Let $\mathcal{H}$ be a family of graphs. An $\mathcal{H}$-packing of a graph $G$ is a set $\{G_1,G_2,\dots,G_k\}$ of disjoint subgraphs of $G$ such that each $G_j$ is isomorphic to some element of $\mathcal{H}$. An $\mathcal{H}$-packing of a graph $G$ that covers the maximum number of vertices of $G$ is called a maximum $\mathcal{H}$-packing of $G$. The $\mathcal{H}$-packing problem seeks to find a maximum $\mathcal{H}$-packing of a graph. Let $i$ be a positive integer. An $i$-star is a complete bipartite graph $K_{1,i}$. This paper investigates the $\mathcal{H}$-packing problem with $\mathcal{H}$ being a family of stars. For an arbitrary family $\mathcal{S}$ of stars, we design a linear-time algorithm for the $\mathcal{S}$-packing problem in trees. Let $t$ be a positive integer. An $\mathcal{H}$-packing is called a $t^+$-star packing if $\mathcal{H}$ consists of all $i$-stars with $i\ge t$. We show that the $t^+$-star packing problem for $t\ge 2$ is NP-hard in bipartite graphs. As a consequence, the $2^+$-star packing problem is NP-hard even in bipartite graphs with maximum degree at most $4$. Let $T$ and $t$ be two positive integers with $T>t$. An $\mathcal{H}$-packing is called a $T\setminus t$-star packing if $\mathcal{H}=\{K_{1,1},K_{1,2},\dots,K_{1,T}\}\setminus \{K_{1,t}\}$. For $t\ge 2$, we present a $\frac{t}{t+1}$-approximation algorithm for the $T\setminus t$-star packing problem that runs in $\mathcal{O}(mn^{1/2})$ time, where $n$ is the number of vertices and $m$ the number of edges of the input graph. We also design a $\frac{1}{2}$-approximation algorithm for the $2^+$-star packing problem that runs in $\mathcal{O}(m)$ time, where $m$ is the number of edges of the input graph. As a consequence, every connected graph with at least $3$ vertices has a $2^+$-star packing that covers at least half of its vertices.
Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines
