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Graph homomorphism problem [3]: GT52 Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph. A related problem is to find a partition that is optimal terms ...
The Subgraph Isomorphism problem is NP-complete. The graph isomorphism problem is suspected to be neither in P nor NP-complete, though it is in NP. This is an example of a problem that is thought to be hard, but is not thought to be NP-complete. This class is called NP-Intermediate problems and exists if and only if P≠NP.
The graph sandwich problem is NP-complete when Π is the property of being a chordal graph, comparability graph, permutation graph, chordal bipartite graph, or chain graph. [2] [4] It can be solved in polynomial time for split graphs, [2] [5] threshold graphs, [2] [5] and graphs in which every five vertices contain at most one four-vertex ...
The graph shows the running time vs. problem size for a knapsack problem of a state-of-the-art, specialized algorithm. The quadratic fit suggests that the algorithmic complexity of the problem is O((log(n)) 2). [24] All of the above discussion has assumed that P means "easy" and "not in P" means "difficult", an assumption known as Cobham's ...
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 special case of finding a long path as an induced subgraph of a hypercube has been particularly well-studied, and is called the snake-in-the-box problem. [3] The maximum independent set problem is also an induced subgraph isomorphism problem in which one seeks to find a large independent set as an induced subgraph of a larger graph, and the maximum clique problem is an induced subgraph ...
For example, the minimum spanning tree of the graph associated with an instance of the Euclidean TSP is a Euclidean minimum spanning tree, and so can be computed in expected O(n log n) time for n points (considerably less than the number of edges). This enables the simple 2-approximation algorithm for TSP with triangle inequality above to ...
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges.