The Geometry of Scheduling

We consider the following general scheduling problem: The input consists of n jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as weighted flow, weighted tardiness, and sum of flow squared. Our main result is a randomized polynomial-time algorithm with an approximation ratio O(log log nP), where P is the maximum job size. We also give an O(1) approximation in the special case when all jobs have identical release times. The main idea is to reduce this scheduling problem to a particular geometric set-cover problem which is then solved using the local ratio technique and Varadarajan's quasi-uniform sampling technique. This general algorithmic approach improves the best known approximation ratios by at least an exponential factor (and much more in some cases) for essentially all of the nontrivial common special cases of this problem. Our geometric interpretation of scheduling may be of independent interest.Comment: Conference version in FOCS 201

Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time

We give a nearly linear time randomized approximation scheme for the Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an undirected edge-weighted graph $G$ on $m$ edges and $\epsilon > 0$, the algorithm outputs in $O(m \log^4n /\epsilon^2)$ time, with high probability, a $(1+\epsilon)$-approximation to the Held-Karp bound on the metric TSP instance induced by the shortest path metric on $G$. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the $O(m^2 \log^2(m)/\epsilon^2)$ running time achieved previously by Garg and Khandekar. The LP solution can be used to obtain a fast randomized $\big(\frac{3}{2} + \epsilon\big)$-approximation for metric TSP which improves upon the running time of previous implementations of Christofides' algorithm

Data-Collection for the Sloan Digital Sky Survey: a Network-Flow Heuristic

The goal of the Sloan Digital Sky Survey is ``to map in detail one-quarter of the entire sky, determining the positions and absolute brightnesses of more than 100 million celestial objects''. The survey will be performed by taking ``snapshots'' through a large telescope. Each snapshot can capture up to 600 objects from a small circle of the sky. This paper describes the design and implementation of the algorithm that is being used to determine the snapshots so as to minimize their number. The problem is NP-hard in general; the algorithm described is a heuristic, based on Lagriangian-relaxation and min-cost network flow. It gets within 5-15% of a naive lower bound, whereas using a ``uniform'' cover only gets within 25-35%.Comment: proceedings version appeared in ACM-SIAM Symposium on Discrete Algorithms (1998