Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs

Even, Guy; Levi, Retsef; Rawitz, Dror; Schieber, Baruch; Shahar, Shimon; Sviridenko, Maxim

doi:10.1145/1367064.1367074

Cited by 34 publications

(45 citation statements)

References 20 publications

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“…Even et al [6] address non-uniform capacities, i.e., each period t has capacity k t such that each batch C with t C = t may stab at most k t intervals, but the interval weights are uniform, for example w I = 1 for each interval I ∈ J. They discuss the hard-and soft-capacitated case.…”

Section: Geometric Interpretation and Dynamic Programmingmentioning

confidence: 99%

Capacitated max -Batching with Interval Graph Compatibilities

Nonner

2010

Lecture Notes in Computer Science

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We consider the problem of partitioning interval graphs into cliques of bounded size. Each interval has a weight, and the cost of a clique is the maximum weight of any interval in the clique. This natural graph problem can be interpreted as a batch scheduling problem. Solving an open question from [7,4,5], we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we give a PTAS based on a novel dynamic programming technique for this case.

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Section: Geometric Interpretation and Dynamic Programmingmentioning

confidence: 99%

Capacitated max -Batching with Interval Graph Compatibilities

Nonner

2010

Lecture Notes in Computer Science

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“…A similar result was obtained by Mecke et al [16]; they give a factor-d approximation algorithm for a problem called d-C1P-Set Cover, which is a generalization of d-Dimensional Rectangle Stabbing. Weighted and capacitated versions of d-Dimensional Rectangle Stabbing have been considered by Even et al [4] and by Xu and Xu [18], also leading to several approximation algorithms. A restricted, but still NP-complete variant of (2-Dimensional) Rectangle Stabbing is called Interval Stabbing; here, every rectangle in the input is intersected by at most one horizontal line (that is, every rectangle is a horizontal interval in the plane).…”

Section: Inputmentioning

confidence: 99%

Parameterized Complexity of Stabbing Rectangles and Squares in the Plane

Dom

Fellows

Rosamond

2009

WALCOM: Algorithms and Computation

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Abstract. The NP-complete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the selected lines. While it is known that the problem can be approximated in polynomial time with a factor of two, its parameterized complexity with respect to the parameter k was open so far-only its generalization to three or more dimensions was known to be W[1]-hard. Giving two fixed-parameter reductions, one from the W[1]-complete problem Multicolored Clique and one to the W[1]-complete problem Short Turing Machine Acceptance, we prove that Rectangle Stabbing is W[1]-complete with respect to the parameter k, which in particular means that there is no hope for fixed-parameter tractability with respect to the parameter k. Our reductions show also the W[1]-completeness of the more general problem Set Cover on instances that "almost have the consecutiveones property", that is, on instances whose matrix representation has at most two blocks of 1s per row. For the special case of Rectangle Stabbing where all rectangles are squares of the same size we can also show W[1]-hardness, while the parameterized complexity of the special case where the input consists of rectangles that do not overlap is open. By giving an algorithm running in (4k + 1) k · n O(1) time, we show that Rectangle Stabbing is fixedparameter tractable in the still NP-hard case where both these restrictions apply.

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“…Hence, each batch C is defined by the parameters t C and w C . Consequently, as in the well-known interval stabbing problem [6,13], we can also think of a batch C as a vertical line with x-coordinate t C that stabs all intervals I ∈ J with w I ≤ w C .…”

Section: Max-batchingmentioning

confidence: 99%

Latency Constrained Aggregation in Chain Networks Admits a PTAS

Nonner

Souza

2009

Algorithmic Aspects in Information and Management

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This paper studies the aggregation of messages in networks that consist of a chain of nodes, and each message is time-constrained such that it needs to be aggregated during a given time interval, called its due interval. The objective is to minimize the maximum sending cost of any node, which is for example a concern in wireless sensor networks, where it is crucial to distribute the energy consumption as equally as possible. First, we settle the complexity of this problem by proving its NP-hardness, even for the case of unit length due intervals. Second, we give a QPTAS, which we extend to a PTAS for the special case that the lengths of the due intervals are constants. This is in particular interesting, since we prove that this problem becomes APX-hard if we consider tree networks instead of chain networks, even for the case of unit length due intervals. Specifically, we show that it cannot be approximated within 4/3 − ǫ for any ǫ > 0, unless P=NP.

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Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs

Cited by 34 publications

References 20 publications

Capacitated max -Batching with Interval Graph Compatibilities

Capacitated max -Batching with Interval Graph Compatibilities

Parameterized Complexity of Stabbing Rectangles and Squares in the Plane

Latency Constrained Aggregation in Chain Networks Admits a PTAS

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