[Paper Review] Scheduling of unit-length jobs with cubic incompatibility graphs on three uniform machines

Hanna Furmańczyk, Marek Kubale|arXiv (Cornell University)|Feb 14, 2015

Scheduling and Optimization Algorithms11 references3 citations

TL;DR

This paper studies scheduling unit-length jobs on three uniform machines with cubic incompatibility graphs, where each job conflicts with exactly three others. It presents an O(n²)-time algorithm for 2-chromatic graphs and an O(n³)-time 4/3-approximation algorithm for 3-chromatic graphs, which is optimal almost surely when 3s₁/4 ≤ s₂ = s₃.

ABSTRACT

In the paper we consider the problem of scheduling $n$ identical jobs on 3 uniform machines with speeds $s_1, s_2,$ and $s_3$ to minimize the schedule length. We assume that jobs are subjected to some kind of mutual exclusion constraints, modeled by a cubic incompatibility graph. We show that if the graph is 2-chromatic then the problem can be solved in $O(n^2)$ time. If the graph is 3-chromatic, the problem becomes NP-hard even if $s_1>s_2=s_3$. However, in this case there exists a $4/3$-approximation algorithm running in $O(n^3)$ time. Moreover, this algorithm solves the problem almost surely to optimality if $3s_1/4 \leq s_2 = s_3$.

Motivation & Objective

• To address scheduling of n identical unit-length jobs on three uniform machines with mutual exclusion constraints modeled by a cubic incompatibility graph.
• To determine the computational complexity of the problem based on the chromatic number of the incompatibility graph.
• To develop efficient algorithms for 2-chromatic and 3-chromatic cubic graphs, with performance guarantees.
• To establish conditions under which the proposed approximation algorithm achieves optimal solutions with high probability.
• To extend the model to disconnected cubic graphs and analyze the feasibility of optimal scheduling in such cases.

Proposed method

• For 2-chromatic graphs, use a polynomial-time algorithm based on equitable 2-coloring of cubic graphs, leveraging that such graphs are bipartite with equal-sized partitions.
• For 3-chromatic graphs, apply a 4/3-approximation algorithm that first finds a large independent set via a greedy approach, then transforms it into a semi-equitable coloring using a vertex-swapping procedure.
• Use the FKR algorithm (Furmańczyk-Kubale-Radziszowski) to iteratively improve the coloring by swapping vertices between independent sets, ensuring balanced load distribution.
• Analyze schedule length using machine speeds s₁, s₂, s₃, with the objective of minimizing C_max = max{P_i(I_i)} where P_i(I_i) = |I_i|/s_i.
• Establish performance bounds by comparing the algorithm’s schedule length to the optimal C_max*, using inequalities derived from independence number bounds and equitable coloring properties.
• Leverage probabilistic results from Frieze and Suen on large independent sets in random cubic graphs to prove almost-sure optimality under specific speed ratios.

Experimental results

Research questions

• RQ1 Can the scheduling problem with unit-length jobs and cubic incompatibility graphs be solved optimally in polynomial time when the graph is 2-chromatic?
• RQ2 What is the computational complexity of the problem when the incompatibility graph is 3-chromatic, and under what conditions is it NP-hard?
• RQ3 Can a 4/3-approximation algorithm be designed for 3-chromatic cubic graphs on three uniform machines, and what is its time complexity?
• RQ4 Under what speed conditions does the approximation algorithm achieve an optimal solution with high probability?
• RQ5 How can the algorithms be extended to handle disconnected cubic graphs, and what is the computational complexity in such cases?

Key findings

• The problem is solvable in O(n²) time when the incompatibility graph is 2-chromatic, as such graphs are bipartite and allow equitable 2-coloring.
• The problem becomes NP-hard when the graph is 3-chromatic, even if s₁ > s₂ = s₃, due to the inherent complexity of equitable 3-coloring in cubic graphs.
• An O(n³)-time 4/3-approximation algorithm is proposed for 3-chromatic graphs, guaranteeing a solution within 4/3 of the optimal schedule length.
• When 3s₁/4 ≤ s₂ = s₃, the algorithm returns an optimal solution with high probability, as large independent sets (≥0.432n) are found in almost all cubic graphs.
• The performance guarantee of 4/3 holds regardless of the speed ratio, provided s₁ > 0, by bounding the ratio of algorithmic to optimal schedule length using independence number and equitable coloring bounds.
• The algorithms can be extended to disconnected cubic graphs: Algorithm 1 for 2-chromatic cases and Algorithm 2 for 3-chromatic cases, both running in O(n²) and O(n³) time, respectively.