The Complexity of the Routing Problem in POR

We have designed and implemented a new protocol for wireless mesh networks called Practical Opportunistic Routing (POR). In this report, we document our finding about the complexity of the routing problem in POR. We prove that the routing problem is NP-hard by reducing a slightly modified version of the MAX-2-SAT problem to the optimal routing problem in POR. I. PRELIMINARIES We have proposed a new packet forwarding and routing protocol for wireless mesh networks called Practical Opportunistic Routing (POR) [2]. As we have mainly focused on the practical design and implementation of POR in [2], in this report, we document our findings about the theoretical routing problem in POR. We begin with a brief review of the routing metric in POR; the details can be found in [2]. Suppose the path is (v1, v2, ..., vL). Any POR path must satisfy the feedback constraint; that is, any node on the path must be able to receive from its next hop node to ensure the correct reception of possible feedbacks from its downstream nodes. POR considers both the forward cost for sending data and backward cost for sending feedback. The path cost calculation is carried out iteratively, starting from the node closest to the destination. Therefore, when calculating the cost of the path, the forward and backward costs of path (vi, vi+1, ..., vL) are known for 2 ≤ i ≤ L, denoted as Cvi and Bvi , respectively. A. Path Cost in a Given State Suppose the links involving v1 are in a certain set of states denoted as τ . 1) Forward Cost: The forward cost of the path is denoted as C v1 , which is defined as the consumed air time in data transmission in order to deliver a unit size block to the destination when the links are in state τ . We denote the BRR of link v1 → vi at rate ρj as μ ρj ,τ i . We use C τ v1,ρj to denote the air time consumed to deliver a unit size block to vL following path (v1, v2, ..., vL), under the condition that v1 transmits at rate ρj . We have C v1,ρj = 1 ρj + ∑L i=2 μ ρj i ∏L t=i+1(1− μ ρj t )Cvi 1− ∏L i=2(1− μ ρj i ) . (1) C v1 is C τ v1,ρ∗ if ρ ∗ has the minimum cost among all rates. 2) Backward Cost: The backward cost at v1 is denoted as B v1 and defined as the consumed air time in feedback transmission in order to deliver a packet to the destination. We let γi = μ ∗,τ i ∏L t=i+1(1− μ ρ∗,τ t ) 1− ∏L i=2(1− μ ρ∗,τ i ) . The backward cost is calculated according to