Question

Suppose that problem $A$ and problem $B$ are two different decision problems. Furthermore, assume that problem $A$ is polynomial-time many-one reducible to problem $B$, If problem $A$ is $NP$ -complete, is problem $B$ NP-complete? Justify your answer.

Short answer

Without additional information, it cannot be definitively stated that problem $B$ is NP-complete just because problem $A$ is NP-complete and $A$ is polynomial-time many-one reducible to $B$. Out of the characteristics of NP-complete problems, it is not given that all problems in NP are reducible to $B$. Thus, $B$ might not be NP-complete.

Step-by-step solution

Understand the Problem Context

Firstly, understand what the problem is asking for. It is known that problem $A$ is NP-complete and it is polynomial-time reducible to problem $B$, meaning every instance of $A$ can be transformed into an instance of $B$ in polynomial time. The question seeks to determine if $B$ is also NP-complete. NP-complete problems are the most complex problems in NP, if any NP-complete problems can be solved in polynomial time, then every problem in NP can be.

Apply the Definition of Reducibility in Complexity Theory

In complexity theory, a problem $A$ is reducible to a problem $B$ if any instance of problem $A$ can be transformed into an instance of problem $B$ in polynomial time. If $A$ is reducible to $B$, and $A$ is NP-complete, then $B$ must be at least as hard as $A$ because if $B$ had an easier or more efficient solution, it would mean that $A$ would also have a more efficient solution.

Determine If Problem B Is NP-Complete

While it is true that $B$ is at least as hard as $A$, that does not necessarily mean $B$ is NP-complete. An NP-complete problem must satisfy two conditions: it must be in NP and all problems in NP must be polynomial time reducible to it. While $B$ might satisfy the first condition because problem $A$ is reducible to it and $A$ is in NP. It is not given that all problems in NP are reducible to $B$. Without this information, it cannot be definitively stated that $B$ is NP-complete.

Key concepts

Polynomial-Time Reducibility

When exploring the landscape of computational problems, it's essential to understand the concept of polynomial-time reducibility. Imagine you have two puzzles, A and B. Now, suppose that every time you solve puzzle A, there's a method to quickly (in polynomial time) transform your solution into a solution for puzzle B. This transformation means that puzzle A is polynomial-time reducible to puzzle B.

In formal terms, if you can take any instance of problem A, and within a number of steps that rise at most polynomially with the size of the input, convert it to an instance of problem B, then A is reducible to B. This relationship is a cornerstone of complexity theory because it helps us understand the relative difficulty of problems. If A is a tough nut to crack and is reducible to B, then B can't be any easier. That's the gist of polynomial-time reducibility. It's a tool for grouping together tough problems under a common roof.

Complexity Theory

At the heart of computational math and computer science lies complexity theory, a field dedicated to classifying and comparing problems based on how difficult they are to solve. This difficulty is often measured by how the time to solve a problem increases with the problem size, known as the time complexity of a problem.

Breaking it down, problems are usually sorted into classes, like P, which contains all the problems solvable in polynomial time—think of fast, efficient solutions. Then there's the famous NP class, which encompasses problems where verifying a given solution can be done quickly, even if finding the solution might be slow or hard. The celebrities of NP are the NP-complete problems: the hardest nuts to crack in NP. If one of them were solved swiftly (in polynomial time), all problems in NP could be solved just as fast. This concept has massive implications across computer science because it means a breakthrough for one problem could revolutionize solutions for countless others.

Decision Problems

Decision problems are the bread and butter of theoretical computer science. Simply put, they are questions that result in a yes or no answer. Classic examples include questions like 'Does this graph have a path that visits each node exactly once?' or 'Is there a combination of items that fits into this knapsack without exceeding its weight limit?'

These problems are crucial because they lay the groundwork for defining the complexity classes like NP, which is made up mainly of decision problems. NP-complete problems, a subset of the broader NP class, are the toughest of decision problems and act as a benchmark. They are the problems to which all other NP problems can be reduced in polynomial time. Understanding decision problems is the first step towards grasping the rest of complexity theory. They're essentially the atoms out of which the molecules of more complex computational problems are constructed.