  The concept of $k$-defective clique, a relaxation of clique by allowing up-to
$k$ missing edges, has been receiving increasing interests recently. Although
the problem of finding the maximum $k$-defective clique is NP-hard, several
practical algorithms have been recently proposed in the literature, with kDC
being the state of the art. kDC not only runs the fastest in practice, but also
achieves the best time complexity. Specifically, it runs in $O^*(\gamma_k^n)$
time when ignoring polynomial factors; here, $\gamma_k$ is a constant that is
smaller than two and only depends on $k$, and $n$ is the number of vertices in
the input graph $G$. In this paper, we propose the kDC-Two algorithm to improve
the time complexity as well as practical performance. kDC-Two runs in $O^*(
(\alpha\Delta)^{k+2} \gamma_{k-1}^\alpha)$ time when the maximum $k$-defective
clique size $\omega_k(G)$ is at least $k+2$, and in $O^*(\gamma_{k-1}^n)$ time
otherwise, where $\alpha$ and $\Delta$ are the degeneracy and maximum degree of
$G$, respectively. In addition, with slight modification, kDC-Two also runs in
$O^*( (\alpha\Delta)^{k+2} (k+1)^{\alpha+k+1-\omega_k(G)})$ time by using the
degeneracy gap $\alpha+k+1-\omega_k(G)$ parameterization; this is better than
$O^*( (\alpha\Delta)^{k+2}\gamma_{k-1}^\alpha)$ when $\omega_k(G)$ is close to
the degeneracy-based upper bound $\alpha+k+1$. Finally, to further improve the
practical performance, we propose a new degree-sequence-based reduction rule
that can be efficiently applied, and theoretically demonstrate its
effectiveness compared with those proposed in the literature. Extensive
empirical studies on three benchmark graph collections show that our algorithm
outperforms the existing fastest algorithm by several orders of magnitude.
