On non-serial dynamic programming

Bertelè, Umberto; Brioschi, Francesco

doi:10.1016/0097-3165(73)90016-2

Cited by 44 publications

(45 citation statements)

References 5 publications

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“…Such NPhard problems include maximum independent set size, minimal dominating set size, chromatic number, Hamiltonian circuit, network reliability and minimum vertex removal forbidden subgraph [5,9]. Several graphs which are important in practice [34], have been shown to be partial k-trees, among them are 1.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Scaling limit of random k-trees

Drmota

Jin

2015

2016 Proceedings of the Thirteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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We consider a random k-tree G n,k that is uniformly selected from the class of labelled k-trees with n + k vertices. Since 1-trees are just trees, it is well-known that Gn,1 (after scaling the distances by 1/(2 √ n)) converges to the Continuum Random Tree Te. Our main result is that for k = 1, the random k-tree G n,k , scaled by (kH k−1 + 1)/(2 √ n) where H k−1 is the (k − 1)-th Harmonic number, converges to the Continuum Random Tree Te, too. In particular this shows that the diameter as well as the typical distance of two vertices in a random k-tree G n,k are of order √ n.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Scaling limit of random k-trees

Drmota

Jin

2015

2016 Proceedings of the Thirteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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“…Indeed, most algorithms leveraging treewidth are dynamic programming algorithms or can be equivalently expressed as such [19][20][21][22][23][24]. Even before dynamic programming on tree-decompositions became an important subject in algorithm design, similar concepts were already used implicitly [25,26]. The sentiment that the table size is the crucial factor in the complexity of dynamic programming algorithms is certainly not new (see e.g., [27]), so it seems natural to provide lower bounds to formalize this intuition.…”

Section: Introductionmentioning

confidence: 99%

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

Chen¹,

Reidl

Rossmanith

et al. 2018

Algorithms

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Treedepth is a well-established width measure which has recently seen a resurgence of interest. Since graphs of bounded treedepth are more restricted than graphs of bounded treeor pathwidth, we are interested in the algorithmic utility of this additional structure. On the negative side, we show with a novel approach that the space consumption of any (single-pass) dynamic programming algorithm on treedepth decompositions of depth d cannot be bounded by (2 −) d • log O(1) n for VERTEX COVER, (3 −) d • log O(1) n for 3-COLORING and (3 −) d • log O(1) n for DOMINATING SET for any > 0. This formalizes the common intuition that dynamic programming algorithms on graph decompositions necessarily consume a lot of space and complements known results of the time-complexity of problems restricted to low-treewidth classes. We then show that treedepth lends itself to the design of branching algorithms. Specifically, we design two novel algorithms for DOMINATING SET on graphs of treedepth d: A pure branching algorithm that runs in time d O(d 2) • n and uses space O(d 3 log d + d log n) and a hybrid of branching and dynamic programming that achieves a running time of O(3 d log d • n) while using O(2 d d log d + d log n) space.

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“…In nonserial dp, methods that describe the connection between variables in the problem using tree representations are used, see for example Bertelè and Brioschi (1973), Moallemi (2007), and Shcherbina (2007). Nonserial dp shares the basic ideas with serial dp, see for example Bertsekas (2000), but can handle more general problem structures.…”

Section: Objectivesmentioning

confidence: 99%

Structure-Exploiting Numerical Algorithms for Optimal Control

Nielsen¹

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Cover illustration: An illustration of the structure of the Karush-Kuhn-Tucker matrix for the unconstrained finite-time optimal control problems considered in this thesis. Thank you Jonas Linder for helping me with the design.Linköping studies in science and technology. Dissertations.No. 1848 Tillägnas hela min underbara familj Structure-Exploiting Numerical Algorithms for Optimal Control AbstractNumerical algorithms for efficiently solving optimal control problems are important for commonly used advanced control strategies, such as model predictive control (mpc), but can also be useful for advanced estimation techniques, such as moving horizon estimation (mhe). In mpc, the control input is computed by solving a constrained finite-time optimal control (cftoc) problem on-line, and in mhe the estimated states are obtained by solving an optimization problem that often can be formulated as a cftoc problem. Common types of optimization methods for solving cftoc problems are interior-point (ip) methods, sequential quadratic programming (sqp) methods and active-set (as) methods. In these types of methods, the main computational effort is often the computation of the second-order search directions. This boils down to solving a sequence of systems of equations that correspond to unconstrained finite-time optimal control (uftoc) problems. Hence, high-performing second-order methods for cftoc problems rely on efficient numerical algorithms for solving uftoc problems. Developing such algorithms is one of the main focuses in this thesis. When the solution to a cftoc problem is computed using an as type method, the aforementioned system of equations is only changed by a low-rank modification between two as iterations. In this thesis, it is shown how to exploit these structured modifications while still exploiting structure in the uftoc problem using the Riccati recursion. Furthermore, direct (non-iterative) parallel algorithms for computing the search directions in ip, sqp and as methods are proposed in the thesis. These algorithms exploit, and retain, the sparse structure of the uftoc problem such that no dense system of equations needs to be solved serially as in many other algorithms. The proposed algorithms can be applied recursively to obtain logarithmic computational complexity growth in the prediction horizon length. For the case with linear mpc problems, an alternative approach to solving the cftoc problem on-line is to use multiparametric quadratic programming (mp-qp), where the corresponding cftoc problem can be solved explicitly off-line. This is referred to as explicit mpc. One of the main limitations with mp-qp is the amount of memory that is required to store the parametric solution. In this thesis, an algorithm for decreasing the required amount of memory is proposed. The aim is to make mp-qp and explicit mpc more useful in practical applications, such as embedded systems with limited memory resources. The proposed algorithm exploits the structure from the qp problem in the parametric solution in order to reduc...

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On non-serial dynamic programming

Cited by 44 publications

References 5 publications

Scaling limit of random k-trees

Scaling limit of random k-trees

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

Structure-Exploiting Numerical Algorithms for Optimal Control

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