An algorithmic version of the Hajnal--Szemer\'edi theorem

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.
Comment: 31 pages