Wireless network design via 3-decompositions

Abstract

We consider some network design problems with applications for wireless networks. The input for these problems is a metric space ( X , d ) and a finite subset U ⊆ X of terminals. In the Steiner Tree with Minimum Number of Steiner Points ( STMSP) problem, the goal is to find a minimum size set S ⊆ X − U of points so that the unit-disc graph of S + U is connected. Let Δ be the smallest integer so that for any finite V ⊆ X for which the unit-disc graph is connected, this graph contains a spanning tree with maximum degree ⩽Δ. The best known approximation ratio for STMSP was Δ − 1 [I.I. Măndoiu, A.Z. Zelikovsky, A note on the MST heuristic for bounded edge-length Steiner trees with minimum number of Steiner points, Information Processing Letters 75 (4) (2000) 165–167]. We improve this ratio to ⌊ ( Δ + 1 ) / 2 ⌋ + 1 + ε . In the Minimum Power Spanning Tree ( MPST) problem, V = X is finite, and the goal is to find a “range assignment” { p ( v ) : v ∈ V } on the nodes so that the edge set { u v ∈ E : d ( u v ) ⩽ min { p ( u ) , p ( v ) } } contains a spanning tree, and ∑ v ∈ V p ( v ) is minimized. We consider a particular case { 0 , 1 } - MPST of MPST when the distances are in { 0 , 1 } ; here the goal is to find a minimum size set S ⊆ V of “active” nodes so that the graph ( V , E 0 + E 1 ( S ) ) is connected, where E 0 = { u v : d ( u v ) = 0 } , and E 1 ( S ) is the set the edges in E 1 = { u v : d ( u v ) = 1 } with both endpoints in S. We will show that the ( 5 / 3 + ε ) -approximation scheme for MPST of [E. Althaus, G. Calinescu, I. Măndoiu, S. Prasad, N. Tchervenski, A. Zelikovsky, Power efficient range assignment for symmetric connectivity in static ad hoc wireless networks, Wireless Networks 12 (3) (2006) 287–299] achieves a ratio 3/2 for { 0 , 1 } -distances. This answers an open question posed in [E. Lloyd, R. Liu, S. Ravi, Approximating the minimum number of maximum power users in ad hoc networks, Mobile Networks and Applications 11 (2006) 129–142].