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Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces

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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}
}

Keywords: Derandomization, Isolation Lemma, Total unimodularity, Near-shortest vectors in Lattices, Regular matroids
Seminar: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue date: 2018
Date of publication: 2018


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