The Complexity of Channel Scheduling in Multi-Radio Multi-Channel Wireless Networks

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 ={c1, c2, · · · , ck} orthogonal channels • ∀ node i ∈ V , 1 ≤ i ≤ n, it is equipped with ri radios and can access a set of Ci ⊆ C channels, where |Ci| = ki.

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 inSR-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 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 Rowi and Colj , respectively.

17. PTAS for OWCS/P-construction • Shifted Dissection: • Partition vertically the network space • by columns of the grids Colj and rows of the grids Rowi, where j | (m+1)= k1 , i |(m+1)= k2, k1 k2 = 0, 1, · · · ,m. Remove all the edges whose both end nodes are in Colj or Rowi • Obtain a number of super-grids with each containing at most m×m grids. Total (m + 1)2 different dissections • Denote each dissection by Pa,b, where a, b indicate that Pa,b is obtained by shifting Col0to column b and Row0to row a.

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

21. PTAS for OWCS/P-algorithm

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

23. PTAS for OWCS/P-performance • For all partition Pa,b • Sa,b is the optimal solution for Ea,b • Let yields ,

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

25. Summary

26. Summary • 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!