Faster optimal algorithms for segment minimization with small maximal value

Biedl, Thérèse; Durocher, Stéphane; Engelbeen, Céline; Fiorini, Samuel; Young, Maxwell

doi:10.1016/j.dam.2012.09.011

Cited by 4 publications

(11 citation statements)

References 16 publications

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“…For the parameter maximum value γ, VE + is known to be fixed-parameter tractable [6]. We complement this result by showing a lower bound on the problem kernel size, and thus demonstrate limitations on the power of preprocessing.…”

Section: Parameterization By Input Smoothnessmentioning

confidence: 60%

“…A significant amount of work has been done to achieve approximation algorithms for minimizing the number of segments which improve on the straightforward factor of two [4] (also see Biedl et al [5]). Improving a previous fixed-parameter tractability result for the parameter "maximum value γ of a vector entry" by Cambazard et al [8], Biedl et al [6] developed a fixed-parameter algorithm solving VE + in polynomial time when γ = O((log n) 2 ) with n being the number of entries in the input vector. Moreover, the parameter "maximum difference between two consecutive vector entries" has been exploited for developing polynomial-time approximation algorithms [5,19].…”

Section: Introductionmentioning

confidence: 99%

“…We take a closer look at these parameterization aspects that help in better understanding and exploiting problem-specific properties. To start with, note that previous work [6,8], motivated by the application in radiation therapy, studied the parameterization by the maximum vector entry γ. They showed fixed-parameter tractability for VE + parameterized by γ, which we complement by showing the non-existence (assuming a standard complexity-theoretic assumption) of a corresponding polynomial-size problem kernel.…”

Section: Introductionmentioning

confidence: 99%

“…VE occurs in the database context and VE + occurs in the radiation therapy context. Motivated by previous work providing polynomial-time solvable special cases [1,4], polynomial-time approximation [5,19] and fixed-parameter tractability results [6,8] (approximation and fixed-parameter algorithms both exploit problem-specific structural parameters), we head on a systematic parameterized and multivariate complexity analysis [13,21] of both problems; see Table 1 for a survey of parameterized complexity results.…”

Section: Introductionmentioning

confidence: 99%

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On Explaining Integer Vectors by Few Homogenous Segments

Bredereck

Hartung

Komusiewicz

et al. 2013

Lecture Notes in Computer Science

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Abstract. We extend previous studies on NP-hard problems dealing with the decomposition of nonnegative integer vectors into sums of few homogeneous segments. These problems are motivated by radiation therapy and database applications. If the segments may have only positive integer entries, then the problem is called Vector Explanation + . If arbitrary integer entries are allowed in the decomposition, then the problem is called Vector Explanation. Considering several natural parameterizations (including maximum vector entry, maximum difference between consecutive vector entries, maximum segment length), we obtain a refined picture of the computational (in-)tractability of these problems. In particular, we show that in relevant cases Vector Explanation + is algorithmically harder than Vector Explanation.

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Section: Parameterization By Input Smoothnessmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Explaining Integer Vectors by Few Homogenous Segments

Bredereck

Hartung

Komusiewicz

et al. 2013

Lecture Notes in Computer Science

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“…As far as we are aware, this approach currently constitutes the fastest exact solution algorithm for the MCSP. An example of discretization of the particle beam emitted by a linear accelerator (from [2]). The matrix on the right encodes the discretized intensity values of the radiation field shown on the left.…”

Section: Introductionmentioning

confidence: 99%

An Integer Linear Programming Formulation for the Minimum Cardinality Segmentation Problem

Catanzaro

Engelbeen

2015

Algorithms

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In this article, we investigate the Minimum Cardinality Segmentation Problem (MCSP), an N P-hard combinatorial optimization problem arising in intensity-modulated radiation therapy. The problem consists in decomposing a given nonnegative integer matrix into a nonnegative integer linear combination of a minimum cardinality set of binary matrices satisfying the consecutive ones property. We show how to transform the MCSP into a combinatorial optimization problem on a weighted directed network and we exploit this result to develop an integer linear programming formulation to exactly solve it. Computational experiments show that the lower bounds obtained by the linear relaxation of the considered formulation improve upon those currently described in the literature and suggest, at the same time, new directions for the development of future exact solution approaches to the problem.

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On explaining integer vectors by few homogeneous segments

Bredereck

Hartung

Komusiewicz

et al. 2015

Journal of Computer and System Sciences

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We extend previous studies on "explaining" a nonnegative integer vector by sums of few homogeneous segments, that is, vectors where all nonzero entries are equal and consecutive. We study two NP-complete variants which are motivated by radiation therapy and database applications. In Vector Positive Explanation, the segments may have only positive integer entries; in Vector Explanation, the segments may have arbitrary integer entries. Considering several natural parameterizations such as the maximum vector entry γ and the maximum difference δ between consecutive vector entries, we obtain a refined picture of the computational (in-)tractability of these problems. For example, we show that Vector Explanation is fixed-parameter tractable with respect to δ, and that, unless NP ⊆ coNP/poly, there is no polynomial kernelization for Vector Positive Explanation with respect to the parameter γ . We also identify relevant special cases where Vector Positive Explanation is algorithmically harder than Vector Explanation.

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Faster optimal algorithms for segment minimization with small maximal value

Cited by 4 publications

References 16 publications

On Explaining Integer Vectors by Few Homogenous Segments

On Explaining Integer Vectors by Few Homogenous Segments

An Integer Linear Programming Formulation for the Minimum Cardinality Segmentation Problem

On explaining integer vectors by few homogeneous segments

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