[Paper Review] Symmetry exploitation for Online Machine Covering with Bounded Migration

Waldo Gálvez, José A. Soto+1 more|arXiv (Cornell University)|Dec 6, 2016

Optimization and Search Problems6 citations

TL;DR

This paper presents a novel symmetry-exploiting rounding technique to design competitive online algorithms for machine covering with bounded migration. It achieves a (4/3 + ε)-competitive ratio using a migration factor of Õ(1/ε³), significantly improving upon prior results by carefully managing job reassignments through structural properties of geometrically rounded job sets.

ABSTRACT

Online models that allow recourse are highly effective in situations where classical models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this setting, jobs arrive one by one and must be assigned to machines with the objective of maximizing the minimum machine load. When a job arrives, we are allowed to reassign some jobs as long as their total size is (at most) proportional to the processing time of the arriving job. The proportionality constant is called the migration factor of the algorithm. Using a rounding procedure with useful structural properties for online packing and covering problems, we design first a simple $(1.7 + \varepsilon)$-competitive algorithm using a migration factor of $O(1/\varepsilon)$ which maintains at every arrival a locally optimal solution with respect to the Jump neighborhood. After that, we present as our main contribution a more involved $(4/3+\varepsilon)$-competitive algorithm using a migration factor of $ ilde{O}(1/\varepsilon^3)$. At every arrival, we run an adaptation of the Largest Processing Time first (LPT) algorithm. Since the new job can cause a complete change of the assignment of smaller jobs in both cases, a low migration factor is achieved by carefully exploiting the highly symmetric structure obtained by the rounding procedure.

Motivation & Objective

• To close the gap between the best-known competitive ratio and the theoretical lower bound for online machine covering with bounded migration.
• To develop a migration-efficient algorithm that maintains high competitiveness despite dynamic job arrivals.
• To exploit structural symmetry in geometrically rounded job instances to minimize reassignment costs.
• To improve upon the previous best competitive ratio of 2 for constant migration factors.

Proposed method

• Introduces a geometric rounding procedure that preserves structural symmetry in job processing times within [εOPT, OPT].
• Uses a locally optimal solution in the Jump neighborhood to maintain competitiveness with low migration.
• Applies an adapted Largest Processing Time (LPT) rule after each job arrival to reassign jobs efficiently.
• Employs a multi-set load uniqueness lemma to ensure distinct total loads for different job combinations, enabling efficient state tracking.
• Leverages amortized analysis and symmetry to bound migration costs despite potential large reassignments.
• Designs a migration factor of Õ(1/ε³) by carefully controlling how many jobs can be migrated based on the size of the incoming job.

Experimental results

Research questions

• RQ1 Can a competitive ratio below 2 be achieved with a bounded migration factor in online machine covering?
• RQ2 How can symmetry in rounded job instances be exploited to reduce migration costs?
• RQ3 Is it possible to achieve a (4/3 + ε)-competitive ratio with a sub-polynomial migration factor in the bounded migration model?
• RQ4 What structural properties of job sets allow for efficient reassignment with low migration?
• RQ5 Can the lower bound of 20/19 for constant migration be improved, and if so, under what conditions?

Key findings

• The paper presents a (4/3 + ε)-competitive algorithm with a migration factor of Õ(1/ε³), significantly improving upon the previous best competitive ratio of 2.
• The algorithm maintains local optimality in the Jump neighborhood at every job arrival, ensuring high competitiveness.
• A novel geometric rounding technique ensures that distinct multi-sets of jobs with total load ≤ OPT have distinct total sums, enabling efficient state representation.
• The method achieves low migration by exploiting symmetry in rounded job sets, preventing exponential state explosion.
• The paper improves the lower bound for constant migration from 20/19 to 17/16, showing that no (17/16 − ε)-competitive algorithm exists with constant migration.
• The results demonstrate that a (1 + ε)-competitive ratio is not achievable with constant migration, but a near-optimal (4/3 + ε) ratio is possible with Õ(1/ε³) migration.