Divide-and-conquer approximation algorithms via spreading metrics

Abstract

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 optimization problem is non-trivial have large diameters. In addition, the spreading metric provides a lower bound, T , 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 0 (min{log Tlog log T , 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 ddimensional 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 OQog nlog log n), where n denotes the number of vertices.For the last two problems we improve the approximation factor from o (min{log T log log T , log nlog log n, log2 le) to 0 (min{log T log log T , log k log log k}), where k denotes the number of source-sink pairs.

ℓ 2 2 Spreading Metrics for Vertex Ordering Problems

“…More specifically, he gave a O(log n log log n)-approximation algorithm for Minimum Feedback Arc Set. Even et al [14] extended the approach used by Seymour to obtain polynomial-time, O(log n log log n)-approximation algorithms for Minimum Linear Arrangement and Minimum Containing Interval Graph and a O(log T log log T )-approximation algorithm for Minimum Storage-Time Product. In fact, they showed similar approximation results for a broader class of graph optimization problems which satisfy their approximation paradigm, i.e., problems for which there exists a spreading metric and which can be handled by their divide-and-conquer approach.…”

“…In a later paper, Even et al [13] extended the spreading metric techniques to graph partitioning problems and obtained simpler recursions that yielded a logarithmic approximation factor for balanced cuts and multiway separators (and not the other problems in [14]). Rao and Richa [28] improved the approximation factor to O(log n) for Minimum Linear Arrangement and Minimum Containing Interval Graph and to O(log T ) for Minimum Storage-Time Product.…”

“…Via the scaling and rounding technique used by [14,28], at the cost of only negligibly increasing the approximation ratio we can and will reduce the case of general weights (represented in binary notation) to the case of a multigraph in which each edge has weight 1. Hence, for the sequel, we will take the input to consist of an unweighted multigraph, though our notation, e.g., "{x, y} ∈ E," won't necessarily reflect the fact that parallel edges are allowed.…”

“…First, we write a vector (semidefinite) program which is similar to the linear program written by Even et al [14], but we enhance it by adding some so-called "triangleinequality" constraints. Adding these new constraints enables us to use the seminal results of [4] (and its subsequent paper by Lee [21]) to show that the solution of the semi-definite relaxation can be rounded to a solution for Minimum Linear Arrangement whose cost is larger by at most a factor of O( √ log n log log n).…”

“…Here is why (1), (2), and (3) Motivated by the linear program in [14], our exponentially large SDP is solvable in polynomial time, for there is a separation oracle. Indeed, it's the same separation oracle used in [14]: For each vertex x, order the vertices in order of increasing distance from x.…”