Reductions: Lower Bound and Upper Bound

The question is from my complexity-theory course.

Explain the concept of polynomial reduction between problems and explain how, and under what circumstances, lower bound and upper bound problems can be determined.

My attempt:

a problem $A$ can be polynomial reduced to $B$ if given an instance $a:A$ an istance $b:B$ can be built such that $a$ is true and $b$ is true.

A decision problem $P_1$ is Karp-reducible to a problem $P_2$ ( $P_1 \leq_m P_2) %$ if and only if exists an algorithm $R$ such that it transforms every instance $x \in I_{P_1}$ if and only if $y \in > I_{P_2}$. $R$ is the Karp-reduction from $P_1$ to $P_2$

One particular case of Karp-reducibility is the polynomial Karp-reducibility:

$P_1$ is polynomial Karp-reducible to $P_2$ ( $P_1 \leq^p_m P_2$) if and only if $P_1$ is Karp-reducible to $P_2$ and the reduction $R$ is an algorithm that runs in polynomial time.

In most cases the complexity of a solution given a particular problem isn't known. For example when we have a huge gap between the lower bound and upper bound. A tipycal situation could be the case of problems when their best lower bound known is $\Omega(n\log n)$ or $\Omega(n^2)$, however the most efficient algorithm has $O(c^n)$ complexity, for $c>1$. In those case the above concepts could be important tools in order to find the complexity relation to different problems.

• You need to give a little more context and detail about your reasoning. As it is now, the question in impossible to answer. – Raphael Sep 3 '17 at 18:10
• @Raphael just added more context and my attempt to solve the question. – Nick Sep 3 '17 at 19:55
• Thanks for the edit! Hint: consider different directions of reduction. – Raphael Sep 3 '17 at 19:57
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