Linear Programming & its Applications to Wireless Networks

1. Linear Programming & its Applications to Wireless Networks Guofeng Deng IMPACT Lab, Arizona State University

2. Outline • Linear programming (LP) • Formulation • Solutions • Flow model • Applications • Maximizing broadcast lifetime • Optimal role assignments • Multicommodity flow • Energy efficient routing in disaster recovery networks • Cross-layer design for lifetime maximization • Minimum power broadcast tree G. Deng

3. LP Summary • LP • Linear objective function • Continuous variables • Linear constraints (equations or inequalities) • Solutions • Simplex methods • Interior-point methods • Software tools • Cplex, GLPK, Matlab • Beyond LP • Integer linear programming (ILP): variables are integers. • It is called mixed integer programming (MIP) if not all variables are integers. • The problem becomes NP-hard. • Approximation methods include branch-and-bound, branch-and-cut. • If removing integer constraints, LP provides a lower/upper bound to a minimization/maximization problem. • Nonlinear programming: some constraints or the objective function is nonlinear. G. Deng

4. App1: Maximizing Broadcast Lifetime using Multiple Trees • Summary: • Problem: Given a set of broadcast trees in the form of power consumption of each node, maximizing broadcast lifetime using multiple trees sequentially. • Variables: Duration of each tree being used. We assume duration is indefinitely divisible. • Constraints: For each node, the overall amount of energy that can be consumed in all the trees is limited by its battery capacity. • Notations: • K: a set of broadcast trees • (): the duration of tree  K • pi(): power of node i on tree  • Ei: battery capacity of node i Objective function: Constraint 1: Constraint 2: G. Deng

5. 3/11 + 3/11 3/11 3/11 + 5/11 B 3 2 1 5/11 3/11 App2: Bounding the Lifetime of Sensor Networks • Summary: • Problem: Given a pair of source and destination nodes and a set of intermediate nodes, maximize the lifetime, i.e., the amount of packets that is transmitted from source to designation. • Variables: f_ij: the flow from i to j. • Constraints: see below. • Notations: • Node 1 is the source and N+1 is the destination. • t: lifetime; e_i: battery capacity of node I • Comment: • - The formulation was later extended to accommodate multiple source and single sink. For any intermediate node, which does not generate any flow, the amount of incoming flow matches the amount of outgoing flow. This is the total amount of flow injected to the network, i.e., the difference between the amount of flow outgoing from source and that incoming to source. Bhardwaj & Chandrakasan, Bounding the Lifetime of Sensor Networks Via Optimal Role Assignments, INFOCOM’02 G. Deng

6. App3: Multicommodity Flow Chang & Tassiulas, Energy conserving routing in wireless ad-hoc networks, INFOCOM’00 Chang & Tassiulas, Maximum lifetime routing in wireless sensor networks, TON, Vol.12 No.4, 2004 Sanka & Liu, Maximum lifetime routing in wireless ad-hoc networks, INFOCOM’04 G. Deng

7. App4: EE Routing in Disaster Recovery Networks \bar{f}_{i,j}: the amount of info transmitted from i to j until time T R: receiver nodes d: destination r_i: the ratio between the rate in which info is generated at badge node i and the maximum possible flow on a link connecting smart badges Zussman & Segall, Energy efficient routing in ad hoc disaster recovery networks, Ad Hoc Networks, Vol.1, 2003 G. Deng

8. App5: Cross-Layer Design for Lifetime Maximization Tv: node lifetime N: number of slots r^n_k: trans rate over link k per unit bandwidth in slot n P^n_k: trans power over link k in slot n P^{max}: maximum trans pwr N_0: noise power non-convex! Madan et al., Cross-layer design for lifetime maximization in interference-limited wireless ad hoc networks, INFOCOM’05 Madan et al., Cross-layer design for lifetime maximization in interference-limited wireless ad hoc networks, IEEE trans. Wireless Communications, Vol.5 No.11, 2006 G. Deng

9. App6: Minimum Power Broadcast Tree Defines relation between continuous and binary variables Source node has to transmit to at least one other node 4 6 1 8 Non-source node at most transmits to one other node 3 5 Source has to transmit in the 1st step. actual trans implicit trans Defines relation between X_ij and X_ijk. 2 7 Variables A non-source node is not allowed to transmit until it is reached actually or implicitly. Non-source node is not allowed to transmit in the 1st step. Y_i: power of node i X_ij: =1 if there is a explicit link from i to j X_ijk: =1 if the kth transmission is i to j Power matrix At most one transmission in each step. Reward matrix R_mn(p)=1 if P_mp ≤ P_mn Each node has to be reached ultimately. Das et al., Minimum Power Broadcast Trees for Wireless Networks: Integer Programming Formulations, INFOCOM’03 Source has to transmit in the 1st step. G. Deng