m {\displaystyle m} clauses and O ( log 2 m ) {\displaystyle
Quasi-polynomial_time
Geometry problem on tiling by hypercubes
Johnson, David S.; Trick, Michael A. (1996), Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, Workshop, October 11–13, 1993
Keller's_conjecture
deciding the satisfiability of propositional logic formula in conjunctive normal form, i.e. for solving the CNF-SAT problem Exact cover problem Min conflicts
List_of_algorithms
Class of algorithms that find approximate solutions to optimization problems
(November 1995). "Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming". J. ACM. 42 (6): 1115–1145
Approximation_algorithm
Model theory concept
number of models of T (up to isomorphism) of cardinality κ. The spectrum problem is to describe the possible behaviors of I(T, κ) as a function of κ. It
Spectrum_of_a_theory
Finite-state machine
Heule and S. Verwer: the minimal DFA identification problem is reduced to deciding the satisfiability of a Boolean formula. The main idea is to build an
Deterministic finite automaton
Deterministic_finite_automaton
Algebraic manipulation of "true" and "false"
is called the Boolean satisfiability problem (SAT), and is of importance to theoretical computer science, being the first problem shown to be NP-complete
Boolean_algebra
On the complexity classes of problems about satisfying a subset of boolean relations
Conjunctive normal form is abbreviated CNF below. X(N)OR-SAT stands for a satisfiability problem which is the AND of several boolean linear equations that can be
Max/min CSP/Ones classification theorems
Max/min_CSP/Ones_classification_theorems
Graph drawing with vertices on a line
exists) may be found in polynomial time by translating the problem into a 2-satisfiability problem, in which the variables represent the placement of each
Arc_diagram
Class of problems solvable in polynomial time
complexity class for problems that are not decision problems (even though, for example, finding the solution to a 2-satisfiability instance in polynomial
P_(complexity)
Estimate of time taken for running an algorithm
{poly}}(n)} . The exponential time hypothesis (ETH) is that 3SAT, the satisfiability problem of Boolean formulas in conjunctive normal form with at most three
Time_complexity
Mathematical proof about the permanent of matrices
interpretation of the permanent. #SAT, a function problem related to the Boolean satisfiability problem, is the problem of counting the number of satisfying assignments
♯P-completeness of 01-permanent
♯P-completeness_of_01-permanent
Cognitive heuristic of searching for an acceptable decision
Rational ignorance Rationality Satisfaction paradox Satisfiability Utility maximization problem Colman, Andrew (2006). A Dictionary of Psychology. New
Satisficing
solving SAT and weighted MAX-SAT problems, Journal of Automated Reasoning, Special Issue on Satisfiability Problems, Kluwer, Vol.24, 2000, 205-223 Mills
Guided_local_search
Award for advancements in discrete mathematics
David P. (1995). "Improved approximation algorithms for the maximum cut and satisfiability probelsm using semi-definite programming". Journal of the ACM
Fulkerson_Prize
Technique for reducing number of solutions
randomized polynomial-time reduction from the satisfiability problem for Boolean formulas to the problem of detecting whether a Boolean formula has a unique
Isolation_lemma
Graph layout on multiple half-planes
transforming the problem into an instance of the Boolean satisfiability problem and applying a SAT solver to the resulting problem. They state that their
Book_embedding
On coloring infinite graphs
studying this problem was to extend from finite to infinite graphs the theorem that, whenever a graph has an orientation with finite maximum out-degree k
De Bruijn–Erdős theorem (graph theory)
De_Bruijn–Erdős_theorem_(graph_theory)
Smallest dimension where a graph can be represented as an intersection graph of boxes
37236/7787, S2CID 119148637. Kratochvil, Jan (1994), "A special planar satisfiability problem and a consequence of its NP–completeness", Discrete Applied Mathematics
Boxicity
selection of some of the problems and fields in which hyper-heuristics have been explored: bin packing problem boolean satisfiability problem educational timetabling
Hyper-heuristic
Infinite cardinal number
{\displaystyle \aleph _{0}} : Every finite set of natural numbers has a maximum, which is also a natural number, and finite unions of finite sets are finite
Aleph_number
concerns the Boolean satisfiability problem for Boolean formulas in conjunctive normal form, with uniform clause size. These problems can be parameterized
Entropy_compression
Generalization of graph theory
useful in modelling such things as satisfiability problems, databases, machine learning, and Steiner tree problems. They have been extensively used in
Hypergraph
Logic that allows infinitely long proofs
constant symbols may be added for each variable with the resulting satisfiability relation remaining the same. To avoid this, some authors use a different
Infinitary_logic
Axiom in the mathematical field of set theory
that the Whitehead problem is independent of ZFC. Martin's axiom has generalizations called the proper forcing axiom and Martin's maximum. Martin, Donald
Martin's_axiom
Representation of a graph as a path graph "thickened" by some amount
Björklund, Andreas; Husfeldt, Thore (2008), "Exact algorithms for exact satisfiability and number of perfect matchings", Algorithmica, 52 (2): 226–249, doi:10
Pathwidth
suitable large cardinal: Proper forcing axiom Open coloring axiom Martin's maximum Existence of 0# Singular cardinals hypothesis Projective determinacy (and
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Most widely known generalized inverse of a matrix
{\displaystyle \|x\|_{2}} among all solutions. If A x = b {\displaystyle Ax=b} is satisfiable, the vector z = A + b {\displaystyle z=A^{+}b} is a solution, and satisfies
Moore–Penrose_inverse
Mathematical set containing no elements
the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element
Empty_set
Canadian-American computer scientist
propositions derived from the maximum clique problem, exponential lower bounds for resolution proofs of dense random 3-satisfiability instances, and subexponential
Toniann_Pitassi
Programming paradigm in which many processes are executed simultaneously
Baran, B. (29 August 2008). "Asynchronous team algorithms for Boolean Satisfiability". 2007 2nd Bio-Inspired Models of Network, Information and Computing
Parallel_computing
Algorithmic technique using hashing
David P. (1995). "Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming". Journal of the ACM. 42
Locality-sensitive_hashing
Awarded every year by the American Mathematical Society
David P. (1995). "Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming". Journal of the ACM. 42
Leroy_P._Steele_Prize
approach to the dynamic optimality problem on online algorithms for binary search trees involves reformulating the problem geometrically, in terms of augmenting
Geometry of binary search trees
Geometry_of_binary_search_trees
Statement in mathematical combinatorics
and 36. This verification was achieved using a combination of Boolean satisfiability (SAT) solving and computer algebra systems (CAS). The proof was generated
Ramsey's_theorem
System including an indeterminate value
xy uses multiplication, and x2 uses exponentiation), or by the minimum/maximum functions: x ∧ y = 1 2 ( x + y − x 2 − y 2 + x y + x 2 y 2 ) = min ( x
Three-valued_logic
Theories in mathematical logic
exists; be satisfiable: there exists a σ-structure for which the sentences of the theory are all true (by the completeness theorem, satisfiability is equivalent
List_of_first-order_theories
Type of cardinal number in mathematics
} are finite. A finite sequence of finite ordinals always has a finite maximum, so ω {\displaystyle \omega } cannot be the limit of any sequence of type
Regular_cardinal
Logic with discrete truth values
hdl:10261/131932. Schockaert, Steven; Janssen, Jeroen; Vermeir, Dirk (2012). "Satisfiability Checking in Łukasiewicz Logic as Finite Constraint Satisfaction". Journal
Finite-valued_logic
Hardware description and hardware verification language
it will require to do so as this is in general an NP-hard problem (boolean satisfiability). In each SystemVerilog class there are 3 predefined methods
SystemVerilog
IEEE Transactions. C (21): 1197–1206. Church, A. (1936). "An unsolvable problem of elementary number theory (first presented on 19 April 1935 to the American
Timeline of artificial intelligence
Timeline_of_artificial_intelligence
Technique used in mathematical logic
sets; They are both unbounded, in other words neither A nor B has either a maximum or a minimum; They are densely ordered, i.e. between any two members there
Back-and-forth_method
Field in logic and theoretical computer science
of as a nondeterministic polynomial-time algorithm for proving non-satisfiability. That is, given a formula ϕ {\displaystyle \phi } , ϕ {\displaystyle
Proof_complexity
Function returning one of only two values
the correlation of that bit with the output of the Boolean function. The maximum (in absolute value) Walsh coefficient is known as the linearity of the
Boolean_function
Mathematical use of "for all" and "there exists"
immediately a problem, since syntax rules are expected to generate finite statements. A succinct equivalent formulation, which avoids these problems, uses universal
Quantifier_(logic)
Function computable with bounded loops
Proper subtraction a ∸ b: If a ≥ b then a−b else 0 Minimum(a1, ... an) Maximum(a1, ... an) Absolute difference: | a−b | =def (a ∸ b) + (b ∸ a) ~sg(a):
Primitive_recursive_function
Population models of evolutionary algorithms
"Combining cellular genetic algorithms and local search for solving satisfiability problems", Proceedings Tenth IEEE International Conference on Tools with
Population model (evolutionary algorithm)
Population_model_(evolutionary_algorithm)
Concept in model theory
isolated types can never be omitted (see below). A model that realizes the maximum possible variety of types is called a saturated model, and the ultrapower
Type_(model_theory)
Axiomatic set theory devised by W.V.O. Quine
the set of nonempty sets of ordinals with α {\displaystyle \alpha } as maximum to cardinals such that If | A | > 1 , τ ( A 1 ) = 2 τ ( A ) {\displaystyle
New_Foundations
MAXIMUM SATISFIABILITY-PROBLEM
MAXIMUM SATISFIABILITY-PROBLEM
Boy/Male
Arabic, French, Muslim
Lucky
Male
Italian
Italian form of Latin Maximus, MASSIMO means "the greatest."
Boy/Male
Arabic
Trusting
Boy/Male
Latin
Greatest.
Boy/Male
American, Australian, French, Latin
Greatest
Male
French
French form of Latin Maximus, MAXIME means "the greatest."Â
Girl/Female
Arabic, Muslim
Increase; Excess; High Degree; Maximum; Feminine of Mazid
Boy/Male
African, Arabic
Far
Boy/Male
Russian American
The greatest.
Boy/Male
Bengali, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Sanskrit
Plenty; Maximum; Intelligent; Young and Dynamic; Earth
Boy/Male
Latin
Greatest.
Boy/Male
Indian
Auspicious, Prosperous
Boy/Male
Italian American
The greatest.
Boy/Male
Muslim
Auspicious, Prosperous
Girl/Female
Latin
The best.
Boy/Male
Latin French
Greatest.
Boy/Male
American, Australian, Chinese, French, German, Greek, Latin, Swedish
Greatest
Male
Spanish
Spanish form of Latin Maximus, MÃXIMO means "the greatest."
Male
Russian
(МакÑим) Variant spelling of Russian Maksim, MAXIM means "the greatest." Compare with another form of Maxim.
Boy/Male
American, Australian, Chinese, Danish, French, German, Latin, Swedish
The Greatest; Form of Maximilian; Great; The Greatest Rival
MAXIMUM SATISFIABILITY-PROBLEM
MAXIMUM SATISFIABILITY-PROBLEM
Boy/Male
American, Australian, Jamaican
Noble; Patrician
Boy/Male
Tamil
Perfect
Girl/Female
Hindu
Wisdom, One with good morals, Good guidance, Righteous
Boy/Male
American, Australian, British, English
From the Ancient Oak Tree
Boy/Male
Hindu
One of the kauravas
Male
Japanese
(æ£ç”·) Japanese name MASAO means "correct man."
Girl/Female
Irish
Dark beauty.
Girl/Female
Teutonic German
noble.
Girl/Female
American, German, Greek
Pearl; Power
Male
Croatian
, golden.
MAXIMUM SATISFIABILITY-PROBLEM
MAXIMUM SATISFIABILITY-PROBLEM
MAXIMUM SATISFIABILITY-PROBLEM
MAXIMUM SATISFIABILITY-PROBLEM
MAXIMUM SATISFIABILITY-PROBLEM
n.
A sewer; as, the Cloaca Maxima of Rome.
n.
The least quantity assignable, admissible, or possible, in a given case; hence, a thing of small consequence; -- opposed to maximum.
n.
A self-registering thermometer, especially one that registers the maximum and minimum during long periods.
n.
The longest note formerly used, equal to two longs, or four breves; a large.
n.
A proposition; a maxim.
n.
A coarse umbelliferous plant of Europe (Tordylium maximum).
n.
Minimum.
n.
A brief reflection or maxim.
pl.
of Minimum
n.
The opinions and maxims of the Stoics.
n.
The greatest quantity or value attainable in a given case; or, the greatest value attained by a quantity which first increases and then begins to decrease; the highest point or degree; -- opposed to minimum.
n.
A popular maxim, adage, or proverb.
n.
Fundamental principle; axiom; maxim.
a.
Greatest in quantity or highest in degree attainable or attained; as, a maximum consumption of fuel; maximum pressure; maximum heat.
n.
An elementary principle or maximum; a short, proverbial rule, in law, ethics, or metaphysics.
a.
Sententious; uttering or containing maxims, or striking detached thoughts; aphoristic.
v. t.
A saying; a proverb; a maxim.
n.
In a curve referred to polar coordinates, any point for which the radius vector is a maximum or minimum.
pl.
of Maximum
n.
An established principle or proposition; a condensed proposition of important practical truth; an axiom of practical wisdom; an adage; a proverb; an aphorism.