  Various classes of induced subgraphs are involved in the deepest results of
graph theory and graph algorithms. A prominent example concerns the {\em
perfection} of $G$ that the chromatic number of each induced subgraph $H$ of
$G$ equals the clique number of $H$. The seminal Strong Perfect Graph Theorem
confirms that the perfection of $G$ can be determined by detecting odd holes in
$G$ and its complement. Chudnovsky et al. show in 2005 an $O(n^9)$ algorithm
for recognizing perfect graphs, which can be implemented to run in
$O(n^{6+\omega})$ time for the exponent $\omega<2.373$ of square-matrix
multiplication. We show the following improved algorithms.
  1. The tractability of detecting odd holes was open for decades until the
major breakthrough of Chudnovsky et al. in 2020. Their $O(n^9)$ algorithm is
later implemented by Lai et al. to run in $O(n^8)$ time, leading to the best
formerly known algorithm for recognizing perfect graphs. Our first result is an
$O(n^7)$ algorithm for detecting odd holes, implying an $O(n^7)$ algorithm for
recognizing perfect graphs.
  2. Chudnovsky et al. extend in 2021 the $O(n^9)$ algorithms for detecting odd
holes (2020) and recognizing perfect graphs (2005) into the first polynomial
algorithm for obtaining a shortest odd hole, which runs in $O(n^{14})$ time. We
reduce the time for finding a shortest odd hole to $O(n^{13})$.
  3. Conforti et al. show in 1997 the first polynomial algorithm for detecting
even holes, running in about $O(n^{40})$ time. It then takes a line of
intensive efforts in the literature to bring down the complexity to
$O(n^{31})$, $O(n^{19})$, $O(n^{11})$, and finally $O(n^9)$. On the other hand,
the tractability of finding a shortest even hole has been open for 16 years
until the very recent $O(n^{31})$ algorithm of Cheong and Lu in 2022. We
improve the time of finding a shortest even hole to $O(n^{23})$.
