1. The Complexity of Channel Scheduling in Multi-Radio Multi-Channel Wireless Networks Wei Cheng & Xiuzhen Cheng The George Washington University Taieb Znati University of Pittsburgh Xicheng Lu & Zexin Lu National University of Defense technology

2. Outline Introduction Network Model The Complexity of OWCS/P PTAS for OWCS/P Summary

3. Introduction – Background Multi-Radio Multi-Channel (MR-MC) to enhance mesh network throughput Equipped with multiple radios, nodes can communicate with multiple neighbors simultaneously over orthogonal channels to improve the network throughput. The key problem is the channel scheduling, which aims to maximize the concurrent traffics without interfering each other.

4. Introduction – Interference Model P(hysical) interference-free model if two nodes want to launch bidirectional communications, any other node whose minimum distance to the two nodes is not larger than the interference range must keep silent. Hop interference-free model (no position) … is no larger than H hops must keep silent.

6. Introduction – Problem Optimal Weighted Channel Scheduling under the Physical distance constraint (OWCS/P) Given an edge-weighted graph G(V,E) representing an MR-MC wireless network, compute an optimal channel scheduling O(G) ∈ E, such that O(G) is P interference-free and the weight of O(G) is maximized Optimal Weighted Channel Scheduling under Hop distance constraint (OWCS/H)

7. Introduction – Motivation Both the physical interference-free model and the hop interference-free model are popular but their relations have never been addressed in literature. Current complexity results for OWCS

8. Related Research Channel allocation, routing, and packet scheduling have been jointly considered as a IP problem Channel Assignment Common channel Default radio for reception Code based approach

9. Related Research The complexity of scheduling in SR-SC networks OWCS/H>=1 is NP hard OWCS/H>=1 has PTAS

10. Network Model Geometric graphs G(V,E), |V | = n a set of C ={c 1, c 2, · · ·, c k } orthogonal channels ∀ node i ∈ V, 1 ≤ i ≤ n, it is equipped with r i radios and can access a set of C i ⊆ C channels, where |C i | = k i.

11. Formal Definition Edge-Physical-Distance Edge-Hop-Distance OWCS/P: Seek an E’ such that any pair of edges in E’ has an Edge-Physical- Distance >P, and E’ is the maximum OWCS/H: Seek an E’ such that any pair of edges in E’ has an Edge-Physical- Distance >H, and E’ is the maximum

12. The Complexity of OWCS/P Lemma : OWCS/P=1 and OWCS/H=1 are equivalent in SR-SC wireless networks. Intuition: the interference graphs of G(V,E) for the cases of P=1 and H=1 are the same Proof: OPT/P=1 is a feasible solution to OWCS/H=1 We can not add another edge to OPT/P=1 for OWCS/H=1 Similarly, OPT/H=1 is optimal to OWCS/P=1

13. The Complexity of OWCS/P Theorem : OWCS/P>=1 is NP-Hard in SR-SC wireless networks. OWCS/H=1 is NP-Hard  OWCS/P=1 is NP- Hard OWCS/P>1 is polynomial time reducible to OWCS/P=1

14. The Complexity of OWCS/P Theorem: OWCS/P>=1 is NP-Hard in MR-MC wireless networks. Known

15. PTAS for OWCS/P Polynomial-Time Approximation Scheme (PTAS) for NP-Hard problem. a polynomial-time approximate solution with a performance ratio (1 − ε) for an arbitrarily small positive number ε. Let Ptas(G) denote the solution given by the PTAS procedure and O(G) the optimal solution for the OWCS/P≥1 problem in a MR- MC network G. We will prove that W(Ptas(G)) ≥ (1 − ε )W(O(G))

16. PTAS for OWCS/P-construction Griding: Partition network space into small grids with each having a size of (P + 2) × (P +2). Label each grid by (a, b), where a, b = 0, 1, · · ·,N − 1, with N the total number of grids at each row or column. The id of the grid at the lower-left corner can be denoted by (0, 0). Denote the ith row and the jth column of the grids by Row i and Col j, respectively.

17. PTAS for OWCS/P-construction Shifted Dissection: Partition vertically the network space by columns of the grids Col j and rows of the grids Row i, where j | (m+1)= k1, i |(m+1)= k 2, k 1 k 2 = 0, 1, · · ·,m. Remove all the edges whose both end nodes are in Col j or Row i Obtain a number of super-grids with each containing at most m×m grids. Total (m + 1) 2 different dissections Denote each dissection by P a,b, where a, b indicate that P a,b is obtained by shifting Col 0 to column b and Row 0 to row a.

20. PTAS for OWCS/P-construction Computation Consider a specific P a,b For each super-grid B in P a,b, compute an maximum weight channel scheduling S B for B. Let S a,b be the union of all SB’s S a,b is a feasible solution for OWCS/P Repeat for all P a,b

21. PTAS for OWCS/P-algorithm

22. PTAS for OWCS/P-complexity Computing S B takes polynomial time. the area of B is at most (m(P + 2) + 2) 2 For a specific channel The number of S B ’s edges in each ((P + 2) 2 ) grid is bounded by O(1). Then the number of edges in S B is bounded by O(m 2 ) Time of computing S B through enumerating is bounded by |E B | O(m2) For all K channels Time of computing S B through enumerating is bounded by |E B | O(m2)K

23. PTAS for OWCS/P-performance For all partition P a,b S a,b is the optimal solution for E a,b Let yields,

24. PTAS for OWCS/P-performance A grid will NOT be included in any super- grid among all (m+ 1) 2 partitions for 2m+ 1 times. An edge will NOT be included in any super-grid among all (m+ 1) 2 partitions for at most 2m+ 1 times.

25. Summary

26. The proposed PTAS for OWCS/P is also a PTAS for OWCS/H in MR-MC wireless networks. Replace P by H

27. Summary OWCS/H=1 is equivalent to OWCS/P>= under the polynomial transformation OWCS/H=1 is equivalent to OWCS/P=1 OWCS/P>1 is polynomial time reducible to OWCS/P=1 Physical interference free model is more precise Need position information

28. Q&A Thanks!