Are these two definitions related to strong NP-hardness equivalent?

Question

Let $P$ be a computational problem whose inputs are integers. Consider the following properties:

(a) There exists a polynomial-time reduction from some strongly-NP-hard problem $Q$ to $P$.

(b) $P$ is NP-hard even when restricted to instances in which the magnitude of the input integers is polynomial in the number of inputs.

Both properties can be seen as plausible definitions of "$P$ is strongly NP-hard". But I could not decide whether or not they are equivalent.

I think (b) is equivalent to: "there exists a polynomial-time reduction from some NP-hard problem $R$ to $P$, where the constructed inputs for $P$ contain only integers with magnitude polynomial in the number of inputs". But I could not decide whether this reduction is equivalent to the reduction in (a).

Are properties (a) and (b) equivalent?

Answer 1

They are not quite equivalent.

Note that every NP-hard problem (also in the weak sense) has by definition a polynomial time reduction from all problems in NP, including from those that are strongly NP-complete.

Hence, Definition (a) would apply to all NP-hard problems, which is clearly not the case as you have the prototype counterexamples like the Knapsack problem.

To show strong NP-hardness you need a different type of reduction:

Let $P, Q \subseteq \Sigma^*$ be decision problems and $f\colon \Sigma^\ast \to \Sigma^\ast$ a function. We say $f$ is pseudo-polynomial reduction from $P$ to $Q$ ($f\colon P \leq_{pp} Q$) if

1. for all $x \in \Sigma^*$, $x \in P$ iff $f(x) \in Q$,
2. for all $x \in \Sigma^*$, $f(x)$ can be computed in time that is polynomially bounded in $|x|$ and the largest number encoded in $x$, call it $\langle x \rangle$,
3. there exists a polynomial $r$ such that for all $x \in \Sigma^*$, $$|x| \leq r(|f(x)|),$$
4. there exists a polynomial $s$ such that for all $x \in \Sigma^*$, $$\langle f(x) \rangle \leq s(\langle x \rangle, |x|).$$

The difference to polynomial reductions is that you have to make sure that the numbers in the output problem of your reduction do not get too large (4.). Condition (3.) seems a bit weird, it is hards to violate this one, you just have to make sure that $|f(x)|$ does not get too small.

With this type of reduction it is quite straight forward to show that for a strongly NP-hard $P$ and a reduction $P \leq_{pp} Q$, you have that $Q$ is strongly NP-hard (in the sense of your Definition (b)).

Reference:
[GJ79] Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-completeness, volume 174. WH Freeman and Company, New York, 1979.