What is NP Hardness?

  • Editor
  • December 29, 2023

What is NP Hardness? It is a cornerstone concept in computational theory and AI and refers to a classification of problems that are at least as hard as the hardest problems in NP (Non-deterministic Polynomial time).

These problems, known for their computational complexity, set the benchmark for identifying the limits of what can be efficiently computed.

Looking to learn more about NP hardness? Read this article written by the AI savants at All About AI.

Why Are NP-Hard Problems Challenging in AI?

The challenge of NP-Hard problems in AI lies in their intrinsic complexity. These problems don’t have known algorithms that can solve them quickly (in polynomial time), making them a significant hurdle in developing efficient AI systems.


Computational Complexity:

One of the primary reasons NP-Hard problems are challenging in AI is their inherent computational complexity.

These problems lack known polynomial-time algorithms for their solution, meaning as the size of the problem increases, the computational resources (like time and memory) required to solve them grow exponentially.

This makes them practically unsolvable for large instances, posing significant challenges in terms of efficiency and feasibility in AI applications.

Scalability Issues:

Scalability is another critical challenge posed by NP-Hard problems. AI systems often need to handle large datasets and complex computations.

NP-Hard problems become increasingly difficult as they scale up in size, creating a barrier in developing AI solutions that can efficiently process and analyze large volumes of data or complex scenarios.

Lack of Optimal Solutions:

Finally, NP-Hard problems are challenging because they often do not have a definitive method for finding the optimal solution. This uncertainty complicates the process of artificial intelligence development, where precision and accuracy are crucial.

AI algorithms must often rely on approximations or heuristic methods, which may not always guarantee the best possible outcome.

What are Common Examples of NP-Hard Problems?

Common examples of NP-Hard problems include the Traveling Salesman Problem (TSP), the Knapsack Problem, and the Boolean Satisfiability Problem (SAT).


Traveling Salesman Problem (TSP):

The Travelling Salesman Problem involves finding the shortest possible route that visits each city exactly once and returns to the original city. It’s a benchmark problem in optimization and computational complexity, with applications in logistics and route planning.

Subset Sum Problem:

The Subset Sum Problem entails determining if there is a subset of a given set of numbers that adds up to a specific sum. This problem is fundamental in cryptography and decision-making processes in AI.

Knapsack Problem:

In the Knapsack Problem, the goal is to select a number of items with given weights and values to maximize the total value without exceeding a weight limit. It’s widely used in resource allocation and inventory management in AI systems.

Clique Problem:

The Clique Problem involves finding a subset of vertices in a graph that are all connected to each other (a clique). This problem has applications in social network analysis and clustering in AI.

Vertex Cover Problem:

This problem involves finding the smallest set of vertices in a graph such that each edge is connected to at least one vertex in the set. It’s important in network design and analysis in AI.

How are NP-Hard Problems Approached in AI?

In AI, NP-Hard problems are approached using heuristic and approximation algorithms. These methods provide feasible solutions without necessarily guaranteeing an optimal solution.

Step 1: Problem Identification

The first step is to accurately identify and define the NP-Hard problem within the AI context. This involves understanding the problem’s parameters and constraints.

Step 2: Heuristic Methods

AI systems often employ heuristic methods to find good-enough solutions for NP-Hard problems. These methods include greedy algorithms, local search, and genetic algorithms, which can provide feasible solutions in a reasonable timeframe.

Step 3: Approximation Algorithms

Approximation algorithms are used where possible. These algorithms guarantee a solution close to the optimal, with a known bound on how far off the solution could be from the best possible one.

Step 4: Advanced Optimization Techniques

AI systems may use advanced optimization techniques like simulated annealing or deep learning models, especially when heuristic methods are insufficient. These techniques can handle larger, more complex instances of NP-Hard problems.

Step 5: Continuous Evaluation and Adaptation

Finally, AI systems continuously evaluate the performance of the applied methods and adapt as necessary. This could mean adjusting parameters, switching algorithms, or incorporating new research findings.

Are There NP-Hard Problems That Have Been Solved?

While NP-Hard problems are inherently difficult, some instances of these problems have been solved using advanced algorithms and computational techniques.


  • Graph Coloring Problem in Specific Cases: There are specific instances where the graph coloring problem has been solved. For example, bipartite graphs can always be colored with two colors. This solution is important in scheduling and network design.
  • TSP for Small Graphs: The Travelling Salesman Problem has been solved for relatively small graphs using sophisticated algorithms like dynamic programming and branch and bound. These solutions are critical in route planning and logistics optimization.
  • Knapsack Problem with Dynamic Programming: For certain types of the Knapsack Problem, dynamic programming approaches have provided exact solutions. This method is effective for problems with discrete, small-scale data sets.
  • Planar Graphs in the Vertex Cover Problem: The Vertex Cover Problem has been solved optimally for planar graphs, which are graphs that can be drawn on a plane without any edges crossing. Solutions in this area are useful in network topology and circuit design.

Want to Read More? Explore These AI Glossaries!

Step into the fascinating world of artificial intelligence with our detailed glossaries. From beginners to seasoned pros, there’s always something intriguing to learn!

  • What is Backpropagation Through Time?: Backpropagation through time is a variant of the standard backpropagation algorithm, tailored specifically for Recurrent Neural Networks (RNNs).
  • What is Backward Chaining?: Backward chaining is an inference method where an AI system starts with a goal or desired outcome and works backward through a series of rules and conditions to find the necessary steps or conditions to achieve that goal.
  • What is the Bag of Words Model?: It is a simplistic yet powerful approach in artificial intelligence, particularly in natural language processing (NLP).
  • What is Batch Normalization?: Batch normalization is an essential technique in artificial intelligence, particularly in neural network training. It involves standardizing the inputs of each layer within a network to have a mean of zero and a standard deviation of one.
  • What is the Bees Algorithm?: It is a nature-inspired computing technique, mirroring the food foraging behavior of honey bees.


In the context of the Traveling Salesman Problem (TSP), NP-Hard refers to the complexity of finding the shortest possible route that visits each city exactly once and returns to the origin city.

NP-Hard classification helps in understanding the computational complexity of problems. It aids in the development of algorithms and approaches to tackle complex problems in computer science and AI.

An NP-hard language in computational theory refers to a decision problem where the answer is a ‘yes’ or ‘no’. These problems are as hard as the hardest problems in NP.

While NP-Hard problems are among the hardest in computational theory, whether they are the absolute hardest is still an open question in computer science.

NP-Hard problems are hardest to solve due to their computational complexity and the lack of known algorithms that can solve them efficiently in polynomial time.

Final Words

Understanding NP-Hardness is crucial in AI, as it frames the boundaries of computational feasibility. While challenging, the ongoing research and development in this area continue to push the frontiers of AI, promising innovative solutions to some of the most complex problems.

This article comprehensively provided an answer to the question, “what is NP hardness,” describing it in the context of AI. If you’re looking to expand your knowledge of the ever-evolving world of AI, read through the rest of the articles in our AI Glossary.



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Dave Andre


Digital marketing enthusiast by day, nature wanderer by dusk. Dave Andre blends two decades of AI and SaaS expertise into impactful strategies for SMEs. His weekends? Lost in books on tech trends and rejuvenating on scenic trails.

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