Power Optimization for Connectivity Problems

1. Power Optimization for Connectivity Problems MohammadTaghi Hajiaghayi, Guy Kortsarz, Vahab S. Mirrokni, Zeev Nutov IPCO 2005

2.  Power Optimization in Fault-Tolerant Topology Control • Wireless multihop networks • Simple low-power devices • Radio transmitters • Power is the main limitation • Power assignment • A power setting for each device • Defines possible communication links • Power versus distance: It takes power rc to transmit a message to distance r for some power attenuation exponent c between 2 and 4.

3. Power Optimization in Fault-Tolerant Topology Control • Goal:Minimize power usage while maintaining key network properties • Connectivity: There is a communication path between any pair of nodes • k-Fault tolerance: Connectivity is maintained in light of at most k-1 failures • Device failures (our focus) • Communication link failures • By k-Fault tolerance,we also have k-disjoint paths and thus higher network capacity

4. Model • A wireless network is modeled as a graph G(V,E) with cost functions d and p on E • V is the set of mobile devices • E is the set of pairs of devices which can communicate bi-directionally • duv is the distance between device u and v • puvis the power needed to transmit between device u and v (usually it is distance to the power attenuation exponent)

5. Model • Conversely, a subgraph H=(V,E’) of the network graph G defines an assignment of power settings: device u transmits at p(u) = max {(u,v) in E’} puv • The power used by a wireless network with power settings defined by H is P(H) = Σu in V p(u)

6. Problem Formulation • Given • A wireless network • Find • An assignment of power settings that guarantees k-fault tolerance while minimizing power usage Recall k-fault tolerance means the network remains connected even when up to k-1 devices (or communication links) fail

7. Related Results for Power Minimization • Connectivity • Cone-based local heuristics [Rodoplu, Meng ’99; Wattenhofer, Li, Bahl, Halpern, Wang ’02] • A 2-approximation based on minimum weight spanning tree [Kerousis, Kranakis, Krizanc, Pelc ’00] • A 1.69-approximation based on minimum weight Steiner tree and a more practical1.875-approximation[Calinescu, Mandoiu, Zelikovsky ’02]

8. Related Results for Power Minimization • 2-Fault tolerance • Heuristic to minimize maximum transmit power [Ramanathan, Rosales-Hain ’00](the only previous result) • Fault tolerance for general k • Pioneered in [Bahramgiri, Hajiaghayi, Mirrokni, WINET’02] and[Hajiaghayi, Immorlica, Mirrokni, MOBICOM’03]

9. Cone-Based Heuristic • Algorithm: • Input: A set of nodes on the plane, with max. power P • Each node increases its power until the angle between any two consecutive neighbors is less than some threshold or it reaches its maximum power P. • Output: two nodes are connected if both can hear each other with the new power assignment • Theorem [BHM’02]: If the network of max. powers is k-connected and the angle between any pair of adjacent neighbors is at most 2π/3k, then the new network is k-connected (2π/3k is almost tight) • Main disadvantage: The algorithm is local and thus does not give any bound on the global goal of minimizing sum of the powers (or the average power)

10. Approximating Connectivity • Recall the powerP(H) of subgraph H is P(H) = Σu in V p(u) where p(u) = max {(u,v) in H(E)} puv • Define the weightW(H) of subgraph H as W(H) = Σ(u,v) in H(E) puv

11. Approximating Connectivity • Theorem[KKKP ’00]: The minimum weight spanning tree MST of G uses at most twice as much power as the minimum power connected subgraph OPT of G. • Lemma 1: For any graph G, P(G) ≤ 2W(G). • Lemma 2: For any tree T, W(T) ≤ P(T). • Lemma 3: OPT is a tree • Proof (of Thm): From the above lemmas, P(MST) ≤ 2W(MST) ≤ 2W(OPT) ≤ 2P(OPT).

12. Approximating k-Connectivity Minimum weight k-connected subgraph: an LP-based algorithm gives a solution of weight at most O(log k) times optimal weight (n is at least 6k2) [Cheriyan, Vempala, Vetta, STOC’02], [Kortsarz, Nutov, STOC’04] Minimum Power k-connected subgraph: Using the above algorithms an O(k)-approximation can be derived [Hajiaghayi, Immorlica, Mirrokni MOBICOM’03]

13. Distributed (Local) Approximation Algorithm: [Hajiaghayi, Immorlica, Mirrokni, MOBICOM’03] • Construct minimum weight spanning tree with O(n log n + m) messages[Gallager, Humbler, Spira, ’83] • Use local augmentation to create a k-connected sub-graph with O(n) messages • Theorem: If puv= (duv)c for all pairs of nodes, then the algorithm is an O(1)-approximation when k is constant.

14. Min Power K-cover • k-edge cover is a subgraph in which the degree of each vertex is at least k.

15. Our Results • Main Result: An min(O(log4n)+2a,k(1+o(1)))-approximation for minimum power k-connected subgraph where a is the approximation factor of minimum weight k-connected subgraph. • An min(O(log4n),k+1)-approximation for min. power k-edge cover subgraph. • An O(√n)-approximation for min. power k-edge connected subgraph. • APX-Hardness of k-edge cover and k-connectivity. • Strong hardness of min. power k-edge disjoint paths.

16. Useful Facts/Lemmas • Fact 1: For any forest F, p(F)>= c(F). • Fact 2: For any graph G, p(G)<= 2c(G). • Thus, approximating cost and power for forests are the same within a constant factor. • Theorem: If G is a k-edge cover and F is an inclusion minimal edge set such that G+F is k-connected, then F is a forest.

17. k-edge cover to k-connected • Theorem: If there are • c-approximation for min. weight k-connected subgraph. • d-approximation for min. power k-edge cover Then we have 2c+d-approximation for min power k-connected subgraph. • c is in O(log n) by previous results. • d is in O(log4n) by a new involved combinatorial algorithm.

18. Algorithm for k-connectivity • Find the minimum power k-edge cover. • Set the weight of the edges in the k-edge cover to zero. • Augment the k-edge cover to a k-connected graph by finding the min. cost k-connected with the new weight function. • Note that the augmentation is a forest.

19. Approximating k-edge cover • A simple k+1-approximation: pick k small edges adjacent to each vertex. • An involvedO(log^4 n)-approximation: • If all weights are the same, then the problem is easy. • Devide the edges into log(n) weight classes. • Devide the vertices into log(n) subsets based on their deficiency….

20. O(√n)-approximation for min. power k-edge connectivity • Augmenting k-edge cover to k-edge connected graph with at most n-1 edges. • Using O(log^4 n)-approximation for min. power k-edge cover and 2-approximation for min. weight k-edge connected subgraph to agument to k-edge connected. • This gives O(√n)-approximation for min. power k-edge connected.

21. Hardness Results • APX-hardness: Reduction from 4-bounded set cover to the minimum power k-edge cover, k-connected and k-edge connected subgraphs. • Stronger inapproximability for minimum power k edge disjoint paths in directed graphs.

22. Open Problems • Closing the gap between the inapproximability and approximation factor of the minimum power k-connected and k-edge connected subgraph. • Better approximation for metric graphs.

23. Thank you.