Power Optimization for Connectivity Problems

Given a graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimization criteria: the Min-Power b-Edge-Cover problem (MPb-EC) where the goal is to find a min-power subgraph so that the degree of every node v is at least some given integer b(v), the MinPower k-node Connected Spanning Subgraph problem (MPk-CSS), MinPower k-edge Connected Spanning Subgraph problem (MPk-ECSS), and finally the Min-Power k-Edge-Disjoint Paths problem in directed graphs (MPk-EDP). We give an O(log n)-approximation algorithm for MPb-EC. This gives an O(log n)-approximation algorithm for MPk-CSS for most values of k, improving the best previously known O(k)-approximation guarantee. In contrast, we obtain an O( √ n) approximation algorithm for MPk-ECSS, and for its variant in directed graphs (i.e., MPk-EDP), we establish the following inapproximability threshold: MPk-EDP cannot be approximated within O(2 1−ε ) for any fixed ε > 0, unless NP-hard problems can be solved in quasi-polynomial time.

[1]  Vahab S. Mirrokni,et al.  Fault-Tolerant and 3-Dimensional Distributed Topology Control Algorithms in Wireless Multi-hop Networks , 2006, Wirel. Networks.

[2]  Guy Kortsarz,et al.  Approximation algorithm for k-node connected subgraphs via critical graphs , 2004, STOC '04.

[3]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[4]  Andrea E. F. Clementi,et al.  Some Recent Theoretical Advances and Open Questions on Energy Consumption in Ad-Hoc Wireless Networks , 2001 .

[5]  Samir Khuller,et al.  Biconnectivity approximations and graph carvings , 1992, STOC '92.

[6]  Samir Khuller,et al.  Improved approximation algorithms for uniform connectivity problems , 1995, STOC '95.

[7]  Paramvir Bahl,et al.  Analysis of a cone-based distributed topology control algorithm for wireless multi-hop networks , 2001, PODC '01.

[8]  W. Mader.  Ecken vom Gradn in minimalenn-fach zusammenhängenden Graphen , 1972 .

[9]  Teresa H. Meng,et al.  Minimum energy mobile wireless networks , 1998, ICC '98. 1998 IEEE International Conference on Communications. Conference Record. Affiliated with SUPERCOMM'98 (Cat. No.98CH36220).

[10]  Santosh S. Vempala,et al.  An Approximation Algorithm for the Minimum-Cost k-Vertex Connected Subgraph , 2003, SIAM J. Comput..

[11]  Joseph Cheriyan,et al.  Fast Algorithms for k-Shredders and k-Node Connectivity Augmentation , 1999, J. Algorithms.

[12]  Guy Kortsarz,et al.  Approximating node connectivity problems via set covers , 2000, APPROX.

[13]  Ernst Althaus,et al.  Power efficient range assignment in ad-hoc wireless networks , 2003, 2003 IEEE Wireless Communications and Networking, 2003. WCNC 2003..

[14]  Andrea E. F. Clementi,et al.  Hardness Results for the Power Range Assignmet Problem in Packet Radio Networks , 1999, RANDOM-APPROX.

[15]  Li Li,et al.  Distributed topology control for power efficient operation in multihop wireless ad hoc networks , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[16]  Madhav V. Marathe,et al.  Algorithmic Aspects of Topology Control Problems for Ad Hoc Networks , 2005, Mob. Networks Appl..

[17]  A. Frank,et al.  An application of submodular flows , 1989 .

[18]  Guy Kortsarz.  On the Hardness of Approximating Spanners , 2001, Algorithmica.

[19]  James B. Orlin.  A Faster Strongly Polynomial Minimum Cost Flow Algorithm , 1993, Oper. Res..

[20]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[21]  Nicole Immorlica,et al.  Power optimization in fault-tolerant topology control algorithms for wireless multi-hop networks , 2007, IEEE/ACM Trans. Netw..

[22]  Peng-Jun Wan,et al.  Range Assignment for High Connectivity in Wireless Ad Hoc Networks , 2003, ADHOC-NOW.

[23]  Andrzej Pelc,et al.  Power consumption in packet radio networks , 2000, Theor. Comput. Sci..

[24]  Mauro Leoncini,et al.  On the Symmetric Range Assignment Problem in Wireless Ad Hoc Networks , 2002, IFIP TCS.

[25]  Ion I. Mandoiu,et al.  Symmetric Connectivity with Minimum Power Consumption in Radio Networks , 2002, IFIP TCS.