NPhard: What is the definition of NPhardness?
by Stephen M. Walker II, CoFounder / CEO
What is the definition of NPhardness?
In computational complexity theory, NPhardness (nondeterministic polynomialtime hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". NP stands for nondeterministic polynomial time, which is a class of problems for which a solution can be verified in polynomial time.
A problem H is considered NPhard when every problem L in NP can be reduced in polynomial time to H. This means that if a solution for H can be found in 1 unit time, H's solution can be used to solve L in polynomial time. If a polynomial time algorithm could be found to solve any NPhard problem, it would provide polynomial time algorithms for all the problems in NP. However, it is suspected that P (the class of problems that can be solved in polynomial time) is not equal to NP, making it unlikely that such an algorithm exists. It's important to note that NPhard problems do not have to be elements of NP; indeed, they may not even be decidable.
Examples of NPhard problems include the subset sum problem, the travelling salesman problem, and the knapsack problem. These problems are difficult to solve because they involve a large number of potential solutions that must be checked in order to find the best one. This can be a timeconsuming process, especially when the problem is large and complex.
The concept of NPhardness is closely related to NPcomplete problems. A problem is NPcomplete if it is in NP and all problems in NP can be reduced to it in polynomial time. All NPcomplete problems are NPhard, but not all NPhard problems are NPcomplete.
What are some examples of NPhard problems?
NPhard problems are a class of problems in computational complexity theory that are "at least as hard as the hardest problems in NP". Here are some examples of NPhard problems:

Travelling Salesman Problem — This problem involves finding the shortest possible route that allows a salesman to visit each city once and return to the origin city. It is NPhard because it requires checking all possible routes to find the shortest one.

Subset Sum Problem — Given a set of integers, the subset sum problem asks whether there is a nonempty subset whose sum is zero. This problem is NPhard because it involves checking all possible subsets of the given set.

Vertex Cover Problem — This problem involves finding the smallest set of vertices that cover all edges in a graph. It is NPhard because it requires checking all possible combinations of vertices.

Hamiltonian Cycle Problem — This problem involves finding a cycle in a given graph that visits each vertex exactly once and returns to the origin vertex. It is NPhard because it requires checking all possible cycles in the graph.

Knapsack Problem — Given a set of items, each with a weight and a value, the knapsack problem involves determining the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It is NPhard because it involves checking all possible combinations of items.

Clique Problem — This problem involves finding the largest clique (a subset of vertices that form a complete graph) in a given graph. It is NPhard because it requires checking all possible subsets of vertices.
Remember, while all NPcomplete problems are NPhard, not all NPhard problems are NPcomplete. NPcomplete problems are those that are both in NP and NPhard.
Why are NPhard problems difficult to solve?
NPhard problems are difficult to solve due to their inherent complexity. In computational complexity theory, a problem is considered NPhard if every problem in NP (nondeterministic polynomial time) can be reduced to it in polynomial time. This means that an NPhard problem is at least as hard as the hardest problems in NP.
The difficulty in solving NPhard problems arises from the fact that they require superpolynomial time to solve, and no known algorithms can solve them in polynomial time. This is significant because polynomial time algorithms are considered efficient, while superpolynomial and exponential time algorithms are considered inefficient due to their high computational cost for large inputs.
Moreover, if a polynomial time algorithm could be found to solve any NPhard problem, it would imply that there are polynomial time algorithms for all problems in NP, effectively proving that P=NP. However, it is widely believed in the computer science community that P≠NP, meaning that there are problems in NP that cannot be solved in polynomial time.
Examples of NPhard problems include the traveling salesman problem and the subset sum problem. These problems are notoriously difficult to solve, and even the best algorithms can take a very long time to find a solution.
It's important to note that while all NPcomplete problems are NPhard, not all NPhard problems are NPcomplete. NPcomplete problems are a subset of NPhard problems that are both in NP and NPhard. This means that they can be verified in polynomial time, but it's unknown whether they can be solved in polynomial time.
The difficulty in solving NPhard problems lies in their computational complexity and the lack of efficient (polynomial time) algorithms to solve them. This makes them a significant area of study in computer science, particularly in fields like optimization and artificial intelligence.
What are some methods for solving NPhard problems?
NPhard problems are a class of problems that are "at least as hard as the hardest problems in NP". While it is suspected that there are no polynomialtime algorithms for NPhard problems, this has not been proven. However, there are several methods to approach solving these problems:

Approximation Algorithms — These are algorithms that find nearoptimal solutions with a guarantee on the proximity to the optimal solution. They are particularly useful when exact solutions are not feasible due to time or resource constraints.

Heuristics — These are ruleofthumb strategies or methods that help in making the problemsolving process quicker or more manageable. They do not guarantee the best solution but often produce good solutions within a reasonable time frame. Heuristics are particularly useful when dealing with very complex problems where finding the optimal solution is too resourceintensive.

PseudopolynomialTime Algorithms — These are algorithms that have a running time that is polynomial in the numeric value of the input, but not polynomial in the length of the input (which is the case for polynomialtime algorithms). They are not truly efficient, but they can be practical for small inputs.

Randomized Algorithms — These are algorithms that employ a degree of randomness as part of their logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits.

Exact Algorithms — These are algorithms that always produce the optimal solution, but they may require nonpolynomial time. Even a faster exact algorithm that does not run in polynomial time can mean that in practice much larger instances can be solved.

Parameterized Algorithms — These are algorithms that depend on certain parameters of the input. They are designed to solve problems more efficiently as the parameter value is small, even if the size of the problem (input size) is large.
It's important to note that the choice of method depends on the specific problem and the resources available. In many realworld applications, it's often more practical to find a "good enough" solution rather than the absolute optimal solution, especially when dealing with large, complex problems.
Are there any NPhard problems that have been solved?
No, there are no NPhard problems that have been definitively solved. NPhard problems are a set of problems to which all problems in NP can be reduced to in polynomial time through a deterministic Turing machine. In essence, if there exists a polynomial time solution to an NPhard problem, then there would exist a polynomial time solution for all problems in NP. This would mean P = NP. As of now, the P vs NP question is one of the seven "Millennium Prize Problems" for which the Clay Mathematics Institute offers a $1 million prize for a correct solution. No one has yet proved whether P = NP or P ≠ NP.
The term "solved" in the context of NPhard problems can be interpreted in two ways: finding an exact solution or finding an approximate solution.
For the first interpretation, it's important to note that NPhard problems are a class of problems that are at least as hard as the hardest problems in NP (nondeterministic polynomialtime). This means that if a polynomialtime algorithm could be found to solve any NPhard problem, it would imply that P=NP, which is a major unsolved question in computer science. As of now, no such algorithm has been found, and it is widely believed that P≠NP, meaning that no polynomialtime algorithm exists for NPhard problems.
However, specific instances of NPhard problems can sometimes be solved exactly. For instance, the partitioning problem, which is NPhard, has a pseudopolynomial time solution using dynamic programming. Additionally, there are heuristics that solve the problem in many cases, sometimes optimally. Another example is the Boolean satisfiability problem, which is NPhard in general, but there are instances called Horn clauses which can be solved in linear time.
For the second interpretation, there are indeed many NPhard problems for which approximate solutions can be found efficiently. These solutions may not be optimal, but they can be "good enough" for practical purposes. For example, the traveling salesman problem is NPhard, but there are heuristic algorithms that can find reasonably good solutions for large instances of the problem.
While no general method exists for solving all NPhard problems exactly in polynomial time, specific instances of these problems can sometimes be solved exactly, and approximate solutions can often be found for practical purposes.
What is the difference between nphard and npincomplete?
In computational complexity theory, NPhard and NPcomplete are two classes of problems that help us understand the difficulty of solving certain problems.
An NPhard problem is a problem that is "at least as hard as the hardest problems in NP". A problem H is considered NPhard if every problem L in NP can be reduced in polynomial time to H. This means that if we can solve H, we can use its solution to solve L in polynomial time. It's important to note that NPhard problems do not have to be in NP, and they may not even be decidable. NPhard problems are not necessarily decision problems, which are problems with a yes or no answer.
On the other hand, an NPcomplete problem is a problem that is both in NP and NPhard. This means that an NPcomplete problem is a decision problem that is as hard as the hardest problems in NP, and a solution to it can be verified in polynomial time. If a polynomial time algorithm could be found to solve any NPcomplete problem, it would mean that P (the class of problems that can be solved in polynomial time) is equal to NP, which is a major open question in computer science.
The main differences between NPhard and NPcomplete problems are:
 NPhard problems are at least as hard as the hardest problems in NP, while NPcomplete problems are the hardest problems in NP.
 NPhard problems do not have to be in NP, while NPcomplete problems must be in NP.
 NPhard problems are not necessarily decision problems, while NPcomplete problems are always decision problems.