Divide-and-conquer approximation algorithms via spreading metrics

We present a novel divide-and-conquer paradigm for approximating NP-hard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divide-and-conquer approach is applicable. Second, a fractional spreading metric is computable in polynomial time. The spreading metric assigns fractional lengths to either edges or vertices of the input graph, such that all subgraphs on which the optimisation problem is non-trivial have large diameters. In addition, the spreading metric provides a lower bound, τ, on the cost of solving the optimization problem. We present a polynomial time approximation algorithm for problems modelled by our paradigm whose approximation factor is O (min{log τ log log τ, log k log log k}), where k denotes the number of “interesting” vertices in the problem instance, and is at most the number of vertices. We present six problems that can be formulated to fit the paradigm. For all these problems our algorithm improves previous results. The problems are: (a) linear arrangement; (b) embedding a graph in a d-dimensional mesh; (c) interval graph completion; (d) minimizing storage-time product; (e) (subset) feedback sets in directed graphs and multicuts in circular networks; (f) symmetric multicuts in directed networks. For the first four problems, we improve the best known approximation factor from O(log² n) to O(log n log log n), where n denotes the number of vertices. For the last two problems we improve the approximation factor from O (min{log τ log log τ, log n log log n,log² k}) to O (min{log τ log log τ, log k log log k}), where k denotes the number of source-sink pairs