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DOI: 10.4230/LIPIcs.STACS.2021.40

Parameterised Counting in Logspace

Authors: Anselm Haak, Arne Meier, Om Prakash, and Raghavendra Rao B. V.

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract

Logarithmic space bounded complexity classes such as L and NL play a central role in space bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space bounded models developed by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined the operators para_W and para_β for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators para_W and para_β by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, para_{W[1]} and para_{βtail}. Then, we consider counting versions of all four operators applied to logspace and obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0,1)-matrices is #para_{βtail} L-hard and can be written as the difference of two functions in #para_{βtail} L. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of #para_{βtail} L under parameterised logspace parsimonious reductions coincides with #para_β L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Also, we show that the counting classes defined can naturally be characterised by parameterised variants of classes based on branching programs in analogy to the classical counting classes. Finally, we underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research.

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Anselm Haak, Arne Meier, Om Prakash, and Raghavendra Rao B. V.. Parameterised Counting in Logspace. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{haak_et_al:LIPIcs.STACS.2021.40,
  author =	{Haak, Anselm and Meier, Arne and Prakash, Om and Rao B. V., Raghavendra},
  title =	{{Parameterised Counting in Logspace}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{40:1--40:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.40},
  URN =		{urn:nbn:de:0030-drops-136859},
  doi =		{10.4230/LIPIcs.STACS.2021.40},
  annote =	{Keywords: Parameterized Complexity, Counting Complexity, Logspace}
}

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DOI: 10.4230/LIPIcs.MFCS.2019.52

Lower Bounds for Multilinear Order-Restricted ABPs

Authors: C. Ramya and B. V. Raghavendra Rao

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract

Proving super-polynomial lower bounds on the size of syntactic multilinear Algebraic Branching Programs (smABPs) computing an explicit polynomial is a challenging problem in Algebraic Complexity Theory. The order in which variables in {x_1,...,x_n} appear along any source to sink path in an smABP can be viewed as a permutation in S_n. In this article, we consider the following special classes of smABPs where the order of occurrence of variables along a source to sink path is restricted: 1) Strict circular-interval ABPs: For every sub-program the index set of variables occurring in it is contained in some circular interval of {1,..., n}. 2) L-ordered ABPs: There is a set of L permutations (orders) of variables such that every source to sink path in the smABP reads variables in one of these L orders, where L <=2^{n^{1/2 -epsilon}} for some epsilon>0. We prove exponential (i.e., 2^{Omega(n^delta)}, delta>0) lower bounds on the size of above models computing an explicit multilinear 2n-variate polynomial in VP. As a main ingredient in our lower bounds, we show that any polynomial that can be computed by an smABP of size S, can be written as a sum of O(S) many multilinear polynomials where each summand is a product of two polynomials in at most 2n/3 variables, computable by smABPs. As a corollary, we show that any size S syntactic multilinear ABP can be transformed into a size S^{O(sqrt{n})} depth four syntactic multilinear Sigma Pi Sigma Pi circuit where the bottom Sigma gates compute polynomials on at most O(sqrt{n}) variables. Finally, we compare the above models with other standard models for computing multilinear polynomials.

Cite as

C. Ramya and B. V. Raghavendra Rao. Lower Bounds for Multilinear Order-Restricted ABPs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ramya_et_al:LIPIcs.MFCS.2019.52,
  author =	{Ramya, C. and Rao, B. V. Raghavendra},
  title =	{{Lower Bounds for Multilinear Order-Restricted ABPs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{52:1--52:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.52},
  URN =		{urn:nbn:de:0030-drops-109963},
  doi =		{10.4230/LIPIcs.MFCS.2019.52},
  annote =	{Keywords: Computational complexity, Algebraic complexity theory, Polynomials}
}

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DOI: 10.4230/LIPIcs.FSTTCS.2016.39

Sum of Products of Read-Once Formulas

Authors: Ramya C. and B. V. Raghavendra Rao

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract

We study limitations of polynomials computed by depth two circuits built over read-once formulas (ROFs). In particular, 1. We prove an exponential lower bound for the sum of ROFs computing the 2n-variate polynomial in VP defined by Raz and Yehudayoff [CC,2009]. 2. We obtain an exponential lower bound on the size of arithmetic circuits computing sum of products of restricted ROFs of unbounded depth computing the permanent of an n by n matrix. The restriction is on the number of variables with + gates as a parent in a proper sub formula of the ROF to be bounded by sqrt(n). Additionally, we restrict the product fan in to be bounded by a sub linear function. This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial. 3. We also show an exponential lower bound for the above model against a polynomial in VP. 4. Finally we observe that the techniques developed yield an exponential lower bound on the size of sums of products of syntactically multilinear arithmetic circuits computing a product of variable disjoint linear forms where the bottom sum gate and product gates at the second level have fan in bounded by a sub linear function. Our proof techniques are built on the measure developed by Kumar et al.[ICALP 2013] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz [STOC 2004].

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Ramya C. and B. V. Raghavendra Rao. Sum of Products of Read-Once Formulas. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{c._et_al:LIPIcs.FSTTCS.2016.39,
  author =	{C., Ramya and Rao, B. V. Raghavendra},
  title =	{{Sum of Products of Read-Once Formulas}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{39:1--39:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.39},
  URN =		{urn:nbn:de:0030-drops-68741},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.39},
  annote =	{Keywords: Arithmetic Circuits, Permanent, Computational Complexity, Algebraic Complexity Theory}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2011.115

Isomorphism testing of read-once functions and polynomials

Authors: Raghavendra Rao B .V. and Jayalal Sarma M. N.

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Abstract

In this paper, we study the isomorphism testing problem of formulas in the Boolean and arithmetic settings. We show that isomorphism testing of Boolean formulas in which a variable is read at most once (known as read-once formulas) is complete for log-space. In contrast, we observe that the problem becomes polynomial time equivalent to the graph isomorphism problem, when the input formulas can be represented as OR of two or more monotone read-once formulas. This classifies the complexity of the problem in terms of the number of reads, as read-3 formula isomorphism problem is hard for \co\NP. We address the polynomial isomorphism problem, a special case of polynomial equivalence problem which in turn is important from a cryptographic perspective[Patarin EUROCRYPT'96, and Kayal SODA'11]. As our main result, we propose a deterministic polynomial time canonization scheme for polynomials computed by constant-free read-once arithmetic formulas. In contrast, we show that when the arithmetic formula is allowed to read a variable twice, this problem is as hard as the graph isomorphism problem.

Cite as

Raghavendra Rao B .V. and Jayalal Sarma M. N.. Isomorphism testing of read-once functions and polynomials. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 115-126, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{raob.v._et_al:LIPIcs.FSTTCS.2011.115,
  author =	{Rao B .V., Raghavendra and Sarma M. N., Jayalal},
  title =	{{Isomorphism testing of read-once functions and polynomials}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{115--126},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.115},
  URN =		{urn:nbn:de:0030-drops-33202},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.115},
  annote =	{Keywords: Isomorphism Problems, Computational Complexity, Read-once formulas, Read-once Polynomials}
}

Document

DOI: 10.4230/DagSemProc.09421.7

Small space analogues of Valiant's classes and the limitations of skew formula

Authors: Meena Mahajan and Raghavendra Rao B. V.

Published in: Dagstuhl Seminar Proceedings, Volume 9421, Algebraic Methods in Computational Complexity (2010)

Abstract

In the uniform circuit model of computation, the width of a boolean circuit exactly characterises the ``space'' complexity of the computed function. Looking for a similar relationship in Valiant's algebraic model of computation, we propose width of an arithmetic circuit as a possible measure of space. We introduce the class VL as an algebraic variant of deterministic log-space L. In the uniform setting, we show that our definition coincides with that of VPSPACE at polynomial width. Further, to define algebraic variants of non-deterministic space-bounded classes, we introduce the notion of ``read-once'' certificates for arithmetic circuits. We show that polynomial-size algebraic branching programs can be expressed as a read-once exponential sum over polynomials in VL, ie $mbox{VBP}inSigma^R cdotmbox{VL}$. We also show that $Sigma^R cdot mbox{VBP} =mbox{VBP}$, ie VBPs are stable under read-once exponential sums. Further, we show that read-once exponential sums over a restricted class of constant-width arithmetic circuits are within VQP, and this is the largest known such subclass of poly-log-width circuits with this property. We also study the power of skew formulas and show that exponential sums of a skew formula cannot represent the determinant polynomial.

Cite as

Meena Mahajan and Raghavendra Rao B. V.. Small space analogues of Valiant's classes and the limitations of skew formula. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{mahajan_et_al:DagSemProc.09421.7,
  author =	{Mahajan, Meena and Rao B. V., Raghavendra},
  title =	{{Small space analogues of Valiant's classes and the limitations of   skew formula}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--23},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{9421},
  editor =	{Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.7},
  URN =		{urn:nbn:de:0030-drops-24126},
  doi =		{10.4230/DagSemProc.09421.7},
  annote =	{Keywords: Algebraic circuits, space bounds, circuit width, nondeterministic circuits, skew formulas}
}

5 Search Results for "Rao, B. V. Raghavendra"

Parameterised Counting in Logspace

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Lower Bounds for Multilinear Order-Restricted ABPs

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Sum of Products of Read-Once Formulas

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Isomorphism testing of read-once functions and polynomials

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Small space analogues of Valiant's classes and the limitations of skew formula

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