An algorithmic version of the Hajnal--Szemerédi theorem

TL;DR

The paper tackles the decision/search problem for the existence of a $K_r$-factor in $n$-vertex graphs under near-Hajnal--Szemerédi degree conditions, proving an algorithmic, fixed-parameter tractable result parameterized by $c$ for constant $r$. It develops a two-pronged approach: in the non-extremal regime, the absorbing method plus an almost-covering argument yields a $K_r$-factor, while in the extremal regime it uses a detailed partitioning, slack, and color-coding based tiling characterizations to either construct a factor or certify nonexistence. Central to the method are the absorbing lemma (constructing a small absorbing set) and the characterization lemmas connecting small $K_r$-tilings to full tilings via admissible tilings, with the color-coding technique enabling efficient search for these tilings. The results establish an exact algorithmic threshold between the fixed-parameter tractable and NP-hard regimes: for constant $c$, the problem is FPT in $c$; for $c$ as a polynomial in $n$, the problem is NP-hard. Overall, the work provides a robust framework combining regularity, hypergraph tilings, and algorithmic derandomization to extend Hajnal–Szemerédi-style guarantees into effective, tractable algorithms with certificates.

Abstract

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--Szemerédi 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.

Theorems & Definitions (68)

• Theorem 1.1
• Theorem 1.3
• Corollary 1.4
• Proposition 1.5
• Lemma 1.6
• Lemma 1.7: Slack Lemma
• Definition 2.1: Sparse set
• Theorem 2.2: Non-extremal case
• Definition 2.3: Absorber
• Definition 2.4: Absorbing set