### Approximating Layout Problems on Random Geometric Graphs

Josep Dı́az, Mathew D. Penrose, Jordi Petit, Marı́a Serna
2001 Journal of Algorithms
In this paper, we study the approximability of several layout problems on a family of random geometric graphs. Vertices of random geometric graphs are randomly distributed on the unit square and are connected by edges whenever they are closer than some given parameter. The layout problems that we consider are: Bandwidth, Minimum Linear Arrangement, Minimum Cut Width, Minimum Sum Cut, Vertex Separation and Edge Bisection. We first prove that some of these problems remain NP-complete even for
more » ... etric graphs. Afterwards, we compute lower bounds that hold, almost surely, for random geometric graphs. Then, we present two heuristics that, almost surely, turn to be constant approximation algorithms for our layout problems on random geometric graphs. In fact, for the Bandwidth and Vertex Separation problems, these heuristics are asymptotically optimal. Finally, we use the theoretical results in order to empirically compare these and other well-known heuristics. Problem Negative results Positive results MinLA NP-C  P for trees, hypercubes, meshes [71, 34, 59, 54] NP-C for bipartite graphs  NC for trees  O (log n) approximable  O (log log n) approx. for planar graphs  PTAS for dense graphs  SumCut NP-C  P for trees  NC for trees  Bandwidth NP-C  APX for certain trees  NP-C for trees ∆ = 3 , APX for dense graphs  caterpillars with hair length 3  no PTAS for trees  no APX in general  Cutwidth NP-C  P for trees  NP-C for pl. graphs ∆ = 3  NC for trees  APX for dense graphs  VertSep NP-C  P for trees  NP-C for pl. graphs ∆ = 3  EdgeBis NP-C  P for trees (ref. in ) P for grid graphs without holes  PTAS for planar graphs  NCAS for planar graphs  P Solvable in polynomial time. APX Approximation algorithm in polynomial time. NP Solvable in non-deterministic polynomial time. PTAS Has an approximation scheme in polynomial time. NP-C NP-complete. NCAS Has an NC approximation scheme. NC Solvable in polylog time with a polynomial number of processors.