approximation algorithms for cluster scheduling

This work addresses the problem of minimizing makespan in parallel task scheduling (PTS) and multi-cluster scheduling (MCS). It resolves a long-standing open question in PTS by presenting the first efficient algorithm that achieves the theoretical approximation bound of (4/3)OPT + p_max. For MCS, the study improves the efficiency of the existing 2-approximation algorithm and generalizes the 9/4-approximation algorithm to arbitrary numbers of clusters. Leveraging combinatorial optimization and scheduling theory, the proposed PTS algorithm employs refined greedy strategies and load-balancing analysis, achieving a time complexity of O(n log n). The MCS algorithms maintain strong theoretical guarantees while significantly enhancing practical applicability. Experimental results demonstrate the high efficiency and scalability of the proposed methods.

This paper studies the NP-hard identical parallel machine scheduling problem with machine capacity constraints: given $n$ jobs (each with processing time $p_j$) and $m$ identical machines (each with capacity $c_i$), find a schedule $sigma: J o M$ minimizing the total load cost $sum_{i=1}^m max{c_i, sum_{j in sigma^{-1}(i)} p_j}$. We propose an enhanced First-Fit Decreasing (FFD) algorithm achieving a tight 1.5-approximation ratio, and further design a polynomial-time approximation scheme (PTAS). Our key innovation lies in unifying bin-packing modeling with mixed-criticality scheduling theory, extending it to heterogeneous criticality settings—thereby significantly improving load balancing and approximation accuracy while respecting resource constraints. Theoretical analysis is complemented by empirical validation, establishing a novel paradigm for real-time systems and cloud resource scheduling.

This study addresses the throughput maximization problem for non-preemptive jobs with time windows on both single and multiple machines—a strongly NP-hard scheduling problem. By integrating combinatorial optimization, approximation algorithm design, and pseudo-polynomial time dynamic programming, the authors significantly improve the best-known approximation ratio for the single-machine case from $1.551+\varepsilon$ to $4/3+\varepsilon$, and further refine it to $5/4+\varepsilon$ in pseudo-polynomial time. These results establish the currently best approximation guarantees for this classical scheduling problem and are successfully extended to the multi-machine setting, offering new theoretical insights and algorithmic advances in scheduling under time-window constraints.

We study the preemptive scheduling problem on $m$ identical parallel machines with class-dependent setup times: $n$ jobs are partitioned into $c$ classes, and switching between classes incurs a setup cost $s_i$ for class $i$. The objective is to minimize the makespan. We present the first polynomial-time approximation algorithm with approximation ratio $4/3 + varepsilon$, where $varepsilon < 1/6$, breaking the previous best-known bound of $3/2$. Our approach decomposes instances into “easy” and “hard” cases, proves the existence of a $4/3,T$-structured schedule (where $T$ denotes the optimal makespan), and designs an algorithm combining preprocessing, dynamic adjustment, and structural schedule analysis. The algorithm runs in $mathcal{O}(n^2 log(1/varepsilon))$ time. This result improves both the theoretical guarantee and practical applicability over prior work.

This paper investigates the computational complexity and efficient solvability of high-multiplicity identical parallel-machine scheduling $P||C_{max}$. Addressing the severe dependence of prior algorithms on the optimal makespan $C_{max}$—with runtime $(log C_{max})^{2^{O(d)}}$—and their inefficiency under large job-type multiplicities, we introduce three key techniques: problem-tailored preprocessing, Frank–Tardos compression encoding, and a novel upper-bound analysis based on the number of vertices of the integer convex hull. We establish, for the first time, an FPT-equivalence between $P||C_{max}$ and $Q||C_{max}$ parameterized by the number $d$ of job and machine types. Our algorithm achieves runtime $(log p_{max})^{2^{O(d)}}$, drastically weakening dependence on $C_{max}$. Furthermore, we provide a tight parameterized lower bound, fully resolving the central open question posed by Mnich and van Bevern regarding FPT status of $d$-type scheduling.

This study investigates the computational complexity of stochastic scheduling on identical parallel machines to minimize the expected total weighted completion time. Despite the abundance of approximation algorithms, a rigorous theoretical foundation has long been lacking, with constant-factor approximations known only under strong distributional assumptions. This work establishes, for the first time, that the problem is #P-hard even in the restricted setting of unit weights and two-point processing time distributions, without relying on the hardness of the deterministic counterpart. Specifically, it proves that both deciding whether a scheduling policy exists whose expected cost meets a given threshold and computing the expected objective value of the classical (W)SEPT greedy policy are #P-hard. The result, obtained via a #P-completeness reduction, fills a longstanding gap in the complexity theory of stochastic scheduling for min-sum objectives.

This work addresses the continuous monotonic moldable job scheduling problem—assigning $n$ jobs with variable parallelism to $m$ identical machines to minimize the makespan—and presents the first approximation algorithm that combines strong theoretical guarantees with practical efficiency. Exploiting the non-increasing nature of job processing time with respect to the number of allocated machines, the algorithm integrates combinatorial optimization techniques to achieve a $(73/50 + \varepsilon) \approx 1.4593 + \varepsilon$ approximation ratio in $O(nm \log(1/\varepsilon))$ time. Compared to existing approaches, the proposed method significantly reduces time complexity while empirical evaluations demonstrate that its real-world performance substantially exceeds the theoretical worst-case bound.

This work proposes the first learning-augmented approximation algorithm for the NP-hard unrelated machine scheduling problem to minimize makespan (R||C_max), introducing prediction-guided strategies into scheduling and resolving an open question posed by Antoniadis et al. The algorithm leverages predictions about the assignment of “heavy” jobs: when predictions are accurate, it achieves a (1+ε)-approximation; as prediction error increases, its performance degrades smoothly to a 2-approximation, matching the best-known worst-case bound. This trade-off yields theoretically tight robustness guarantees. Empirical evaluations demonstrate the algorithm’s practical effectiveness in real-world scenarios.

This work addresses the problem of minimizing makespan for scheduling a large set of jobs on $m$ identical machines, proposing two sublinear-time approximation algorithms tailored to scenarios where the total number of jobs is either known or unknown. Leveraging weighted random sampling and an adaptive multi-round sampling strategy, the algorithms efficiently construct an approximately optimal scheduling sketch using only $O(\log n)$ uniformly drawn samples. The approach achieves a $(1+3\varepsilon)$-approximation ratio—the first of its kind—with a running time of $\widetilde{O}(m^5/\varepsilon^4 \cdot \sqrt{n} + A(\lceil m/\varepsilon \rceil, \varepsilon))$, where $A(k, \varepsilon)$ denotes the complexity of a $(1+\varepsilon)$-approximation algorithm for instances of size $k$. This result strikes a favorable balance between theoretical guarantees and practical efficiency.

In the bottleneck multiple knapsack problem, we are given a set of items and a set of knapsacks, where each item has a profit and a weight, and each knapsack has a capacity. Our goal is to assign items to knapsacks so as to maximize the minimum profit received by any knapsack subject to the capacity constraint. When all knapsacks have identical capacity, we give a $(\frac{2}{3} - \varepsilon)$-approximation algorithm for any constant $\varepsilon > 0$. This result almost matches the $(\frac{2}{3} + \varepsilon)$ inapproximability bound for the bottleneck multiple subset sum problem (Caprara et al., 2000). When the knapsacks can have arbitrary capacities, we propose a $(\frac{1}{2} - \varepsilon)$-approximation algorithm for any constant $\varepsilon > 0$. We also prove a hardness bound of $(\frac{1}{2} + \varepsilon)$ for any constant $\varepsilon > 0$.