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Documents authored by Nanoti, Saraswati Girish

Document
DOI: 10.4230/LIPIcs.STACS.2024.31
On the Power of Border Width-2 ABPs over Fields of Characteristic 2

Authors: Pranjal Dutta, Christian Ikenmeyer, Balagopal Komarath, Harshil Mittal, Saraswati Girish Nanoti, and Dhara Thakkar

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract
The celebrated result by Ben-Or and Cleve [SICOMP92] showed that algebraic formulas are polynomially equivalent to width-3 algebraic branching programs (ABP) for computing polynomials. i.e., VF = VBP₃. Further, there are simple polynomials, such as ∑_{i = 1}⁸ x_i y_i, that cannot be computed by width-2 ABPs [Allender and Wang, CC16]. Bringmann, Ikenmeyer and Zuiddam, [JACM18], on the other hand, studied these questions in the setting of approximate (i.e., border complexity) computation, and showed the universality of border width-2 ABPs, over fields of characteristic ≠ 2. In particular, they showed that polynomials that can be approximated by formulas can also be approximated (with only a polynomial blowup in size) by width-2 ABPs, i.e., VF ̅ = VBP₂ ̅. The power of border width-2 algebraic branching programs when the characteristic of the field is 2 was left open. In this paper, we show that width-2 ABPs can approximate every polynomial irrespective of the field characteristic. We show that any polynomial f with 𝓁 monomials and with at most t odd-power indeterminates per monomial can be approximated by 𝒪(𝓁⋅ (deg(f)+2^t))-size width-2 ABPs. Since 𝓁 and t are finite, this proves universality of border width-2 ABPs. For univariate polynomials, we improve this upper-bound from O(deg(f)²) to O(deg(f)). Moreover, we show that, if a polynomial f can be approximated by small formulas, then the polynomial f^d, for some small power d, can be approximated by small width-2 ABPs. Therefore, even over fields of characteristic two, border width-2 ABPs are a reasonably powerful computational model. Our construction works over any field.
Cite as

Pranjal Dutta, Christian Ikenmeyer, Balagopal Komarath, Harshil Mittal, Saraswati Girish Nanoti, and Dhara Thakkar. On the Power of Border Width-2 ABPs over Fields of Characteristic 2. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dutta_et_al:LIPIcs.STACS.2024.31,
  author =	{Dutta, Pranjal and Ikenmeyer, Christian and Komarath, Balagopal and Mittal, Harshil and Nanoti, Saraswati Girish and Thakkar, Dhara},
  title =	{{On the Power of Border Width-2 ABPs over Fields of Characteristic 2}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{31:1--31:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.31},
  URN =		{urn:nbn:de:0030-drops-197419},
  doi =		{10.4230/LIPIcs.STACS.2024.31},
  annote =	{Keywords: Algebraic branching programs, border complexity, characteristic 2}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.68
Spartan Bipartite Graphs Are Essentially Elementary

Authors: Neeldhara Misra and Saraswati Girish Nanoti

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
We study a two-player game on a graph between an attacker and a defender. To begin with, the defender places guards on a subset of vertices. In each move, the attacker attacks an edge. The defender must move at least one guard across the attacked edge to defend the attack. The defender wins if and only if the defender can defend an infinite sequence of attacks. The smallest number of guards with which the defender has a winning strategy is called the eternal vertex cover number of a graph G and is denoted by evc(G). It is clear that evc(G) is at least mvc(G), the size of a minimum vertex cover of G. We say that G is Spartan if evc(G) = mvc(G). The characterization of Spartan graphs has been largely open. In the setting of bipartite graphs on 2n vertices where every edge belongs to a perfect matching, an easy strategy is to have n guards that always move along perfect matchings in response to attacks. We show that these are essentially the only Spartan bipartite graphs.
Cite as

Neeldhara Misra and Saraswati Girish Nanoti. Spartan Bipartite Graphs Are Essentially Elementary. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{misra_et_al:LIPIcs.MFCS.2023.68,
  author =	{Misra, Neeldhara and Nanoti, Saraswati Girish},
  title =	{{Spartan Bipartite Graphs Are Essentially Elementary}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{68:1--68:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.68},
  URN =		{urn:nbn:de:0030-drops-186025},
  doi =		{10.4230/LIPIcs.MFCS.2023.68},
  annote =	{Keywords: Bipartite Graphs, Eternal Vertex Cover, Perfect Matchings, Elementary, Spartan}
}
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