  In the \textsc{Maximum Degree Contraction} problem, input is a graph $G$ on
$n$ vertices, and integers $k, d$, and the objective is to check whether $G$
can be transformed into a graph of maximum degree at most $d$, using at most
$k$ edge contractions. A simple brute-force algorithm that checks all possible
sets of edges for a solution runs in time $n^{\mathcal{O}(k)}$. As our first
result, we prove that this algorithm is asymptotically optimal, upto constants
in the exponents, under Exponential Time Hypothesis (\ETH).
  Belmonte, Golovach, van't Hof, and Paulusma studied the problem in the realm
of Parameterized Complexity and proved, among other things, that it admits an
\FPT\ algorithm running in time $(d + k)^{2k} \cdot n^{\mathcal{O}(1)} =
2^{\mathcal{O}(k \log (k+d) )} \cdot n^{\mathcal{O}(1)}$, and remains \NP-hard
for every constant $d \ge 2$ (Acta Informatica $(2014)$). We present a
different \FPT\ algorithm that runs in time $2^{\mathcal{O}(dk)} \cdot
n^{\mathcal{O}(1)}$. In particular, our algorithm runs in time
$2^{\mathcal{O}(k)} \cdot n^{\mathcal{O}(1)}$, for every fixed $d$. In the same
article, the authors asked whether the problem admits a polynomial kernel, when
parameterized by $k + d$. We answer this question in the negative and prove
that it does not admit a polynomial compression unless $\NP \subseteq
\coNP/poly$.
