Polytime Decomposition of Generalized Submodular Base Polytopes with Efficient Sampling

Saha, A (2020) Polytime Decomposition of Generalized Submodular Base Polytopes with Efficient Sampling. In: 12th Asian Conference on Machine Learning, ACML 2020, 18 - 20 Nov 2020, Bangkok, pp. 625-640.

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Abstract

We consider the problem of efficient decomposition of a given point x in an n-dimensional convex polytope into convex combination of its extreme points. Besides the widespread scopes of the problem in theory of convex polytopes in mathematics, the problem also has applications in online combinatorial optimization problems. Towards this we first propose a general class of convex polytopes-Generalized Submodular Base Polytopes (GSBPs)-that includes several well known convex polytopes as its special case including permutahedron, k-forest, spanning tree, combinatorial subset choice polytopes. We next propose a general decomposition algorithm for the above class of GSBPs that uses the novel idea of first decomposing the given point into at most n face centers, and further decomposing each face center into extreme points of their corresponding faces. In addition, we discover a few special class of partition-respecting and symmetric GSBPs for which the above two steps could be performed in respectively O(n2 + nT(f)) and O(n2T(f))1 time. We also give a complete characterization of the underlying submodular function f, for which the associated GSBP satisfies the above properties. One interesting fact is that we show that the support of the resulting decomposition with our proposed algorithm is only poly(n) in the number of extreme points which respects efficient sampling from the resulting distribution. Finally we corroborate our theoretical results with empirical evaluations. © 2020 A. Saha.

Item Type:	Conference Paper
Publication:	Proceedings of Machine Learning Research
Publisher:	ML Research Press
Additional Information:	The copyright for this article belongs to ML Research Press.
Keywords:	Decomposition; Topology, Combinatorial optimization problems; Convex combinations; Convex polytopes; Efficient decomposition; Efficient sampling; Extreme points; General class; K forests; Polytopes; Submodular, Combinatorial optimization
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	28 Jun 2023 09:29
Last Modified:	28 Jun 2023 09:29
URI:	https://eprints.iisc.ac.in/id/eprint/82205

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