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Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing

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

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ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
Keywords
  • Prefix discrepancy
  • smoothed analysis
  • combinatorial vector balancing

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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 ℝ^d, 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+ log log 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.

Cite As Get BibTex

Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla, and Makrand Sinha. Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.13

Author Details

Nikhil Bansal
  • University of Michigan, Ann Arbor, MI, USA
Haotian Jiang
  • University of Washington, Seattle, WA, USA
Raghu Meka
  • University of California, Los Angeles, CA, USA
Sahil Singla
  • Georgia Institute of Technology, Atlanta, GA, USA
Makrand Sinha
  • Simons Institute, Berkeley, CA, USA
  • University of California, Berkeley, CA, USA

Funding

  • Bansal, Nikhil: Supported in part by the NWO VICI grant 639.023.812. The work was done when the author was at CWI, Amsterdam.
  • Jiang, Haotian: Supported in part by the National Science Foundation, Grant Number CCF-1749609, CCF-1740551, DMS-1839116.
  • Meka, Raghu: Supported by NSF Grant CCF-1553605.
  • Singla, Sahil: The work was partly done when the author was at Princeton University.
  • Sinha, Makrand: Work done while at CWI Amsterdam and supported by the NWO VICI grant 639.023.812.

Acknowledgements

The authors would like to thank the anonymous reviewers of ITCS 2022 for helpful comments.

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