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In computer science, the problem of finding a minimum vertex cover is a 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 – it cannot be approximated up to a factor smaller than 2 if the unique games conjecture is true.
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", [1] Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete [2] (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction ...
The problem of deciding the satisfiability of a given conjunction of Horn clauses is called Horn-satisfiability, or HORN-SAT. It can be solved in polynomial time by a single step of the unit propagation algorithm, which produces the single minimal model of the set of Horn clauses (w.r.t. the set of literals assigned to TRUE).
Boolean satisfiability problem (SAT). [2] [3]: 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. [3]: p. 48 Circuit satisfiability problem; Conjunctive Boolean query [3]: SR31
There is often only a small difference 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-complete, whereas the slightly more restricted 2-satisfiability problem is in P (specifically, it is NL-complete), but the slightly more general max ...
These problems include graph k-colorability, finding Hamiltonian cycles, maximum cliques, maximum independent sets, and vertex cover on -vertex graphs. Conversely, if any of these problems has a subexponential algorithm, then the exponential time hypothesis could be shown to be false. [7] [6]
A simple example of an approximation algorithm is one for the minimum vertex cover problem, where the goal is to choose the smallest set of vertices such that every edge in the input graph contains at least one chosen vertex. One way to find a vertex cover is to repeat the following process: find an uncovered edge, add both its endpoints to the ...
The W2SAT problem includes as a special case the vertex cover problem, of finding a set of k vertices that together touch all the edges of a given undirected graph. For any given instance of the vertex cover problem, one can construct an equivalent W2SAT problem with a variable for each vertex of a graph.