Packing $K_r$s in bounded degree graphs

TL;DR

This work provides a complete complexity classification for fixed-$r$ packing problems in bounded-degree graphs, distinguishing vertex-disjoint and edge-disjoint variants. The authors introduce thresholds on the maximum degree $oldsymbol{ abla}$, showing linear-time solvability when $oldsymbol{ abla} < 3oldsymbol{r}/2 - 1$, polynomial-time solvability when $oldsymbol{ abla} < 5oldsymbol{r}/3 - 1$, and APX-hardness when $oldsymbol{ abla} \\ge \\lceil 5oldsymbol{r}/3 ceil - 1$, with the same regimes extending to EDKr for $oldsymbol{r} \\ge 6$ and adapting to $oldsymbol{r} \\le 5$ via an additional bound $oldsymbol{ abla} \\le 2oldsymbol{r}-2$. The core methodology leverages intersection graphs $oldsymbol{ his$K_r^G}$ and $oldsymbol{ his$K'_r^G}$ and, when claw-free, polynomial-time maximum independent-set algorithms to obtain packing solutions; APX-hardness is established via L-reductions from MIS-3-TF and Max 2-SAT$_{\le 3}$. These results yield a tight complexity landscape for fixed $r$ and advance the understanding of clique packing in bounded-degree graphs, with potential extensions to weighted variants and parameterized analyses.

Abstract

We study the problem of finding a maximum-cardinality set of $r$-cliques in an undirected graph of fixed maximum degree $Δ$, subject to the cliques in that set being either vertex-disjoint or edge-disjoint. It is known for $r=3$ that the vertex-disjoint (edge-disjoint) problem is solvable in linear time if $Δ=3$ ($Δ=4$) but APX-hard if $Δ\geq 4$ ($Δ\geq 5$). We generalise these results to an arbitrary but fixed $r \geq 3$, and provide a complete complexity classification for both the vertex- and edge-disjoint variants in graphs of maximum degree $Δ$. Specifically, we show that the vertex-disjoint problem is solvable in linear time if $Δ< 3r/2 - 1$, solvable in polynomial time if $Δ< 5r/3 - 1$, and APX-hard if $Δ\geq \lceil 5r/3 \rceil - 1$. We also show that if $r\geq 6$ then the above implications also hold for the edge-disjoint problem. If $r \leq 5$, then the edge-disjoint problem is solvable in linear time if $Δ< 3r/2 - 1$, solvable in polynomial time if $Δ\leq 2r - 2$, and APX-hard if $Δ> 2r - 2$.

Figures (2)

• Figure 1: The reduction from Max $2\text{SAT}_{\leq 3}$ to EDK4
• Figure 2: The reduction from Max $2\text{SAT}_{\leq 3}$ to EDK5

Theorems & Definitions (53)

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