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Register a new userParallel Online Algorithms for the Bin Packing Problem
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Added by
Arne Schmidt
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We study emph{parallel} online algorithms: For some fixed integer $k$, a collective of $k$ parallel processes that perform online decisions on the same sequence of events forms a $k$-emph{copy algorithm}. For any given time and input sequence, the overall performance is determined by the best of the $k$ individual total results. Problems of this type have been considered for online makespan minimization; they are also related to optimization with emph{advice} on future events, i.e., a number of bits available in advance. We develop textsc{Predictive Harmonic}$_3$ (PH3), a relatively simple family of $k$-copy algorithms for the online Bin Packing Problem, whose joint competitive factor converges to 1.5 for increasing $k$. In particular, we show that $k=6$ suffices to guarantee a factor of $1.5714$ for PH3, which is better than $1.57829$, the performance of the best known 1-copy algorithm textsc{Advanced Harmonic}, while $k=11$ suffices to achieve a factor of $1.5406$, beating the known lower bound of $1.54278$ for a single online algorithm. In the context of online optimization with advice, our approach implies that 4 bits suffice to achieve a factor better than this bound of $1.54278$, which is considerably less than the previous bound of 15 bits.
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In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio $rho$ of the online bounded space variant is $Omega(log d)$ and $O(d/log d)$, and conjectured that it is $Theta(log d)$. We show that $rho$ is in fact $Theta(d/log d)$, using probabilistic arguments.
We consider the online problem of packing circles into a square container. A sequence of circles has to be packed one at a time, without knowledge of the following incoming circles and without moving previously packed circles. We present an algorithm that packs any online sequence of circles with a combined area not larger than 0.350389 0.350389 of the squares area, improving the previous best value of {pi}/10 approx 0.31416; even in an offline setting, there is an upper bound of {pi}/(3 + 2 sqrt{2}) approx 0.5390. If only circles with radii of at least 0.026622 are considered, our algorithm achieves the higher value 0.375898. As a byproduct, we give an online algorithm for packing circles into a 1times b rectangle with b geq 1. This algorithm is worst case-optimal for b geq 2.36.
We tackle the Online 3D Bin Packing Problem, a challenging yet practically useful variant of the classical Bin Packing Problem. In this problem, the items are delivered to the agent without informing the full sequence information. Agent must directly pack these items into the target bin stably without changing their arrival order, and no further adjustment is permitted. Online 3D-BPP can be naturally formulated as Markov Decision Process (MDP). We adopt deep reinforcement learning, in particular, the on-policy actor-critic framework, to solve this MDP with constrained action space. To learn a practically feasible packing policy, we propose three critical designs. First, we propose an online analysis of packing stability based on a novel stacking tree. It attains a high analysis accuracy while reducing the computational complexity from $O(N^2)$ to $O(N log N)$, making it especially suited for RL training. Second, we propose a decoupled packing policy learning for different dimensions of placement which enables high-resolution spatial discretization and hence high packing precision. Third, we introduce a reward function that dictates the robot to place items in a far-to-near order and therefore simplifies the collision avoidance in movement planning of the robotic arm. Furthermore, we provide a comprehensive discussion on several key implemental issues. The extensive evaluation demonstrates that our learned policy outperforms the state-of-the-art methods significantly and is practically usable for real-world applications.
The bin covering problem asks for covering a maximum number of bins with an online sequence of $n$ items of different sizes in the range $(0,1]$; a bin is said to be covered if it receives items of total size at least 1. We study this problem in the advice setting and provide tight bounds for the size of advice required to achieve optimal solutions. Moreover, we show that any algorithm with advice of size $o(log log n)$ has a competitive ratio of at most 0.5. In other words, advice of size $o(log log n)$ is useless for improving the competitive ratio of 0.5, attainable by an online algorithm without advice. This result highlights a difference between the bin covering and the bin packing problems in the advice model: for the bin packing problem, there are several algorithms with advice of constant size that outperform online algorithms without advice. Furthermore, we show that advice of size $O(log log n)$ is sufficient to achieve a competitive ratio that is arbitrarily close to $0.53bar{3}$ and hence strictly better than the best ratio $0.5$ attainable by purely online algorithms. The technicalities involved in introducing and analyzing this algorithm are quite different from the existing results for the bin packing problem and confirm the different nature of these two problems. Finally, we show that a linear number of bits of advice is necessary to achieve any competitive ratio better than 15/16 for the online bin covering problem.
We introduce and study a general version of the fractional online knapsack problem with multiple knapsacks, heterogeneous constraints on which items can be assigned to which knapsack, and rate-limiting constraints on the assignment of items to knapsacks. This problem generalizes variations of the knapsack problem and of the one-way trading problem that have previously been treated separately, and additionally finds application to the real-time control of electric vehicle (EV) charging. We introduce a new algorithm that achieves a competitive ratio within an additive factor of one of the best achievable competitive ratios for the general problem and matches or improves upon the best-known competitive ratio for special cases in the knapsack and one-way trading literatures. Moreover, our analysis provides a novel approach to online algorithm design based on an instance-dependent primal-dual analysis that connects the identification of worst-case instances to the design of algorithms. Finally, we illustrate the proposed algorithm via trace-based experiments of EV charging.
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