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Directed Hypercube Routing, a Generalized Lehman-Ron Theorem, and Monotonicity Testing

Authors Deeparnab Chakrabarty , C. Seshadhri

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Author Details

Deeparnab Chakrabarty
  • Dartmouth College, Hanover, NH, USA
C. Seshadhri
  • University of California, Santa Cruz, CA, USA

Funding

  • Chakrabarty, Deeparnab: Supported by NSF-CAREER CCF-2041920 and CCF-2402571.
  • Seshadhri, C.: Supported by NSF CCF-1740850, CCF-1839317, CCF-2402572, and DMS-2023495.

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Deeparnab Chakrabarty and C. Seshadhri. Directed Hypercube Routing, a Generalized Lehman-Ron Theorem, and Monotonicity Testing. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.34

Abstract

Motivated by applications to monotonicity testing, Lehman and Ron (JCTA, 2001) proved the existence of a collection of vertex disjoint paths between comparable sub-level sets in the directed hypercube. The main technical contribution of this paper is a new proof method that yields a generalization of their theorem: we prove the existence of two edge-disjoint collections of vertex disjoint paths. Our main conceptual contributions are conjectures on directed hypercube flows with simultaneous vertex and edge capacities of which our generalized Lehman-Ron theorem is a special case. We show that these conjectures imply directed isoperimetric theorems, and in particular, the robust directed Talagrand inequality due to Khot, Minzer, and Safra (SIAM J. on Comp, 2018). These isoperimetric inequalities, that relate the directed surface area (of a set in the hypercube) to its distance to monotonicity, have been crucial in obtaining the best monotonicity testers for Boolean functions. We believe our conjectures pave the way towards combinatorial proofs of these directed isoperimetry theorems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Monotonicity testing
  • isoperimetric inequalities
  • hypercube routing

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