NP-Hard Class

A ‘P‘ problem is said to be NP-Hard when all ‘Q’ belonging in NP can be reduced in polynomial time ( 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 least as hard as the hardest problems in NP”. P’s solution can be used to solve ‘Q’ in polynomial time. For example, the Subset Sum Problem, and Turing Halting problem is an NP-Hard problem.

Features of NP-Hard:

• All NP-hard problems are not in NP.
• It takes a long time to check them. This means if a solution for an NP-hard problem is given, it takes a long time to check whether it is right.
• A problem ‘P’ is NP-hard if there is a polynomial-time reduction from ‘Q’ to ‘P’ for every problem ‘Q’ in NP.

Real-World Applications:

• Approximate Computing: In approximate computing, the goal is to trade off accuracy for computational efficiency. Many optimization problems in this domain, like optimizing approximation algorithms, can be NP-hard. This implies that finding optimal solutions might be impractical, and researchers need to devise heuristic or approximation methods to get close-to-optimal results.
• Cryptography: While most cryptographic algorithms involve polynomial-time operations, certain cryptographic problems, such as solving certain lattice-based problems in post-quantum cryptography, are believed to be NP-hard. This is crucial for ensuring the security of cryptographic systems.
• Data Mining: NP-hard problems often arise in data mining tasks like clustering, feature selection, and association rule mining. As data mining involves finding patterns and relationships in large datasets, it’s common to encounter optimization problems that are difficult to solve exactly.
• Routing/Vehicle Routing: Routing problems, like finding the shortest path or optimizing routes for vehicles, are often NP-hard. These problems arise in logistics, transportation, and networking. Practical solutions involve approximation algorithms and heuristics due to the complexity of real-world scenarios.

Problems under NP-Hard Class:

• Traveling Salesman Problem (TSP)
• Subset Sum problem
• Knapsack Problem
• Boolean Satisfiability Problem (SAT)
• Graph Coloring Problem
• Independent Set Problem
• 3-SAT Problem
• Vertex Cover Problem
• Bin Packing Problem
• Hamiltonian Cycle Problem
• Partition Problem
• Knights Tour Problem
• Quadratic Assignment Problem
• Clique Problem