Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing

Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, Makrand Sinha

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

A well-known result of Banaszczyk in discrepancy theory concerns the prefix discrepancy problem (also known as the signed series problem): given a sequence of T unit vectors in Rd, find ± signs for each of them such that the signed sum vector along any prefix has a small ℓ∞-norm? This problem is central to proving upper bounds for the Steinitz problem, and the popular Komlós problem is a special case where one is only concerned with the final signed sum vector instead of all prefixes. Banaszczyk gave an O(√log d + log T) bound for the prefix discrepancy problem. We investigate the tightness of Banaszczyk's bound and consider natural generalizations of prefix discrepancy: - We first consider a smoothed analysis setting, where a small amount of additive noise perturbs the input vectors. We show an exponential improvement in T compared to Banaszczyk's bound. Using a primal-dual approach and a careful chaining argument, we show that one can achieve a bound of O(√log d + loglog T) with high probability in the smoothed setting. Moreover, this smoothed analysis bound is the best possible without further improvement on Banaszczyk's bound in the worst case. - We also introduce a generalization of the prefix discrepancy problem to arbitrary DAGs. Here, vertices correspond to unit vectors, and the discrepancy constraints correspond to paths on a DAG on T vertices - prefix discrepancy is precisely captured when the DAG is a simple path. We show that an analog of Banaszczyk's O(√log d + log T) bound continues to hold in this setting for adversarially given unit vectors and that the √log T factor is unavoidable for DAGs. We also show that unlike for prefix discrepancy, the dependence on T cannot be improved significantly in the smoothed case for DAGs. - We conclude by exploring a more general notion of vector balancing, which we call combinatorial vector balancing. In this problem, the discrepancy constraints are generalized from paths of a DAG to an arbitrary set system. We obtain near-optimal bounds in this setting, up to poly-logarithmic factors.

Original languageEnglish (US)
Title of host publication13th Innovations in Theoretical Computer Science Conference, ITCS 2022
EditorsMark Braverman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772174
DOIs
StatePublished - Jan 1 2022
Externally publishedYes
Event13th Innovations in Theoretical Computer Science Conference, ITCS 2022 - Berkeley, United States
Duration: Jan 31 2022Feb 3 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume215
ISSN (Print)1868-8969

Conference

Conference13th Innovations in Theoretical Computer Science Conference, ITCS 2022
Country/TerritoryUnited States
CityBerkeley
Period1/31/222/3/22

Keywords

  • Combinatorial vector balancing
  • Prefix discrepancy
  • Smoothed analysis

ASJC Scopus subject areas

  • Software

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