[Paper Review] A PTAS for Scheduling with Tree Assignment Restrictions

Ulrich M. Schwarz|arXiv (Cornell University)|Sep 23, 2010

Scheduling and Optimization Algorithms6 references3 citations

TL;DR

This paper presents a Polynomial-Time Approximation Scheme (PTAS) for scheduling jobs with tree-hierarchical assignment restrictions, where each job can only be assigned to machines along a path from a node to the root in a rooted tree. By combining geometric rounding of job sizes, configuration-based dynamic programming, and hierarchical bottom-up processing, the algorithm achieves a $(1+4\epsilon)$-approximation in time polynomial in the input size and exponential in $1/\epsilon$, improving upon the prior $4/3$-approximation.

ABSTRACT

Scheduling with assignment restrictions is an important special case of scheduling unrelated machines which has attracted much attention in the recent past. While a lower bound on approximability of 3/2 is known for its most general setting, subclasses of the problem admit polynomial-time approximation schemes. This note provides a PTAS for tree-like hierarchical structures, improving on a recent 4/3-approximation by Huo and Leung.

Motivation & Objective

• To close the gap between the known $4/3$-approximation and the optimal solution for scheduling with tree-hierarchical assignment restrictions.
• To develop a polynomial-time approximation scheme (PTAS) for this special case of unrelated machine scheduling with assignment constraints.
• To extend the class of assignment structures for which a PTAS exists beyond nested and interval restrictions.
• To show that tree-hierarchical structures admit arbitrarily good approximations, despite being NP-hard.

Proposed method

• Rounding all large jobs (larger than $\epsilon C$) to sizes of the form $\epsilon C \cdot (1+\epsilon)^k$ to reduce the number of distinct sizes to $O(\log_{1+\epsilon} 1/\epsilon)$, a constant depending only on $\epsilon$.
• Representing job subsets at each node by configuration tuples: counts of large job sizes and total size of small jobs in multiples of $\epsilon C$, rounded up.
• Using a bottom-up dynamic programming approach on the tree, where each node aggregates feasible configurations from its children and combines them with its own job set.
• For each node, maintaining a set of feasible configuration combinations that can be pushed upward, including the option to schedule a subconfiguration on the node and push the remainder up.
• Handling small jobs greedily by filling available space up to the next multiple of $\epsilon C$, accepting a maximum $\epsilon C$ overage per machine.
• Using binary search on the makespan $C$ and verifying feasibility via the dynamic programming procedure to achieve a $(1+4\epsilon)$-approximate schedule.

Experimental results

Research questions

• RQ1 Can a PTAS be designed for scheduling with tree-hierarchical assignment restrictions, where each job is restricted to machines on a root-to-leaf path in a tree?
• RQ2 Does the tree-hierarchical structure allow for better approximation guarantees than the previously known $4/3$-approximation by Huo and Leung?
• RQ3 Is it possible to achieve arbitrarily good approximation ratios for this problem class, despite its NP-hardness?
• RQ4 Can the configuration-based dynamic programming approach be adapted to hierarchical tree structures with bounded job size rounding?

Key findings

• The paper establishes a PTAS for scheduling with tree-hierarchical assignment restrictions, achieving a $(1+4\epsilon)$-approximation for any $\epsilon > 0$.
• The running time is polynomial in the input size and exponential in $1/\epsilon$, satisfying the definition of a PTAS.
• The approach uses geometric rounding of large jobs and configuration-based dynamic programming on the tree structure to maintain feasibility and approximation guarantees.
• Small jobs are handled by greedy filling of space in multiples of $\epsilon C$, introducing at most an additional $\epsilon C$ per machine, which is bounded within the overall approximation factor.
• The algorithm correctly identifies a feasible schedule of length at most $(1+4\epsilon)C$ whenever a schedule of length $C$ exists, via a relaxed decision procedure.
• The result improves upon the prior $4/3$-approximation by Huo and Leung, and shows that tree-hierarchical structures admit arbitrarily good approximations.