[Paper Review] Approximation algorithms for two-machine flow-shop scheduling with a conflict graph

Yinhui Cai, Guangting Chen+5 more|arXiv (Cornell University)|Mar 7, 2018

Scheduling and Optimization Algorithms3 citations

TL;DR

This paper presents approximation algorithms for two-machine flow-shop scheduling with job conflicts modeled as a conflict graph. For unit jobs, it introduces a 4/3-approximation algorithm based on a polynomial-time solvable variant of the path cover problem; for conflict graphs that are the union of two disjoint cliques, it proposes a 3/2-approximation algorithm using job aggregation and Johnson's rule, both with tight performance guarantees.

ABSTRACT

Path cover is a well-known intractable problem that finds a minimum number of vertex disjoint paths in a given graph to cover all the vertices. We show that a variant, where the objective function is not the number of paths but the number of length-$0$ paths (that is, isolated vertices), turns out to be polynomial-time solvable. We further show that another variant, where the objective function is the total number of length-$0$ and length-$1$ paths, is also polynomial-time solvable. Both variants find applications in approximating the two-machine flow-shop scheduling problem in which job processing has constraints that are formulated as a conflict graph. For the unit jobs, we present a $4/3$-approximation algorithm for the scheduling problem with an arbitrary conflict graph, based on the exact algorithm for the variants of the path cover problem. For the arbitrary jobs while the conflict graph is the union of two disjoint cliques, that is, all the jobs can be partitioned into two groups such that the jobs in a group are pairwise conflicting, we present a simple $3/2$-approximation algorithm.

Motivation & Objective

• To develop efficient approximation algorithms for two-machine flow-shop scheduling with job conflict constraints.
• To address the challenge of minimizing makespan when jobs cannot be processed concurrently due to pairwise conflicts.
• To explore special cases where the conflict graph is a union of two disjoint cliques or consists of unit-length jobs.
• To establish tight approximation ratios for these special cases through novel graph-theoretic techniques.
• To bridge the gap between intractable path cover variants and practical scheduling approximation algorithms.

Proposed method

• Introduces a variant of the path cover problem where the objective is to minimize the number of isolated vertices (0-paths) and edges (1-paths), which is shown to be polynomial-time solvable.
• Uses the exact solution of this path cover variant to design a 4/3-approximation algorithm for unit jobs with arbitrary conflict graphs.
• For conflict graphs that are the union of two disjoint cliques, aggregates jobs into two super-jobs based on their conflict group.
• Applies Johnson’s algorithm to the two aggregated jobs to produce a schedule with a 3/2-approximation guarantee.
• Proves the performance ratio by comparing the algorithm’s makespan to the optimal makespan, considering all possible processing time orderings.
• Employs combinatorial analysis and case distinctions based on relative processing times to establish the approximation bound.

Experimental results

Research questions

• RQ1 Can the path cover problem be modified such that minimizing the number of 0-paths and 1-paths becomes polynomial-time solvable?
• RQ2 What is the best possible approximation ratio for two-machine flow-shop scheduling with unit jobs and an arbitrary conflict graph?
• RQ3 Can a better approximation ratio be achieved when the conflict graph is the union of two disjoint cliques?
• RQ4 Is the 4/3-approximation ratio for unit jobs asymptotically tight?
• RQ5 Can the idle time in the aggregated job scheduling approach be reduced to improve the approximation ratio?

Key findings

• The variant of the path cover problem that minimizes the number of 0-paths (isolated vertices) is solvable in polynomial time.
• The variant that minimizes the total number of 0-paths and 1-paths (isolated vertices and edges) is also polynomial-time solvable.
• A 4/3-approximation algorithm is achieved for two-machine flow-shop scheduling with unit jobs and an arbitrary conflict graph.
• The 4/3-approximation ratio for unit jobs is asymptotically tight, meaning no better ratio is possible under standard complexity assumptions.
• A 3/2-approximation algorithm is presented for the case where the conflict graph is the union of two disjoint cliques.
• The 3/2-approximation algorithm runs in O(m) time, where m is the number of edges in the conflict graph, and is based on job aggregation and Johnson’s rule.