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Since 2000, the field of machine scheduling—an integral part of computer science and operations research—has seen significant advancements. This paper explores the dynamic progression of machine scheduling, offering a detailed overview of its past advancements, current practices, and future directions. Anchoring the research in robust data analysis and statistical methodologies, the paper reveals the subtle yet impactful changes that have characterized the field in the last two decades. It examines the prominence of various scheduling problems, identifies leading research journals, and highlights international contributions and collaborations, thereby offering a thorough guide to the machine scheduling ecosystem. The study delves into specific problem characteristics and assesses performance criteria and solution methods to provide an in-depth view of the field's multifaceted nature. Ultimately, this paper captures the essence of machine scheduling's evolution and suggests new paths for exploration. The insights gained contribute significantly to academic discussions and equip practitioners with a comprehensive understanding of the dynamic landscape of machine scheduling. [ABSTRACT FROM AUTHOR]

Fair division of indivisible resources is a fundamental problem in many disciplines, including computer science, economy, operations research, etc. The existence of PMMS allocations of goods is a major open problem in fair division, even for additive valuations. At the state of the art, there is no identified setting where PMMS allocations are deemed impossible, and the known existing results regarding their existence are limited to highly constrained settings. In this paper, we provide an algorithm for computing a PMMS allocation for identical variant. We also use the algorithm to find the PMMS allocations for 3 agents in line graphs. Moreover, we show that a 4 5 -PMMS allocation can be computed in polynomial time when agents agree on the ordinal ranking of the goods. A quantitative gauge for assessing the impact on social welfare is known as the price of fairness , which quantifies the maximum possible reduction in social welfare due to the imposition of fairness constraints. Originally explored in the context of divisible goods, the concept of the price of fairness continues to be a pivotal metric for evaluating the effect of fairness considerations on overall social welfare. Consequently, we study the efficiency decrease under PMMS allocations and prove a tight upper bound on the price of PMMS allocations. • We present an algorithm to compute a PMMS allocation if agents' valuation functions are identical. • We analyze an existing algorithm terminates with a 4 5 -PMMS allocation in polynomial time when agents have identical ordinal ranking values of the goods. • We also examine the efficiency loss associated with PMMS allocations and establish their precise price bound. [ABSTRACT FROM AUTHOR]