Learning Small Decision Trees with Few Outliers: A Parameterized Perspective
This paper studies learning small decision trees tolerant to at most $t$ misclassifications, focusing on two core variants: minimizing tree size (DTSO) and minimizing tree depth (DTDO). Within the parameterized complexity framework, we establish for the first time that both problems are W[1]-hard with respect to $s+y$ and $d+y$, where $y$ is the number of attributes. However, when parameterized by the misclassification tolerance $t$, both become fixed-parameter tractable (FPT), and we present the first FPT algorithm explicitly depending on $t$. We systematically characterize kernelization complexity, providing tight polynomial kernel existence and impossibility results, thereby completing the kernelization classification for DTSO and DTDO. Our main contributions are: (i) establishing precise computational complexity boundaries; (ii) revealing the “complexity-reducing” role of $t$, which shifts hardness from W[1]-hardness to FPT; and (iii) delivering theoretically complete algorithms and matching lower bounds.