License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2018.74
URN: urn:nbn:de:0030-drops-90782
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9078/
Gurjar, Rohit ;
Thierauf, Thomas ;
Vishnoi, Nisheeth K.
Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces
Abstract
We present a geometric approach towards derandomizing the {Isolation Lemma} by Mulmuley, Vazirani, and Vazirani. In particular, our approach produces a quasi-polynomial family of weights, where each weight is an integer and quasi-polynomially bounded, that can isolate a vertex in any 0/1 polytope for which each face lies in an affine space defined by a totally unimodular matrix. This includes the polytopes given by totally unimodular constraints and generalizes the recent derandomization of the Isolation Lemma for {bipartite perfect matching} and {matroid intersection}. We prove our result by associating a {lattice} to each face of the polytope and showing that if there is a totally unimodular kernel matrix for this lattice, then the number of vectors of length within 3/2 of the shortest vector in it is polynomially bounded. The proof of this latter geometric fact is combinatorial and follows from a polynomial bound on the number of circuits of size within 3/2 of the shortest circuit in a regular matroid. This is the technical core of the paper and relies on a variant of Seymour's decomposition theorem for regular matroids. It generalizes an influential result by Karger on the number of minimum cuts in a graph to regular matroids.
BibTeX - Entry
@InProceedings{gurjar_et_al:LIPIcs:2018:9078,
author = {Rohit Gurjar and Thomas Thierauf and Nisheeth K. Vishnoi},
title = {{Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {74:1--74:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-076-7},
ISSN = {1868-8969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9078},
URN = {urn:nbn:de:0030-drops-90782},
doi = {10.4230/LIPIcs.ICALP.2018.74},
annote = {Keywords: Derandomization, Isolation Lemma, Total unimodularity, Near-shortest vectors in Lattices, Regular matroids}
}
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Keywords: |
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Derandomization, Isolation Lemma, Total unimodularity, Near-shortest vectors in Lattices, Regular matroids |
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Collection: |
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45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) |
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Issue Date: |
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2018 |
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Date of publication: |
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04.07.2018 |
