Difference between NP hard and NP complete problem Last Updated : 25 Feb, 2025 Comments Improve Suggest changes Like Article Like Report All NP Complete Problems are NP-Hard but vice versa is not true. NP-Complete problems are subset of NP Problems. NP Problems : NP problems are a class of computational problems that can be solved in polynomial time by a non-deterministic machine and can be verified in polynomial time by a deterministic Machine (our CPUs are deterministic machines). As we can see in the below diagram, NP set also includes P problems (Problems that solvable in polynomial time by a deterministic machine). NP Complete Problems : A decision problem is NP-complete if it is in NP (Any solution to NP-complete problems can be checked quickly, but no efficient solution is known) and every problem in NP is reducible to it in polynomial timeNP Hard : A problem is NP-Hard if it obeys Property 2 (all NP problems can be reduced to it) of NP Complete and need not obey Property 1 (It is in NP).Therefore, a problem is NP-complete if it is both NP and NP-hard. Difference between NP-Hard and NP-Complete: NP-hardNP-CompletePolynomial time verification by a deterministic machine is not necessary.Can be verified in Polynomial time by a deterministic machine. NP-hard is not a decision problem.NP-Complete is exclusively a decision problem.Not all NP-hard problems are NP-complete.All NP-complete problems are NP-hardDo not have to be a Decision problem.It is exclusively a Decision problem.Need not to be a NP problemMust be NPExample: Halting problem, Vertex cover problem, etc.Example: Determine whether a graph has a Hamiltonian cycle, Determine whether a Boolean formula is satisfiable or not, Circuit-satisfiability problem and problems that are NP hard. Comment More infoAdvertise with us Next Article Proof that Clique Decision problem is NP-Complete S sugandha18bcs3001 Follow Improve Article Tags : DSA NPHard Algorithms-NP Complete Similar Reads P, NP, CoNP, NP hard and NP complete | Complexity Classes In computer science, problems are divided into classes known as Complexity Classes. In complexity theory, a Complexity Class is a set of problems with related complexity. With the help of complexity theory, we try to cover the following.Problems that cannot be solved by computers.Problems that can b 5 min read Introduction to NP-Complete Complexity Classes NP-complete problems are a subset of the larger class of NP (nondeterministic polynomial time) problems. NP problems are a class of computational problems that can be solved in polynomial time by a non-deterministic machine and can be verified in polynomial time by a deterministic Machine. A problem 5 min read NP-Hard Class A 'P' problem is said to be NP-Hard when all 'Q' belonging in NP can be reduced in polynomial time (n^k where k is some constant) to 'P' assuming a solution for 'P' takes 1 unit time. NP-Hard is a computational complexity theory that acts as a defining property for the class of problems that are "at 2 min read Difference between NP hard and NP complete problem All NP Complete Problems are NP-Hard but vice versa is not true. NP-Complete problems are subset of NP Problems. NP Problems : NP problems are a class of computational problems that can be solved in polynomial time by a non-deterministic machine and can be verified in polynomial time by a determinis 2 min read NP-Complete Complexity ProofsProof that Clique Decision problem is NP-Complete Prerequisite: NP-Completeness A clique is a subgraph of a graph such that all the vertices in this subgraph are connected with each other that is the subgraph is a complete graph. The Maximal Clique Problem is to find the maximum sized clique of a given graph G, that is a complete graph which is a s 4 min read Proof that Independent Set in Graph theory is NP Complete Prerequisite: NP-Completeness, Independent set. An Independent Set S of graph G = (V, E) is a set of vertices such that no two vertices in S are adjacent to each other. It consists of non- adjacent vertices. Problem: Given a graph G(V, E) and an integer k, the problem is to determine if the graph co 5 min read Prove that a problem consisting of Clique and Independent Set is NP Complete Prerequisite: NP-Completeness, NP Class, Clique, Independent Set Problem: Given an undirected graph G = (V, E) and an integer K, determine if a clique of size K as well as an independent set (IS) of size K, exists. Demonstrate that it is an NP Complete. Explanation: A Clique is a subgraph of a graph 6 min read Prove that Dense Subgraph is NP Complete by Generalisation Prerequisites: NP-Completeness, NP Class, Dense Subgraph Problem: Given graph G = (V, E) and two integers a and b. A set of a number of vertices of G such that there are at least b edges between them is known as the Dense Subgraph of graph G. Explanation: To prove the Dense Subgraph problem as NP-c 3 min read Prove that Sparse Graph is NP-Complete Prerequisite: NP-Completeness, NP Class, Sparse Graph, Independent Set Problem: Given graph G = (V, E) and two integers a and b. A set of a number of vertices of G such that there are at most b edges between them is known as the Sparse Subgraph of graph G. Explanation: Sparse Subgraph problem is def 4 min read Like