Technology Category
Constraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Combinatorial OptimizationPlanning, Routing, and Scheduling: Scheduling under Uncertainty
Improving the Parameter Dependency for High-Multiplicity Scheduling on Uniform Machines
🤖 AI Summary
This paper investigates the parameterized complexity of high-multiplicity scheduling on uniform machines, focusing on classical objectives—makespan minimization (Q||C_max), Santa Claus (Q||C_min), and envy minimization (Q||C_envy)—formulated as n-fold integer programs. We propose a generalized load-feasibility testing method based on a divide-and-conquer framework, integrating structural analysis of machine speeds, optimized multiplicity encoding, and high-dimensional dynamic programming. Our approach reduces the algorithm’s parameter dependence on the maximum processing time from p_max^{O(d²)} to p_max^{O(d)} under reasonable speed assumptions—partially resolving Koutecký–Zink’s open question on improving the d-dependence from quadratic to linear. The resulting polynomial-time algorithms achieve the current best parameter bounds for all three problems, significantly advancing theoretical efficiency in high-multiplicity uniform-machine scheduling.
Technology Category
Application Category
📝 Abstract
We address scheduling problems on uniform machines with high-multiplicity encoding, introducing a divide and conquer approach to assess the feasibility of a general Load Balancing Problem (LBP). Via reductions, our algorithm can also solve the more well-known problems $Q|C_{max}$ (makespan minimization), $Q|C_{min}$ (santa claus) and $Q|C_{ ext{envy}}$ (envy minimization). State-of-the-art algorithms for these problems, e.g. by Knop et al. (Math. Program. '23), have running times with parameter dependency $p_{max}^{O(d^2)}$, where $p_{max}$ is the largest processing time and $d$ is the number of different processing times. We partially answer the question asked by Kouteck'y and Zink (ISAAC'20) about whether this quadratic dependency of $d$ can be improved to a linear one: Under the natural assumption that the machines are similar in a way that $s_{max}/s_{min} leq p_{max}^{O(1)}$ and $ auleq p_{max}^{O(1)}$, our proposed algorithm achieves parameter dependency $p_{max}^{O(d)}$ for the problems ${Q|{C_{max},C_{min},C_{ ext{envy}}}}$. Here, $ au$ is the number of distinct machine speeds. Even without this assumption, our running times achieve a state-of-the-art parameter dependency and do so with an entirely different approach.
Problem
Research questions and friction points this paper is trying to address.
Optimizing high-multiplicity scheduling on uniform machines
Solving block-structured ILPs with specific matrix constraints
Improving parameter dependency for graph imbalance problems
Innovation
Methods, ideas, or system contributions that make the work stand out.
Divide and conquer for n-fold ILPs
Bounded right-hand sides calculation
Efficient combination of smaller solutions
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