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

By a T-star we mean a complete bipartite graph K-1,K-t for some t <= T. For an undirected graph G, a 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 >= 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 root 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 9 T/4 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 root n log n) time. The node-weighted problem with arbitrary weights is more difficult. We prove that it is NP-hard for T >= 3 and is solvable in strongly-polynomial time for T = 2.