Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2408.14310

The Power of Proportional Fairness for Non-Clairvoyant Scheduling under Polyhedral Constraints

Sven Jäger·Alexander Lindermayr·Nicole Megow
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Polytope Scheduling Problem (PSP) was introduced by Im, Kulkarni, and Munagala (JACM 2018) as a very general abstraction of resource allocation over time and captures many well-studied problems including classical unrelated machine scheduling, multidimensional scheduling, and broadcast scheduling. In PSP, jobs with different arrival times receive processing rates that are subject to arbitrary packing constraints. An elegant and well-known algorithm for instantaneous rate allocation with good fairness and efficiency properties is the Proportional Fairness algorithm (PF), which was analyzed for PSP by Im et al. We drastically improve the analysis of the PF algorithm for both the general PSP and several of its important special cases subject to the objective of minimizing the sum of weighted completion times. We reduce the upper bound on the competitive ratio from 128 to 27 for general PSP and to 4 for the prominent class of monotone PSP. For certain heterogeneous machine environments we even close the substantial gap to the lower bound of 2 for non-clairvoyant scheduling. Our analysis also gives the first polynomial-time improvements over the nearly 30-year-old bounds on the competitive ratio of the doubling framework by Hall, Shmoys, and Wein (SODA 1996) for clairvoyant online preemptive scheduling on unrelated machines. Somewhat surprisingly, we achieve this improvement by a non-clairvoyant algorithm, thereby demonstrating that non-clairvoyance is not a (significant) hurdle. Our improvements are based on exploiting monotonicity properties of PSP, providing tight dual fitting arguments on structured instances, and showing new additivity properties on the optimal objective value for scheduling on unrelated machines. Finally, we establish new connections of PF to matching markets, and thereby provide new insights on equilibria and their computational complexity.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Sven Jäger, Alexander Lindermayr, Nicole Megow. 2024-08-26. The Power of Proportional Fairness for Non-Clairvoyant Scheduling under Polyhedral Constraints. https://arxiv.org/abs/2408.14310

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗