Generating All Vertices of a Polyhedron Is Hard

Khachiyan, Leonid; Boros, Endre; Borys, Konrad; Elbassioni, Khaled; Gurvich, Vladimir

doi:10.1007/s00454-006-1259-6

Cited by 40 publications

(56 citation statements)

References 2 publications

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“…So again, writing all vertices down, let alone computing them, cannot usually be done in polynomial time. We remark that for polyhedra it is generally impossible to compute the vertices even in time polynomial in the output, that is, in the number of vertices [60]. However, in this chapter we do overcome these difficulties and, following the general outline of this strategy, solve the problem in polynomial time in two different broad situations.…”

Section: Find and Output A Feasible Pointmentioning

confidence: 94%

“…The problem of vertex enumeration of a polyhedron presented by linear inequalities, which is of different flavor, has been studied extensively in the literature. An output-sensitive algorithm for vertex enumeration is given in [5] by Avis and Fukuda, and the hardness of the problem for possibly unbounded polyhedra is shown in [60]. Theorem 2.19 on convex integer maximization over a system defined by a totally unimodular matrix given the circuits of the matrix can be extended to other classes of matrices which define integer polyhedra.…”

Section: Notesmentioning

confidence: 99%

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Nonlinear Discrete Optimization

Onn

2010

Zurich Lectures in Advanced Mathematics

113

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This monograph develops an algorithmic theory of nonlinear discrete optimization. It introduces a simple and useful setup which enables the polynomial time solution of broad fundamental classes of nonlinear combinatorial optimization and integer programming problems in variable dimension. An important part of this theory is enhanced by recent developments in the algebra of Graver bases. The power of the theory is demonstrated by deriving the first polynomial time algorithms in a variety of application areas within operations research and statistics, including vector partitioning, matroid optimization, experimental design, multicommodity flows, multiindex transportation and privacy in statistical databases. The monograph is based on twelve lectures which I gave within the Nachdiplom Lectures series at ETH Zürich in Spring 2009, broadly extending and updating five lectures on convex discrete optimization which I gave within the Séminaire de Mathématiques Supérieures series at Université de Montréal in June 2006. I thank the support of the Israel Science Foundation and the support and hospitality of ETH Zürich, Université de Montréal, and Mathematisches Forschungsinstitut Oberwolfach, where parts of the research underlying the monograph took place. I am indebted to Jesus De Loera, Raymond Hemmecke, Jon Lee, Uriel G. Rothblum and Robert Weismantel for their collaboration in developing the theory described herein. I am also indebted to Komei Fukuda for his interest and very helpful suggestions, and to many other colleagues including my co-authors on articles cited herein and my audience at the Nachdiplom Lectures at ETH. Finally, I am especially grateful to my wife Ruth, our son Amos, and our daughter Naomi, for their support throughout the writing of this monograph. Haifa, May 2010 Shmuel Onn n i=1 w i x i is linear, which is the case considered in most literature on discrete optimization and is already hard and often intractable. The matrix W is given explicitly, and the computational complexity of the problem depends on the encoding of its entries (binary versus unary). Nonlinear matroid optimization. Given a matroid M = (N, B) on ground set N := {1,. .. , n}, an integer d × n matrix W , and a function f : Z d → R, solve max{f (W x) : x ∈ S} with S ⊆ {0, 1} n the set of (indicators of) bases or independent sets of M :

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Section: Find and Output A Feasible Pointmentioning

confidence: 94%

Section: Notesmentioning

confidence: 99%

Nonlinear Discrete Optimization

Onn

2010

Zurich Lectures in Advanced Mathematics

113

View full text Add to dashboard Cite

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“…First, an idea is to make the assumption that the preferences of the agent obey a weighted sum aggregation scheme, whose weights are unknown. Then, we might think of finding a sampling of systems of weights summing to 1 that are compatible with the constraints induced by E. But, enumerating the vertices of the polytope defined by the system of inequations corresponding to the preferences in E is a NP hard problem that is not easy at all to handle in practice [13]. Indeed given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard.…”

Section: Multiple Criteria-based Preference and Analogical Proportionsmentioning

confidence: 99%

Predicting Preferences by Means of Analogical Proportions

Bounhas¹,

Pirlot

Prade

2018

Case-Based Reasoning Research and Development

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Open Archive Toulouse Archive OuverteOATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible To cite this version: Bounhas, Myriam and Pirlot, Marc and Prade, Henri Predicting preferences by means of analogical proportions. (Abstract. It is assumed that preferences between two items, described in terms of criteria values belonging to a finite scale, are known for a limited number of pairs of items, which constitutes a case base. The problem is then to predict the preference between the items of a new pair. A new approach based on analogical proportions is presented. Analogical proportions are statements of the form "a is to b as c is to d". If the change between item-1 and item-2 is the same as the change between item-3 and item-4, and a similar statement holds for item'-1, item'-2, item'-3, item'-4, then one may plausibly assume that the preference between item-1 and item'-1 is to the preference between item-2 and item'-2 as the preference between item-3 and item'-3 is to the preference between item-4 and item'-4. This offers a basis for a plausible prediction of the fourth preference if the three others are known. This approach fits well with the postulates underlying weighted averages. Two algorithms are proposed that look for triples of preferences appropriate for a prediction. The first one only exploits the given set of examples. The second one completes this set with new preferences deducible from this set under a monotony assumption. This completion is limited to the generation of preferences that are useful for the requested prediction. The predicted preferences should fit with the assumption that known preferences agree with a unique unknown weighted average. The reported experiments suggest the effectiveness of the proposed approach.

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“…We observe that there is a difference between the above version of the MOLP problem and the problem of enumerating all Pareto optimal basic feasible solutions: The latter problem cannot be solved in output‐polynomial time unless P = N P , because it subsumes the vertex enumeration problem even when allowing only two objectives. Khachiyan, Boros, Borys, Elbassioni, and Gurvich () proved that the vertex enumeration problem for general polyhedra cannot be solved in output‐polynomial time unless P = N P . But we conjecture that there exists an output‐sensitive algorithm for the MOLP problem as in Definition .…”

Section: Enumerating Supported Efficient Solutions and Nondominated Ementioning

confidence: 99%

Output-sensitive complexity of multiobjective combinatorial optimization

Bökler

Ehrgott

Morris

et al. 2017

J. Multi-Crit. Decis. Anal.

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We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is output-sensitive, that is, its running time is bounded by a polynomial in the input and the output size. We provide both practical examples of multiobjective combinatorial optimization problems for which such an efficient algorithm exists as well as problems for which no efficient algorithm exists under mild complexity theoretic assumptions. KEYWORDScombinatorial optimization, linear programming, multiobjective optimization, output-sensitive complexity 1 1.1 gives us problems, which are not harder than a variant of the problem where we also want to find solutions. Hence, proving hardness of the problem as defined in Definition 1.1 will lead to a hardness result for the problem including finding of a representative solution.

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Generating All Vertices of a Polyhedron Is Hard

Cited by 40 publications

References 2 publications

Nonlinear Discrete Optimization

Nonlinear Discrete Optimization

Predicting Preferences by Means of Analogical Proportions

Output-sensitive complexity of multiobjective combinatorial optimization

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