We study the problem of finding a maximum-cardinality set of $r$-cliques in an undirected graph of fixed maximum degree $\Delta$, 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 $\Delta=3$ ($\Delta=4$) but APX-hard if $\Delta \geq 4$ ($\Delta \geq 5$). 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 $\Delta$. Specifically, we show that the vertex-disjoint problem is solvable in linear time if $\Delta < 3r/2 - 1$, solvable in polynomial time if $\Delta < 5r/3 - 1$, and APX-hard if $\Delta \geq \lceil 5r/3 \rceil - 1$. 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 $\Delta < 3r/2 - 1$, solvable in polynomial time if $\Delta \leq 2r - 2$, and APX-hard if $\Delta > 2r - 2$.
翻译:我们研究在固定最大度$$\Delta$\Delta=4$的非方向图中找到最高心心力的集合值为$2 美元(Delta=4美元)的问题, 但问题在于该图中的螺旋值要么是顶端分解, 要么是边缘分解, 要么是3美元。 以 $=3 =Delta=3$(Delta=4美元), 以 APX- hard $2 elta\ge $4$(Delta\geq 5美元) 。 我们将这些结果概括为任意但固定的 美元(Geqqq 3美元) 。 以 $=3r=3美元为著称, 顶端分解问题在线性时间上是可溶解的。 具体地说, 如果 $=xxl=xxxl=lx 问题, 我们显示, lex 3\ 3r/2 美元(Del-r=3美元) 问题在线上是可溶解的。