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Conference Papers Year : 2011

New Exact and Approximation Algorithms for the Star Packing Problem in Undirected Graphs

Abstract

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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Dates and versions

hal-00573617 , version 1 (05-03-2011)

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  • HAL Id : hal-00573617 , version 1

Cite

Maxim Babenko, Alexey Gusakov. New Exact and Approximation Algorithms for the Star Packing Problem in Undirected Graphs. Symposium on Theoretical Aspects of Computer Science (STACS2011), Mar 2011, Dortmund, Germany. pp.519-530. ⟨hal-00573617⟩

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