Some Algorithmic Problems in Polytope Theory

Kaibel, Volker; Pfetsch, Marc E.

doi:10.1007/978-3-662-05148-1_2

Cited by 52 publications

(38 citation statements)

References 63 publications

(93 reference statements)

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“…The vertex enumeration problem is NP-hard in general and it is not known whether for the special case of bounded polytopes (like W(C)) an algorithm exists with PTIME input-output complexity. We also observe that, for any fixed value of d, the number of vertices q is at most O(c d/2 ) (see [10] and references therein). Table 3 summarizes our results.…”

Section: Considerations About Complexitymentioning

confidence: 74%

Reconciling skyline and ranking queries

Ciaccia

Martinenghi

2017

Proc. VLDB Endow.

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Traditionally, skyline and ranking queries have been treated separately as alternative ways of discovering interesting data in potentially large datasets. While ranking queries adopt a specific scoring function to rank tuples, skyline queries return the set of non-dominated tuples and are independent of attribute scales and scoring functions. Ranking queries are thus less general, but usually cheaper to compute and widely used in data management systems.We propose a framework to seamlessly integrate these two approaches by introducing the notion of restricted skyline queries (R-skylines). We propose R-skyline operators that generalize both skyline and ranking queries by applying the notion of dominance to a set of scoring functions of interest. Such sets can be characterized, e.g., by imposing constraints on the function's parameters, such as the weights in a linear scoring function. We discuss the formal properties of these new operators, show how to implement them efficiently, and evaluate them on both synthetic and real datasets.

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Section: Considerations About Complexitymentioning

confidence: 74%

Reconciling skyline and ranking queries

Ciaccia

Martinenghi

2017

Proc. VLDB Endow.

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“…). However, we can obtain a tighter bound by observing that counting the number of corner weights given a CCS is equivalent to vertex enumeration, which is the dual problem of facet enumeration, i.e., counting the number of vertices given the corner weights (Kaibel & Pfetsch, 2003).…”

Section: Discussionmentioning

confidence: 99%

Computing Convex Coverage Sets for Faster Multi-objective Coordination

Roijers¹,

Whiteson²,

Oliehoek³

2015

jair

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In this article, we propose new algorithms for multi-objective coordination graphs (MOCoGs). Key to the efficiency of these algorithms is that they compute a convex coverage set (CCS) instead of a Pareto coverage set (PCS). Not only is a CCS a sufficient solution set for a large class of problems, it also has important characteristics that facilitate more efficient solutions. We propose two main algorithms for computing a CCS in MO-CoGs. Convex multi-objective variable elimination (CMOVE) computes a CCS by performing a series of agent eliminations, which can be seen as solving a series of local multi-objective subproblems. Variable elimination linear support (VELS) iteratively identifies the single weight vector w that can lead to the maximal possible improvement on a partial CCS and calls variable elimination to solve a scalarized instance of the problem for w. VELS is faster than CMOVE for small and medium numbers of objectives and can compute an ε-approximate CCS in a fraction of the runtime. In addition, we propose variants of these methods that employ AND/OR tree search instead of variable elimination to achieve memory efficiency. We analyze the runtime and space complexities of these methods, prove their correctness, and compare them empirically against a naive baseline and an existing PCS method, both in terms of memory-usage and runtime. Our results show that, by focusing on the CCS, these methods achieve much better scalability in the number of agents than the current state of the art.

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“…The representation of reachable sets does not only have to be compact, but more importantly, relevant operations have to be efficient with respect to the system dimension n, which are: linear transformation, Minkowski addition, box enclosure, and convex hull computation of linearly transformed sets [9]. For polytopes, Minkowski addition and convex hull computation, which is denoted by opCH(), are generally limited to problems with up to 4 − 6 dimensions [21], [22] and tend to run into numerical problems unless infinite precision arithmetic is used [23]. Unlike polytopes, zonotopes are numerically stable and operations for reachability analysis have a maximum complexity of O(n 3 ).…”

Section: Definition 3 (G-representation Of a Zonotope)mentioning

confidence: 99%

Zonotope bundles for the efficient computation of reachable sets

Althoff

Krogh

2011

IEEE Conference on Decision and Control and European Control Conference

109

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We introduce zonotope bundles for computing the set of states reachable by a dynamical system, also known as the reachable set. Reachable set computations suffer from the curse of dimensionality, which has been successfully addressed by using zonotopes for linear systems. However, zonotopes are not closed under intersection leading to challenges when applying them to nonlinear and hybrid problems. We introduce zonotope bundles as the intersection of zonotopes (without explicitly computing the intersection). Zonotope bundles are closed under intersection, while inheriting many positive properties of zonotopes. This is demonstrated for linear, nonlinear, and hybrid systems. A further property of zonotope bundles is that their computation can be easily parallelized.

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Some Algorithmic Problems in Polytope Theory

Abstract: In this section problems are collected whose input are geometrical data, i.e., the Hor V-description of a polytope. Some problems which are also given by geometrical data appear in Sections 4 and 5.

Cited by 52 publications

References 63 publications

Reconciling skyline and ranking queries

Reconciling skyline and ranking queries

Computing Convex Coverage Sets for Faster Multi-objective Coordination

Zonotope bundles for the efficient computation of reachable sets

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