Local approximation schemes for ad hoc and sensor networks

We present two local approaches that yield polynomial-time approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a <i>(1+ε)</i>-approximation to the problems at hand for any given <i>ε > 0</i>. The time complexity of both algorithms is <i>O(T<inf>MIS</inf> + log<sup>*</sup>! n/ε<sup>O(1)</sup>)</i>, where <i>T<inf>MIS</inf></i> is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pair-wise independent nodes in every <i>r</i>-neighborhood is at most polynomial in <i>r</i>. Such <i>graphs of polynomially bounded growth</i> are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs.

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