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The Geometry of Scheduling
Nikhil Bansal, Kirk Pruhs
TL;DR
Scheduling with arbitrary job release times, sizes, and non-decreasing completion costs includes several natural objectives, while important flow-based cases have limited approximation guarantees. The paper reduces scheduling to geometric set cover and gives a randomized polynomial-time O(log log nP)-approximation, with an O(1)-approximation when release times are identical.
Problem
For flow-based scheduling objectives, approximation guarantees were limited: no nontrivial ratios were previously known for flow time squared or weighted tardiness with arbitrary release dates.
Source: §1 · p. 2Method
The paper reduces scheduling to capacitated geometric set cover and rounds a strengthened LP using knapsack cover inequalities and geometric covering techniques.
Source: §1.1 · pp. 2–3Results
The randomized algorithm gives a polynomial-time O(log log nP)-approximation for GSP and an O(1)-approximation when all release times are identical.
Source: §1.1 · p. 2, §1.2 · p. 3Takeaways & Limitations
The geometric reduction connects scheduling to weighted geometric set-cover techniques, including quasi-uniform sampling for instances with low union complexity.
Source: §1.1 · pp. 2–3, §1.3 · p. 4
Abstract
from arXiv · showhide
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.
1 Introduction
The introduction frames scheduling with arbitrary release times, job sizes, and non-decreasing job costs, then presents approximation algorithms for the general problem and identical-release case. Its main approach maps scheduling to geometric set cover and applies geometric covering methods.
1 Introduction
- GSP allows arbitrary job-specific costs that are non-decreasing in completion time, so completing a job earlier cannot increase its cost.The introduction notes that the formulation includes weighted flow, weighted tardiness, and flow time squared.Source: §1 · pp. 1–2
1.3 Related Results
- The paper situates its contribution amid limited approximation guarantees for flow-based objectives and geometric set-cover methods designed to improve the O(log n) bound.It notes that no nontrivial approximation ratios were previously known for flow time squared or weighted tardiness with arbitrary release dates.Source: §1 · p. 2, §1.3 · p. 4
1.1 Our Results
- A randomized polynomial-time algorithm approximates GSP within O(log log nP), with an O(1)-approximation when all release times are identical.The stated running time is polynomial in n, log P, and log W, provided the times when a weight function doubles can be determined in polynomial time.Source: §1.1 · p. 2
- The reduction maps GSP to capacitated geometric set cover R2C with only a constant-factor loss, then strengthens its LP with knapsack cover inequalities.The rounding approach further reduces the problem to priority set cover and set multicover instances.Source: §1.1 · p. 2
- For heavy points, the reduction uses R3U instances with union complexity O(k log P), enabling an O(log log nP)-approximation via Varadarajan’s quasi-uniform sampling.Heavy points are those extensively covered in the LP by rectangles whose capacity exceeds the point’s demand.Source: §1.1 · p. 3
- For light points, the method solves log P weighted geometric multicover instances with local ratio, achieving total cost O(log log nP) times the LP cost.Each instance corresponds to one possible job size.Source: §1.1 · p. 3
1.2 Identical Release Times
- With identical release times, the geometric instances become generalized caching, for which a polynomial-time 4-approximation yields an O(1)-approximation for GSP.Source: §1.2 · p. 3
2 The Reduction from GSP to R2C
The reduction maps scheduling instances to R2C with a fourfold objective-value loss in one direction and no shrinkage in the other. Consequently, an R2C α-approximation yields a 4α-approximation for GSP.
2 The Reduction from GSP to R2C
- A polynomial-time α-approximation for R2C gives a polynomial-time 4α-approximation for GSP.The reduction’s objective-value increase is bounded by four, while the reverse conversion does not increase cost.Source: §2 · pp. 4–5
- Each job’s cost classes become weighted rectangles of capacity equal to its size, while time intervals become points demanding excess released job size over interval length.The rectangle weights are 2^k − 1, and point demands are max(0, P(X) − |X|).Source: §2 · p. 5
- With identical release times, the reduction becomes generalized caching, for which a polynomial-time 4-approximation yields an O(1)-approximation for GSP.In this case, relevant intervals begin at zero and have demand equal to total job size minus the interval endpoint.Source: §2 · p. 6
3 The LP Formulation for R2C
The R2C integer program selects weighted rectangles to satisfy point demands. Knapsack cover inequalities strengthen its LP relaxation, which is then simplified and divided into heavy- and light-point cases.
3 The LP Formulation for R2C
- The integer program uses one indicator variable per rectangle to select rectangles meeting each point’s demand, but its natural LP relaxation can have a large integrality gap.The gap persists even for a single point, where the problem is knapsack cover.Source: §3 · p. 6
- Knapsack cover inequalities require remaining rectangles to cover the demand left after any chosen subset’s capacity is accounted for.A feasible (1 + ε)-approximate LP solution can be found with the Ellipsoid algorithm, with the constraints satisfied exactly.Source: §3 · p. 7
- The rounding preparation selects rectangles whose LP values are at least β, updates residual demands, scales the remaining solution, and classifies points as heavy or light.The heavy and light cases are handled by different algorithms.Source: §3 · pp. 7–8
4 Covering Heavy Points
Heavy points are covered by reducing their geometric covering problem to R3U. Low union complexity and quasi-uniform sampling yield the stated approximation guarantee.
4 Covering Heavy Points
- The reduction represents heavy points and eligible rectangles as points and cuboids in R3U, preserving the correspondence between covers.Each cuboid has the rectangle’s weight and extends through the capacity dimension.Source: §4 · p. 8
- The union complexity of any k cuboids in the resulting instances is O(k log P), and the paper notes this bound is tight for these cuboids.The bound uses the O(k) union complexity of the underlying axis-parallel rectangles and O(log P) distinct heights.Source: §4 · p. 8
- O(log log nP) approximation suffices to cover all heavy points in polynomial time.The bound follows from applying Varadarajan’s quasi-uniform sampling technique to the resulting R3U instances.Source: §4 · pp. 8–9
5 Covering Light Points
The light-point problem is decomposed into capacity-class multicovers. A logarithmic-demand reduction and local-ratio set-cover rounding provide the stated approximation approach.
5 Covering Light Points
- O(log log m) = O(log log nP) approximation is obtained for an R2M instance, using a 2-approximation for set cover relative to its fractional solution.The paper also notes that adapting quasi-uniform sampling directly to multicovers could give better results, but uses the simpler approach.Source: §5 · pp. 9–10
- The light-point problem is split into log P R2M instances, one for each rectangle-capacity class, and their selected rectangles together satisfy the original light-point demands.Each instance uses the original points and rectangles of its designated capacity class.Source: §5 · pp. 9–10
- The maximum demand in a multicovers instance can be assumed to be O(log m) at a loss of an O(1) approximation factor.The proof first samples sets with probability min(1, 2x_j), then handles the residual instance.Source: §5 · p. 10
- An LP-based α-approximation for set cover yields an α log d approximation for a multicovers problem with maximum demand d.The reduction proceeds in d rounds, covering elements with their current demand in each round.Source: §5 · p. 10