An efficient cost-sharing mechanism for the prize-collecting Steiner forest problem

In an instance of the <i>prize-collecting Steiner forest</i> problem (PCSF) we are given an undirected graph <i>G</i> = (<i>V, E</i>), non-negative edge-costs <i>c</i>(<i>e</i>) for all <i>e</i> ε <i>E</i>, terminal pairs <i>R</i> = {(<i>s<inf>i</inf></i>, <i>t<inf>i</inf></i>)}1≤<i>i</i>≤<i>k</i>, and penalties π<inf>1</inf>,...,π<i><inf>k</inf></i>. A feasible solution (<i>F, Q</i>) consists of a forest <i>F</i> and a subset <i>Q</i> of terminal pairs such that for all (<i>s<inf>i</inf></i>, <i>t<inf>i</inf></i>) ε <i>R</i> either <i>s<inf>i</inf></i>, <i>t<inf>i</inf></i> are connected by <i>F</i> or (<i>s<inf>i</inf></i>, <i>t<inf>i</inf></i>) ε <i>Q</i>. The objective is to compute a feasible solution of minimum cost <i>c</i>(<i>F</i>) + π (<i>Q</i>). A game-theoretic version of the above problem has <i>k</i> players, one for each terminal-pair in <i>R</i>. Player <i>i's</i> ultimate goal is to connect <i>s<inf>i</inf></i> and <i>t<inf>i</inf></i>, and the player derives a privately held <i>utility u<inf>i</inf></i> ≥ 0 from being connected. A service provider can connect the terminals <i>s<inf>i</inf></i> and <i>t<inf>i</inf></i> of player <i>i</i> in two ways: (1) by buying the edges of an <i>s<inf>i</inf></i>, <i>t<inf>i</inf></i>-path in <i>G</i>, or (2) by buying an alternate connection between <i>s<inf>i</inf></i> and <i>t<inf>i</inf></i> (maybe from some other provider) at a cost of π<i><inf>i</inf></i>. In this paper, we present a simple 3-budget-balanced and group-strategyproof mechanism for the above problem. We also show that our mechanism computes client sets whose social cost is at most <i>O</i>(log<sup>2</sup> <i>k</i>) times the minimum social cost of any player set. This matches a lower-bound that was recently given by Roughgarden and Sundararajan (STOC '06).

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