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

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

4. Model A wireless network is modeled as a graph G(V,E) with cost functions d and p on E 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 V is the set of mobile devices E is the set of pairs of devices which can communicate bi-directionally E is the set of pairs of devices which can communicate bi-directionally d uv is the distance between device u and v d uv is the distance between device u and v p uv is the power needed to transmit between device u and v (usually it is distance to the power attenuation exponent) p uv is 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 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’} p uv The power used by a wireless network with power settings defined by H is The power used by a wireless network with power settings defined by H is P(H) = Σ u in V p(u)

6. Problem Formulation Given Given A wireless network A wireless network Find Find An assignment of power settings that guarantees k-fault tolerance while minimizing power usage 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 Connectivity Cone-based local heuristics Cone-based local heuristics [Rodoplu, Meng ’99; Wattenhofer, Li, Bahl, Halpern, Wang ’02] [Rodoplu, Meng ’99; Wattenhofer, Li, Bahl, Halpern, Wang ’02] A 2-approximation based on minimum weight spanning tree A 2-approximation based on minimum weight spanning tree [Kerousis, Kranakis, Krizanc, Pelc ’00] [Kerousis, Kranakis, Krizanc, Pelc ’00] A 1.69-approximation based on minimum weight Steiner tree and a more practical 1.875- approximation [Calinescu, Mandoiu, Zelikovsky ’02] A 1.69-approximation based on minimum weight Steiner tree and a more practical 1.875- approximation [Calinescu, Mandoiu, Zelikovsky ’02]

8. Related Results for Power Minimization 2-Fault tolerance 2-Fault tolerance Heuristic to minimize maximum transmit power Heuristic to minimize maximum transmit power [Ramanathan, Rosales-Hain ’00] (the only previous result) [Ramanathan, Rosales-Hain ’00] (the only previous result) Fault tolerance for general k Fault tolerance for general k Pioneered in [Bahramgiri, Hajiaghayi, Mirrokni, WINET’02] and [Hajiaghayi, Immorlica, Mirrokni, MOBICOM’03] 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 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) 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 power P(H) of subgraph H is Recall the power P(H) of subgraph H is P(H) = Σ u in V p(u) where p(u) = max {(u,v) in H(E)} p uv where p(u) = max {(u,v) in H(E)} p uv Define the weight W(H) of subgraph H as Define the weight W(H) of subgraph H as W(H) = Σ (u,v) in H(E) p uv

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. 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 1: For any graph G, P(G) ≤ 2W(G). Lemma 2: For any tree T, W(T) ≤ P(T). Lemma 2: For any tree T, W(T) ≤ P(T). Lemma 3: OPT is a tree Lemma 3: OPT is a tree Proof (of Thm): From the above lemmas, 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 6k 2 ) [Cheriyan, Vempala, Vetta, STOC’02], [Kortsarz, Nutov, STOC’04] 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 6k 2 ) [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] 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] Construct minimum weight spanning tree with O(n log n + m) messages [Gallager, Humbler, Spira, ’83] Use local augmentation to create a Use local augmentation to create a k-connected sub-graph with O(n) messages k-connected sub-graph with O(n) messages Theorem: If p uv = (d uv ) c for all pairs of nodes, then the algorithm is an O(1)-approximation when k is constant. Theorem: If p uv = (d uv ) 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. 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(log 4 n)+2a,k(1+o(1)))- approximation for minimum power k-connected subgraph where a is the approximation factor of minimum weight k-connected subgraph. Main Result: An min(O(log 4 n)+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(log 4 n),k+1)-approximation for min. power k-edge cover subgraph. An min(O(log 4 n),k+1)-approximation for min. power k-edge cover subgraph. An O(√n)-approximation for min. power k-edge connected subgraph. An O(√n)-approximation for min. power k-edge connected subgraph. APX-Hardness of k-edge cover and k-connectivity. APX-Hardness of k-edge cover and k-connectivity. Strong hardness of min. power k-edge disjoint paths. Strong hardness of min. power k-edge disjoint paths.

16. Useful Facts/Lemmas Fact 1: For any forest F, p(F)>= c(F). Fact 1: For any forest F, p(F)>= c(F). Fact 2: For any graph G, p(G)<= 2c(G). Fact 2: For any graph G, p(G)<= 2c(G). Thus, approximating cost and power for forests are the same within a constant factor. 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. 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 Theorem: If there are c-approximation for min. weight k-connected subgraph. c-approximation for min. weight k-connected subgraph. d-approximation for min. power k-edge cover 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. c is in O(log n) by previous results. d is in O(log 4 n) by a new involved combinatorial algorithm. d is in O(log 4 n) by a new involved combinatorial algorithm.

18. Algorithm for k-connectivity Find the minimum power k-edge cover. Find the minimum power k-edge cover. Set the weight of the edges in the k-edge cover to zero. 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. 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. 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. A simple k+1-approximation: pick k small edges adjacent to each vertex. An involved O(log^4 n)-approximation: An involved O(log^4 n)-approximation: If all weights are the same, then the problem is easy. If all weights are the same, then the problem is easy. Devide the edges into log(n) weight classes. Devide the edges into log(n) weight classes. Devide the vertices into log(n) subsets based on their deficiency…. 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. 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. 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. 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. 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. 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. 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. Better approximation for metric graphs.

23. Thank you. Thank you.