A $K_r$-factor of a graph $G$ is a collection of vertex disjoint $r$-cliques covering $V(G)$. We prove the following algorithmic version of the classical Hajnal--Szemer\'edi Theorem in graph theory, when $r$ is considered as a constant. Given $r, c, n\in \mathbb{N}$ such that $n\in r\mathbb N$, let $G$ be an $n$-vertex graph with minimum degree at least $(1-1/r)n - c$. Then there is an algorithm with running time $2^{c^{O(1)}} n^{O(1)}$ that outputs either a $K_r$-factor of $G$ or a certificate showing that none exists, namely, this problem is fixed-parameter tractable in $c$. On the other hand, it is known that if $c = n^{\varepsilon}$ for fixed $\varepsilon \in (0,1)$, the problem is \texttt{NP-C}. We indeed establish characterization theorems for this problem, showing that the existence of a $K_r$-factor is equivalent to the existence of certain class of $K_r$-tilings of size $o(n)$, whose existence can be searched by the color-coding technique developed by Alon--Yuster--Zwick.

翻译：图 $G$ 的 $K_r$-因子是指覆盖 $V(G)$ 的一组顶点互不相交的 $r$-团。当 $r$ 被视为常数时，我们证明了图论中经典 Hajnal--Szemerédi 定理的如下算法版本。给定满足 $n\in r\mathbb N$ 的 $r, c, n\in \mathbb{N}$，令 $G$ 为最小度至少为 $(1-1/r)n - c$ 的 $n$ 顶点图。则存在一个运行时间为 $2^{c^{O(1)}} n^{O(1)}$ 的算法，该算法要么输出 $G$ 的一个 $K_r$-因子，要么输出一个证明其不存在的证书，即该问题在参数 $c$ 上是固定参数可解的。另一方面，已知若 $c = n^{\varepsilon}$ 且 $\varepsilon \in (0,1)$ 固定，则该问题是 \texttt{NP-C} 的。我们确实建立了该问题的刻画定理，表明 $K_r$-因子的存在性等价于存在某类规模为 $o(n)$ 的 $K_r$-平铺，其存在性可通过 Alon--Yuster--Zwick 所发展的颜色编码技术进行搜索。