Generating All Vertices of a Polyhedron Is Hard

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

doi:10.1007/s00454-008-9050-5

Cited by 95 publications

(34 citation statements)

References 31 publications

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“…Corollary 5.5 gives rise to the following approach of computing minimal witnessing subsystems: enumerate all vertices in the corresponding polytope and choose one with a maximal amount of zeros. Vertex enumeration of polytopes has been studied extensively [11,12,14,20,21,34,35,39,40,62,67] and has been shown to be computationally hard [54,Corollary 2].…”

Section: Computing Witnessing Subsystemsmentioning

confidence: 99%

Farkas Certificates and Minimal Witnesses for Probabilistic Reachability Constraints

Funke

Jantsch

Baier

2020

Lecture Notes in Computer Science

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This paper introduces Farkas certificates for lower and upper bounds on minimal and maximal reachability probabilities in Markov decision processes (MDP), which we derive using an MDP-variant of Farkas' Lemma. The set of all such certificates is shown to form a polytope whose points correspond to witnessing subsystems of the model and the property. This allows translating the problem of finding minimal witnesses to the problem of finding vertices with a maximal number of zeros. While computing such vertices is computationally hard in general, we derive new heuristics from our formulations that exhibit competitive performance compared to state-of-the-art techniques and apply to more situations. As an argument that asymptotically better algorithms cannot be hoped for, we show that the decision version of finding minimal witnesses is NP-complete even for acyclic Markov chains.

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Section: Computing Witnessing Subsystemsmentioning

confidence: 99%

Farkas Certificates and Minimal Witnesses for Probabilistic Reachability Constraints

Funke

Jantsch

Baier

2020

Lecture Notes in Computer Science

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“…This result has important theoretical implications regarding the computational complexity of circuit enumeration: the existence of a polynomial total-time extreme ray or vertex enumeration algorithm would immediately imply the existence a polynomial total-time algorithm for general circuit enumeration. Although vertex enumeration is provably hard for unbounded polyhedra, it is open whether or not a polynomial total-time enumeration scheme exists for bounded polytopes such as P A,B [23].…”

Section: Resultsmentioning

confidence: 99%

“…Since a polyhedron may have exponentially many circuits, complete circuit enumeration is hard in general. However, it is open whether or not the problem is solvable in polynomial total-time [23], i.e., if the output can be generated in time that is polynomial in both the input and output sizes. On the other hand, methods to directly optimize over circuits could provide implementations of these circuit augmentation schemes without needing to completely enumerate the set of circuits.…”

Section: Introductionmentioning

confidence: 99%

“…Algorithm 3 also has important theoretical implications. For both the circuit enumeration problem and the vertex enumeration problem for polytopes, it is open whether or not there exists a polynomial total-time algorithm [23]-one whose running time is polynomial in both the input and the output sizes. However, Theorem 4 implies that a polynomial total-time algorithm for vertex enumeration would immediately yield a polynomial total-time scheme for circuit enumeration.…”

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confidence: 99%

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A polyhedral model for enumeration and optimization over the set of circuits

Borgwardt

Viss

2022

Discrete Applied Mathematics

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Circuits play a fundamental role in polyhedral theory and linear programming. For instance, circuits are used as step directions in various augmentation schemes for solving linear programs or to leave degenerate vertices while running the simplex method. However, there are significant challenges when implementing these approaches: The set of circuits of a polyhedron may be of exponential size and is highly sensitive to the representation of the polyhedron. In this paper, we provide a universal framework for enumerating the set of circuits and optimizing over sets of circuits of a polyhedron in any representation-we propose a polyhedral model in which the circuits of the original polyhedron are encoded as extreme rays or vertices. Many methods in the literature and software assume that a polyhedron is in standard form; our framework is a direct generalization. We demonstrate its value by showing that the conversion of a general representation to standard form may introduce exponentially many new circuits. We then discuss the main advantages of the generalized polyhedral model. It enables the direct enumeration of useful subsets of circuits such as strictly feasible circuits or sign-compatible circuits, as well as optimization over these sets. In particular, this leads to the efficient computation of a steepest-descent circuit, which can be used in an augmentation scheme for solving linear programs or the construction of sign-compatible circuit walks with additional properties.

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“…If the running time of an algorithm is bounded by a polynomial in the size of the input plus the size of the output, then the algorithm is called output-polynomial. A large number of such algorithms have been given over the last 30 years; many of them solving problems on graphs and hypergraphs [9,10,11,14,20,21,22,24]. It is also possible to show that certain enumeration problems have no output-polynomial time algorithm unless P = NP [20,21,22].…”

Section: Introductionmentioning

confidence: 99%

Output-Polynomial Enumeration on Graphs of Bounded (Local) Linear MIM-Width

et al. 2017

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The linear induced matching width (LMIM-width) of a graph is a width parameter defined by using the notion of branch-decompositions of a set function on ternary trees. In this paper we study output-polynomial enumeration algorithms on graphs of bounded LMIM-width and graphs of bounded local LMIM-width. In particular, we show that all 1-minimal and all 1-maximal (σ, ρ)-dominating sets, and hence all minimal dominating sets, of graphs of bounded LMIM-width can be enumerated with polynomial (linear) delay using polynomial space. Furthermore, we show that all minimal dominating sets of a unit square graph can be enumerated in incremental polynomial time.A preliminary version of this paper appeared as an extended abstract in the proceedings of ISAAC 2015.

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Generating All Vertices of a Polyhedron Is Hard

Cited by 95 publications

References 31 publications

Farkas Certificates and Minimal Witnesses for Probabilistic Reachability Constraints

Farkas Certificates and Minimal Witnesses for Probabilistic Reachability Constraints

A polyhedral model for enumeration and optimization over the set of circuits

Output-Polynomial Enumeration on Graphs of Bounded (Local) Linear MIM-Width

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