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DOI: 10.4230/LIPIcs.CCC.2016.29

Identity Testing for Constant-Width, and Commutative, Read-Once Oblivious ABPs

Authors: Rohit Gurjar, Arpita Korwar, and Nitin Saxena

Published in: LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

Abstract

We give improved hitting-sets for two special cases of Read-once Oblivious Arithmetic Branching Programs (ROABP). First is the case of an ROABP with known variable order. The best hitting-set known for this case had cost (nw)^{O(log(n))}, where n is the number of variables and w is the width of the ROABP. Even for a constant-width ROABP, nothing better than a quasi-polynomial bound was known. We improve the hitting-set complexity for the known-order case to n^{O(log(w))}. In particular, this gives the first polynomial time hitting-set for constant-width ROABP (known-order). However, our hitting-set works only over those fields whose characteristic is zero or large enough. To construct the hitting-set, we use the concept of the rank of partial derivative matrix. Unlike previous approaches whose starting point is a monomial map, we use a polynomial map directly. The second case we consider is that of commutative ROABP. The best known hitting-set for this case had cost d^{O(log(w))}(nw)^{O(log(log(w)))}, where d is the individual degree. We improve this hitting-set complexity to (ndw)^{O(log(log(w)))}. We get this by achieving rank concentration more efficiently.

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Rohit Gurjar, Arpita Korwar, and Nitin Saxena. Identity Testing for Constant-Width, and Commutative, Read-Once Oblivious ABPs. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gurjar_et_al:LIPIcs.CCC.2016.29,
  author =	{Gurjar, Rohit and Korwar, Arpita and Saxena, Nitin},
  title =	{{Identity Testing for Constant-Width, and Commutative, Read-Once Oblivious ABPs}},
  booktitle =	{31st Conference on Computational Complexity (CCC 2016)},
  pages =	{29:1--29:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-008-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{50},
  editor =	{Raz, Ran},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.29},
  URN =		{urn:nbn:de:0030-drops-58438},
  doi =		{10.4230/LIPIcs.CCC.2016.29},
  annote =	{Keywords: PIT, hitting-set, constant-width ROABPs, commutative ROABPs}
}

Document

DOI: 10.4230/LIPIcs.CCC.2015.323

Deterministic Identity Testing for Sum of Read-once Oblivious Arithmetic Branching Programs

Authors: Rohit Gurjar, Arpita Korwar, Nitin Saxena, and Thomas Thierauf

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

Abstract

A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity n^(O(log(n))). In both the cases, our time complexity is double exponential in the number of ROABPs. ROABPs are a generalization of set-multilinear depth-3 circuits. The prior results for the sum of constantly many set-multilinear depth-3 circuits were only slightly better than brute-force, i.e. exponential-time. Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).

Cite as

Rohit Gurjar, Arpita Korwar, Nitin Saxena, and Thomas Thierauf. Deterministic Identity Testing for Sum of Read-once Oblivious Arithmetic Branching Programs. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 323-346, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{gurjar_et_al:LIPIcs.CCC.2015.323,
  author =	{Gurjar, Rohit and Korwar, Arpita and Saxena, Nitin and Thierauf, Thomas},
  title =	{{Deterministic Identity Testing for Sum of Read-once Oblivious Arithmetic Branching Programs}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{323--346},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.323},
  URN =		{urn:nbn:de:0030-drops-50647},
  doi =		{10.4230/LIPIcs.CCC.2015.323},
  annote =	{Keywords: PIT, Hitting-set, Sum of ROABPs, Evaluation Dimension, Rank Concentration}
}

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Documents authored by Korwar, Arpita

Identity Testing for Constant-Width, and Commutative, Read-Once Oblivious ABPs

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Deterministic Identity Testing for Sum of Read-once Oblivious Arithmetic Branching Programs

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