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Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces

Authors Rohit Gurjar, Thomas Thierauf, Nisheeth K. Vishnoi

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ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Mathematics of computing → Matroids and greedoids
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Derandomization
  • Isolation Lemma
  • Total unimodularity
  • Near-shortest vectors in Lattices
  • Regular matroids

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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.

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Rohit Gurjar, Thomas Thierauf, and Nisheeth K. Vishnoi. Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 74:1-74:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.74

Author Details

Rohit Gurjar
  • California Institute of Technology, USA
Thomas Thierauf
  • Aalen University, Germany
Nisheeth K. Vishnoi
  • École Polytechnique Fédérale de Lausanne (EPFL), Switzerland

Funding

  • Gurjar, Rohit: The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 257575 and from the Israel Science Foundation (grant number 552/16).
  • Thierauf, Thomas: Supported by DFG grant TH 472/4.

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