Ashwin Ganesan

An assumption that researchers have often used to model interference in a wireless network is the unit disk graph model. While many theoretical results and performance guarantees have been obtained under this model, an open research direction is to extend these results to hypergraph interference models. Motivated by recent results that the worst-case performance of the distributed maximal scheduling algorithm is characterized by the interference degree of the hypergraph, in the present work we investigate properties of the interference degree of the hypergraph and the structure of hypergraphs arising from physical constraints. We show that the problem of computing the interference degree of a hypergraph is NP-hard and we prove some properties and results concerning this hypergraph invariant. We investigate which hypergraphs are realizable, i.e. which hypergraphs arise in practice, based on physical constraints, as the interference model of a wireless network. In particular, a question that arises naturally is: what is the maximal value of $r$ such that the hypergraph $K_{1,r}$ is realizable? We determine this quantity for various integral and nonintegral values of the path loss exponent of signal propagation. We also investigate hypergraphs generated by line networks.

An assumption that has often been used by researchers to model the interference in a wireless network is the unit disk graph model. While many theoretical results and performance guarantees have been obtained under this model, an open research direction is to extend these results to hypergraph interference models. Motivated by recent results that the worst-case performance of the distributed maximal scheduling algorithm is characterized by the interference degree of the hypergraph, in the present work we investigate properties of the interference degree of the hypergraph and the structure of hypergraphs arising from physical constraints. We show that the problem of computing the interference degree of a hypergraph is NP-hard and we prove some properties and results concerning this hypergraph invariant. We investigate which hypergraphs are realizable, i.e. which hypergraphs arise in practice, based on physical constraints, as the interference model of a wireless network. In particular, a question that arises naturally is: what is the maximal value of $r$ such that the hypergraph $K_{1,r}$ is realizable? We determine this quantity for various integral and nonintegral values of the path loss exponent of signal propagation. We also investigate hypergraphs generated by line networks.

An open problem is to extend the results in the literature on unit disk graphs to hypergraph models. Motivated by recent results that the worst-case performance of the distributed maximal scheduling algorithm is characterized by the interference degree of the hypergraph, in the present work we investigate properties of the interference degree of the hypergraph and the structure of hypergraphs arising from physical constraints. We show that the problem of computing the interference degree of a hypergraph is NP-hard and we prove some properties and results concerning this hypergraph invariant. We then investigate which hypergraphs are realizable, i.e. which hypergraphs arise in practice, based on physical constraints, as the interference model of a wireless network. In particular, given the results on the worst-case performance of the maximal scheduling algorithm, a question that arises naturally is: what is the maximal value of $r$ such that the hypergraph $K_{1,r}$ is realizable? We show that this value is $r=4$.

If each node in a wireless network has information about only its $1$-hop neighborhood, then what are the limits to performance? This problem is considered for wireless networks where each communication link has a minimum bandwidth quality-of-service (QoS) requirement. Links in the same vicinity contend for the shared wireless medium. The conflict graph captures which pairs of links interfere with each other and depends on the MAC protocol. In IEEE 802.11 MAC protocol-based networks, when communication between nodes $i$ and $j$ takes place, the neighbors of both $i$ and $j$ remain silent. This model of interference is called the $2$-hop interference model because the distance in the network graph between any two links that can be simultaneously active is at least $2$. In the admission control problem studied in the present paper, the objective is to determine, using only localized information, whether a given set of flow rates is feasible. While distance-$d$ distributed algorithms have been analyzed for the $1$-hop interference model, an open problem in the literature is to extend these results to the $K$-hop interference model, and the present work initiates the generalization to the $K$-hop interference model.We show that the centralized version of the problem is NP-hard and then investigate distributed, low-complexity solutions for this problem. We propose a distributed algorithm for this problem where each node has information about only its $1$-hop neighborhood. The worst-case performance of the distributed algorithm, i.e. the largest factor by which the performance of this distributed algorithm is away from that of an optimal, centralized algorithm, is analyzed. Lower and upper bounds on the suboptimality of the distributed algorithm are obtained, and both bounds are shown to be tight. The exact worst-case performance is obtained for some ring topologies. The performance of the distance-$1$ distributed algorithm is compared with that of the row constraints, and these two distributed algorithms are shown to be incomparable.

Consider a wireless network where each communication link has a minimum bandwidth quality-of-service requirement. Certain pairs of wireless links interfere with each other due to being in the same vicinity, and this interference is modeled by a conflict graph. Given the conflict graph and link bandwidth requirements, the objective is to determine, using only localized information, whether the demands of all the links can be satisfied. At one extreme, each node knows the demands of only its neighbors; at the other extreme, there exists an optimal, centralized scheduler that has global information. The present work interpolates between these two extremes by quantifying the tradeoff between the degree of decentralization and the performance of the distributed algorithm. This open problem is resolved for the primary interference model, and the following general result is obtained: if each node knows the demands of all links in a ball of radius $d$ centered at the node, then there is a distributed algorithm whose performance is away from that of an optimal, centralized algorithm by a factor of at most $(2d+3)/(2d+2)$. The tradeoff between performance and complexity of the distributed algorithm is also analyzed. It is shown that for line networks under the protocol interference model, the row constraints are a factor of at most $3$ away from optimal. Both bounds are best possible.