We present some unique challenges in cognitive radio ad-hoc networks (CRAHNs) that are not present in conventional single-channel or multi-channel wireless ad-hoc networks. We first briefly survey these challenges and their potential impact on the design of efficient algorithms for several fundamental problems in CRAHNs. Then, we describe our recent contributions to the capacity maximization problem~cite{capacity-mass} and the connectivity problem~cite{connectivity-algosensors}. The capacity maximization problem is to maximize the overall throughput utility among multiple unicast sessions, the connectivity problem is to find a connected subgraph from the given cognitive radio network where each secondary node is equipped with multiple radios. By assuming the physical interference model and asynchronous communications, we reformulate the above two problems where the capacity maximization problem is to find the maximum number of simultaneously transmitting links in secondary networks, and the connectivity problem is to construct a spanning tree over secondary networks using the fewest timeslots. We discuss the challenging issues for designing distributed approximation algorithms and give a preliminary framework for solving these two problems.
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