Complexity results on labeled shortest path problems from wireless routing metrics

Metrics to assess the cost of paths through networks are critical to ensuring the efficiency of network routing. This is particularly true in multi-radio multi-hop wireless networks. Effective metrics for these networks must measure the cost of a wireless path based not only on traditional measures such as throughput, but also on the distribution of wireless channels used. In this paper, we argue that routing metrics over such networks may be viewed as a class of existing shortest path problems, the formal language constrained path problems. On this basis, we describe labeled path problems corresponding to two multi-radio wireless routing metrics: Weighted Cumulative Expected Transmission Time (WCETT), developed by Draves et al., and Metric for Interference and Channel-switch (MIC), developed by Yang et al. For the first, we give a concise proof that calculating shortest WCETT paths is strongly NP-Complete for a variety of graph classes. We also show that the existing heuristic given by Draves et al. is an approximator. For the second, we show that calculating loop-free (simple) shortest MIC paths is NP-Complete, and additionally show that the optimization version of the problem is NPO PB-Complete. This result implies that shortest simple MIC paths are only poorly approximable in the worst case. Furthermore, we demonstrate how the polynomial-time algorithm for shortest MIC paths is derivable from an existing language constrained shortest path algorithm. We use this as a basis to exhibit the general utility of viewing multi-channel wireless routing metrics as labeled graph problems, and discuss how a class of related polynomial-time computable metrics are derivable from this algorithm.