In the Demand Strip Packing problem (DSP), we are given a time interval and a collection of tasks, each characterized by a processing time and a demand for a given resource (such as electricity, computational power, etc.). A feasible solution consists of a schedule of the tasks within the mentioned time interval. Our goal is to minimize the peak resource consumption, i.e. the maximum total demand of tasks executed at any point in time.
It is known that DSP is NP-hard to approximate below a factor 3/2, and standard techniques for related problems imply a (polynomial-time) 2-approximation. Our main result is a (5/3+eps)-approximation algorithm for any constant eps>0. We also achieve best-possible approximation factors for some relevant special cases.
We study the well-known two-dimensional strip packing problem. Given is a set of rectangular axis-parallel items and a strip of width $W$ with infinite height. The objective is to find a packing of these items into the strip, which minimizes the packing height. Lately, it has been shown that the lower bound of $3/2$ of the absolute approximation ratio can be beaten when we allow a pseudo-polynomial running-time of type $(n W)^{f(1/\varepsilon)}$. If $W$ is polynomially bounded by the number of items, this is a polynomial running-time. We present a pseudo-polynomial algorithm with approximation…
We consider the relaxed online strip packing problem: Rectangular items arrive online and have to be packed without rotations into a strip of fixed width such that the packing height is minimized. Thereby, repacking of previously packed items is allowed. The amount of repacking is measured by the migration factor, defined as the total size of repacked items divided by the size of the arriving item. First, we show that no algorithm with constant migration factor can produce solutions with asymptotic ratio better than 4/3. Against this background, we allow amortized migration, i.e. to save migra…
In the Strip Packing problem (SP), we are given a vertical half-strip $[0,W]\times[0,\infty)$ and a set of $n$ axis-aligned rectangles of width at most $W$. The goal is to find a non-overlapping packing of all rectangles into the strip such that the height of the packing is minimized. A well-studied and frequently used practical constraint is to allow only those packings that are guillotine separable, i.e., every rectangle in the packing can be obtained by recursively applying a sequence of edge-to-edge axis-parallel cuts (guillotine cuts) that do not intersect any item of the solution. In thi…
We study Nonpreemptive Peak Demand Minimization (NPDM) problem, where we are given a set of jobs, specified by their processing times and energy requirements. The goal is to schedule all jobs within a fixed time period such that the peak load (the maximum total energy requirement at any time) is minimized. This problem has recently received significant attention due to its relevance in smart-grids. Theoretically, the problem is related to the classical strip packing problem (SP). In SP, a given set of axis-aligned rectangles must be packed into a fixed-width strip, such that the height of the …
Strip packing is a classical packing problem, where the goal is to pack a set of rectangular objects into a strip of a given width, while minimizing the total height of the packing. The problem has multiple applications, e.g. in scheduling and stock-cutting, and has been studied extensively. When the dimensions of objects are allowed to be exponential in the total input size, it is known that the problem cannot be approximated within a factor better than $3/2$, unless $\mathrm{P}=\mathrm{NP}$. However, there was no corresponding lower bound for polynomially bounded input data. In fact, Nadirad…
We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of a fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3/2 is possible for strip packing in polynomial time (assuming P$\neq$NP). Nadiradze and Wiese [SODA16] overcame th…
Iterative rounding has enjoyed tremendous success in elegantly resolving open questions regarding the approximability of problems dominated by covering constraints. Although iterative rounding methods have been applied to packing problems, no single method has emerged that matches the effectiveness and simplicity afforded by the covering case. We offer a simple iterative packing technique that retains features of Jain's seminal approach, including the property that the magnitude of the fractional value of the element rounded during each iteration has a direct impact on the approximation guaran…
We study the Multiple Cluster Scheduling problem and the Multiple Strip Packing problem. For both problems, there is no algorithm with approximation ratio better than $2$ unless $P = NP$. In this paper, we present an algorithm with approximation ratio $2$ and running time $O(n)$ for both problems. While a $2$ approximation was known before, the running time of the algorithm is at least $Ω(n^{256})$ in the worst case. Therefore, an $O(n)$ algorithm is surprising and the best possible. We archive this result by calling an AEPTAS with approximation guarantee $(1+\varepsilon)OPT +p_{\max}$ and run…
In the Two-Bar Charts Packing Problem (2-BCPP), it is required to pack the bar charts (BCs) consisting of two bars into the horizontal unit-height strip of minimal length. The bars may move vertically within the strip, but it is forbidden to change the order and separate the chart's bars. Recently, for this new problem, which is a generalization of the Bin Packing Problem (BPP), Strip Packing Problem (SPP), and 2-Dimensional Vector Packing Problem (2-DVPP), several approximation algorithms with guaranteed estimates were proposed. However, after a preliminary analysis of the solutions construct…
We consider a Two-Bar Charts Packing Problem (2-BCPP), in which it is necessary to pack two-bar charts (2-BCs) in a unit-height strip of minimum length. The problem is a generalization of the Bin Packing Problem (BPP). Earlier, we proposed an $O(n^2)$-time algorithm that constructs the packing which length at most $2\cdot OPT+1$, where $OPT$ is the minimum length of the packing of $n$ 2-BCs. In this paper, we propose an $O(n^4)$-time 3/2-approximate algorithm when each BC has at least one bar greater than 1/2.
We study the d-dimensional hypercube knapsack problem where we are given a set of d-dimensional hypercubes with associated profits, and a knapsack which is a unit d-dimensional hypercube. The goal is to find an axis-aligned non-overlapping packing of a subset of hypercubes such that the profit of the packed hypercubes is maximized. For this problem, Harren (ICALP'06) gave an algorithm with an approximation ratio of (1+1/2^d+epsilon). For d=2, Jansen and Solis-Oba (IPCO'08) showed that the problem admits a polynomial-time approximation scheme (PTAS); Heydrich and Wiese (SODA'17) further improve…
Online algorithms that allow a small amount of migration or recourse have been intensively studied in the last years. They are essential in the design of competitive algorithms for dynamic problems, where objects can also depart from the instance. In this work, we give a general framework to obtain so called robust online algorithms for these dynamic problems: these online algorithm achieve an asymptotic competitive ratio of $γ+ε$ with migration $O(1/ε)$, where $γ$ is the best known offline asymptotic approximation ratio. In order to use our framework, one only needs to construct a suitable on…
The cloud computing industry has grown rapidly over the last decade, and with this growth there is a significant increase in demand for compute resources. Demand is manifested in the form of Virtual Machine (VM) requests, which need to be assigned to physical machines in a way that minimizes resource fragmentation and efficiently utilizes the available machines. This problem can be modeled as a dynamic version of the bin packing problem with the objective of minimizing the total usage time of the bins (physical machines). Earlier works on dynamic bin packing assumed that no knowledge is availa…
In rectangle packing problems we are given the task of placing axis-aligned rectangles in a given plane region, so that they do not overlap with each other. In Maximum Weight Independent Set of Rectangles (MWISR), their position is given and we can only select which rectangles to choose, while trying to maximize their total weight. In Strip Packing (SP), we have to pack all the given rectangles in a rectangular region of fixed width, while minimizing its height. In 2-Dimensional Geometric Knapsack (2DGK), the target region is a square of a given size, and our goal is to select and pack a subse…
We consider a Bar Charts Packing Problem (BCPP), in which it is necessary to pack bar charts (BCs) in a strip of minimum length. The problem is, on the one hand, a generalization of the Bin Packing Problem (BPP), and, on the other hand, a particular case of the Project Scheduling Problem with multidisciplinary jobs and one limited non-accumulative resource. Earlier, we proposed a polynomial algorithm that constructs the optimal package for a given order of non-increasing BCs. This result generalizes a similar result for BPP. For Two-Bar Charts Packing Problem (2-BCPP), when each BC consists of…
In $k$-hypergraph matching, we are given a collection of sets of size at most $k$, each with an associated weight, and we seek a maximum-weight subcollection whose sets are pairwise disjoint. More generally, in $k$-hypergraph $b$-matching, instead of disjointness we require that every element appears in at most $b$ sets of the subcollection. Our main result is a linear-programming based $(k-1+\tfrac{1}{k})$-approximation algorithm for $k$-hypergraph $b$-matching. This settles the integrality gap when $k$ is one more than a prime power, since it matches a previously-known lower bound. When the …
We explore approximation algorithms for the $d$-dimensional geometric bin packing problem ($d$BP). Caprara (MOR 2008) gave a harmonic-based algorithm for $d$BP having an asymptotic approximation ratio (AAR) of $T_{\infty}^{d-1}$ (where $T_{\infty} \approx 1.691$). However, their algorithm doesn't allow items to be rotated. This is in contrast to some common applications of $d$BP, like packing boxes into shipping containers. We give approximation algorithms for $d$BP when items can be orthogonally rotated about all or a subset of axes. We first give a fast and simple harmonic-based algorithm ha…
Indian Institute of Science（印度科学学院）；Vrije Universiteit, Amsterdam（阿姆斯特丹自由大学）
We study ROUND-UFP and ROUND-SAP, two generalizations of the classical BIN PACKING problem that correspond to the unsplittable flow problem on a path (UFP) and the storage allocation problem (SAP), respectively. We are given a path with capacities on its edges and a set of tasks where for each task we are given a demand and a subpath. In ROUND-UFP, the goal is to find a packing of all tasks into a minimum number of copies (rounds) of the given path such that for each copy, the total demand of tasks on any edge does not exceed the capacity of the respective edge. In ROUND-SAP, the tasks are con…
Throughput is a main performance objective in communication networks. This paper considers a fundamental maximum throughput routing problem -- the all-or-nothing multicommodity flow (ANF) problem -- in arbitrary directed graphs and in the practically relevant but challenging setting where demands can be (much) larger than the edge capacities. Hence, in addition to assigning requests to valid flows for each routed commodity, an admission control mechanism is required which prevents overloading the network when routing commodities. We make several contributions. On the theoretical side we obtain…
Time and Supply Fairness in Electricity Distribution using $k$-times bin packing
Ariel University（阿里埃尔大学）；UPES（乌塔拉克汉德人民教育学会大学）；Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine（乌克兰国家科学院皮德里斯特哈赫应用力学与数学问题研究所）
We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges form a planar graph. By planar duality this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem.
Effective placement of charging stations plays a key role in Electric Vehicle (EV) adoption. In the placement problem, given a set of candidate sites, an optimal subset needs to be selected with respect to the concerns of both (a) the charging station service provider, such as the demand at the candidate sites and the budget for deployment, and (b) the EV user, such as charging station reachability and short waiting times at the station. This work addresses these concerns, making the following three novel contributions: (i) a supervised multi-view learning framework using Canonical Correlation…
In conventional public clouds, designing a suitable initial cluster for a given application workload is important in reducing the computational foot-print during run-time. In edge or on-premise clouds, cold-start rightsizing the cluster at the time of installation is crucial in avoiding the recurrent capital expenditure. In both these cases, rightsizing has to balance cost-performance trade-off for a given application with multiple tasks, where each task can demand multiple resources, and the cloud offers nodes with different capacity and cost. Multidimensional bin-packing can address this col…
