New Exact and Approximation Algorithms for the Star Packing Problem in Undirected Graphs
Résumé
By a \emph{$T$-star} we mean a complete bipartite graph $K_{1,t}$ for some $t \le T$. For an undirected graph~$G$, a \emph{$T$-star packing} is a collection of node-disjoint $T$-stars in $G$. For example, we get ordinary matchings for $T = 1$ and packings of paths of length 1 and 2 for $T = 2$. Hereinafter we assume that $T \ge 2$. Hell and Kirkpatrick devised an ad-hoc augmenting algorithm that finds a $T$-star packing covering the maximum number of nodes. The latter algorithm also yields a min-max formula. We show that $T$-star packings are reducible to network flows, hence the above problem is solvable in $O(m\sqrt{n})$ time (hereinafter $n$ denotes the number of nodes in $G$, and $m$~--- the number of edges). For the edge-weighted case (in which weights may be assumed positive) finding a maximum $T$-packing is NP-hard. A novel $\frac{9}{4} \frac{T}{T + 1}$-factor approximation algorithm is presented. For non-negative node weights the problem reduces to a special case of a max-cost flow. We develop a divide-and-conquer approach that solves it in $O(m\sqrt{n} \log n)$ time. The node-weighted problem with arbitrary weights is more difficult. We prove that it is NP-hard for $T \ge 3$ and is solvable in strongly-polynomial time for $T = 2$.
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