712 research outputs found
Approximating Bin Packing within O(log OPT * log log OPT) bins
For bin packing, the input consists of n items with sizes s_1,...,s_n in
[0,1] which have to be assigned to a minimum number of bins of size 1. The
seminal Karmarkar-Karp algorithm from '82 produces a solution with at most OPT
+ O(log^2 OPT) bins.
We provide the first improvement in now 3 decades and show that one can find
a solution of cost OPT + O(log OPT * log log OPT) in polynomial time. This is
achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation
using the Entropy Method from discrepancy theory. The result is constructive
via algorithms of Bansal and Lovett-Meka
Adjustable Robust Reinforcement Learning for Online 3D Bin Packing
Designing effective policies for the online 3D bin packing problem (3D-BPP)
has been a long-standing challenge, primarily due to the unpredictable nature
of incoming box sequences and stringent physical constraints. While current
deep reinforcement learning (DRL) methods for online 3D-BPP have shown
promising results in optimizing average performance over an underlying box
sequence distribution, they often fail in real-world settings where some
worst-case scenarios can materialize. Standard robust DRL algorithms tend to
overly prioritize optimizing the worst-case performance at the expense of
performance under normal problem instance distribution. To address these
issues, we first introduce a permutation-based attacker to investigate the
practical robustness of both DRL-based and heuristic methods proposed for
solving online 3D-BPP. Then, we propose an adjustable robust reinforcement
learning (AR2L) framework that allows efficient adjustment of robustness
weights to achieve the desired balance of the policy's performance in average
and worst-case environments. Specifically, we formulate the objective function
as a weighted sum of expected and worst-case returns, and derive the lower
performance bound by relating to the return under a mixture dynamics. To
realize this lower bound, we adopt an iterative procedure that searches for the
associated mixture dynamics and improves the corresponding policy. We integrate
this procedure into two popular robust adversarial algorithms to develop the
exact and approximate AR2L algorithms. Experiments demonstrate that AR2L is
versatile in the sense that it improves policy robustness while maintaining an
acceptable level of performance for the nominal case.Comment: Accepted to NeurIPS202
Minimal proper non-IRUP instances of the one-dimensional Cutting Stock Problem
We consider the well-known one dimensional cutting stock problem (1CSP).
Based on the pattern structure of the classical ILP formulation of Gilmore and
Gomory, we can decompose the infinite set of 1CSP instances, with a fixed
demand n, into a finite number of equivalence classes. We show up a strong
relation to weighted simple games. Studying the integer round-up property we
computationally show that all 1CSP instances with are proper IRUP,
while we give examples of a proper non-IRUP instances with . A gap larger
than 1 occurs for . The worst known gap is raised from 1.003 to 1.0625.
The used algorithmic approaches are based on exhaustive enumeration and integer
linear programming. Additionally we give some theoretical bounds showing that
all 1CSP instances with some specific parameters have the proper IRUP.Comment: 14 pages, 2 figures, 2 table
High-throughput machine learning algorithms
The field of machine learning has become strongly compute driven, such that emerging research and applications require larger amounts of specialised hardware or smarter algorithms to advance beyond the state-of-the-art. This thesis develops specialised techniques and algorithms for a subset of computationally difficult machine learning problems. The applications under investigation are quantile approximation in the limited-memory data streaming setting, interpretability of decision tree ensembles, efficient sampling methods in the space of permutations, and the generation of large numbers of pseudorandom permutations. These specific applications are investigated as they represent significant bottlenecks in real-world machine learning pipelines, where improvements to throughput have significant impact on the outcomes of machine learning projects in both industry and research. To address these bottlenecks, we discuss both theoretical improvements, such as improved convergence rates, and hardware/software related improvements, such as optimised algorithm design for high throughput hardware accelerators.
Some contributions include: the evaluation of bin-packing methods for efficiently scheduling small batches of dependent computations to GPU hardware execution units, numerically stable reduction operators for higher-order statistical moments, and memory bandwidth optimisation for GPU shuffling. Additionally, we apply theory of the symmetric group of permutations in reproducing kernel Hilbert spaces, resulting in improved analysis of Monte Carlo methods for Shapley value estimation and new, computationally more efficient algorithms based on kernel herding and Bayesian quadrature. We also utilise reproducing kernels over permutations to develop a novel statistical test for the hypothesis that a sample of permutations is drawn from a uniform distribution.
The techniques discussed lie at the intersection of machine learning, high-performance computing, and applied mathematics. Much of the above work resulted in open source software used in real applications, including the GPUTreeShap library [38], shuffling primitives for the Thrust parallel computing library [2], extensions to the Shap package [31], and extensions to the XGBoost library [6]
Semidefinite optimization in discrepancy theory
Recently, there have been several new developments in discrepancy theory based on connections to semidefinite programming. This connection has been useful in several ways. It gives efficient polynomial time algorithms for several problems for which only non-constructive results were previously known. It also leads to several new structural results in discrepancy itself, such as tightness of the so-called determinant lower bound, improved bounds on the discrepancy of the union of set systems and so on. We will give a brief survey of these results, focussing on the main ideas and the techniques involved
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-HĂŒbner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro PezzĂ©, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
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