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Documents authored by Sarma, Jayalal

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Sarma, Jayalal M. N.

Document
DOI: 10.4230/LIPIcs.STACS.2013.622
Pebbling, Entropy and Branching Program Size Lower Bounds

Authors: Balagopal Komarath and Jayalal M. N. Sarma

Published in: LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Abstract
We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced in (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012). Proving an exponential lower bound for the size of the non-deterministic thrifty branching programs would separate NL from P under the thrifty hypothesis. In this context, we consider a restriction of non-deterministic thrifty branching programs called bitwise-independence. We show that any bitwise-independent non-deterministic thrifty branching program solving BT_2(h,k) must have at least 1/2 k^{h/2} states. Prior to this work, lower bounds were known for general branching programs only for fixed heights h=2,3,4 (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012). Our lower bounds are also tight (up to a factor of k), since the known (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012) non-deterministic thrifty branching programs for this problem of size O(k^{h/2+1}) are bitwise-independent. We prove our results by associating a fractional pebbling strategy with any bitwise-independent non-deterministic thrifty branching program solving the Tree Evaluation Problem. Such a connection was not known previously even for fixed heights. Our main technique is the entropy method introduced by Jukna and Zak (S. Jukna and S. Žák, 2003) originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds known (Stephen A. Cook, Pierre McKenzie, Dustin Wehr, Mark Braverman, and Rahul Santhanam, 2012) for deterministic branching programs for Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any k-way deterministic branching program solving Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.
Cite as

Balagopal Komarath and Jayalal M. N. Sarma. Pebbling, Entropy and Branching Program Size Lower Bounds. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 622-633, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{komarath_et_al:LIPIcs.STACS.2013.622,
  author =	{Komarath, Balagopal and Sarma, Jayalal M. N.},
  title =	{{Pebbling, Entropy and Branching Program Size Lower Bounds}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{622--633},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-50-7},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{20},
  editor =	{Portier, Natacha and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.622},
  URN =		{urn:nbn:de:0030-drops-39709},
  doi =		{10.4230/LIPIcs.STACS.2013.622},
  annote =	{Keywords: Pebbling, Entropy Method, Branching Programs, Size Lower Bounds.}
}

Sarma, Jayalal

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.62
Avoiding Range via Turan-Type Bounds

Authors: Neha Kuntewar and Jayalal Sarma

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
Given a circuit C : {0,1}^n → {0,1}^m from a circuit class 𝒞, with m > n, finding a y ∈ {0,1}^m such that ∀ x ∈ {0,1}ⁿ, C(x) ≠ y, is the range avoidance problem (denoted by C-Avoid). Deterministic polynomial time algorithms (even with access to NP oracles) solving this problem are known to imply explicit constructions of various pseudorandom objects like hard Boolean functions, linear codes, PRGs etc. Deterministic polynomial time algorithms are known for NC⁰₂-Avoid when m > n, and for NC⁰₃-Avoid when m ≥ n²/log n, where NC⁰_k is the class of circuits with bounded fan-in which have constant depth and the output depends on at most k of the input bits. On the other hand, it is also known that NC⁰₃-Avoid when m = n+O(n^{2/3}) is at least as hard as explicit construction of rigid matrices. In fact, algorithms for solving range avoidance for even NC⁰₄ circuits imply new circuit lower bounds. In this paper, we propose a new approach to solving the range avoidance problem via hypergraphs. We formulate the problem in terms of Turan-type problems in hypergraphs of the following kind: for a fixed k-uniform hypergraph H, what is the maximum number of edges that can exist in H_C, which does not have a sub-hypergraph isomorphic to H? We show the following: - We first demonstrate the applicability of this approach by showing alternate proofs of some of the known results for the range avoidance problem using this framework. - We then use our approach to show (using several different hypergraph structures for which Turan-type bounds are known in the literature) that there is a constant c such that Monotone-NC⁰₃-Avoid can be solved in deterministic polynomial time when m > cn². - To improve the stretch constraint to linear, more precisely, to m > n, we show a new Turan-type theorem for a hypergraph structure (which we call the loose X_{2ℓ}-cycles). More specifically, we prove that any connected 3-uniform linear hypergraph with m > n edges must contain a loose X_{2ℓ} cycle. This may be of independent interest. - Using this, we show that Monotone-NC⁰₃-Avoid can be solved in deterministic polynomial time when m > n, thus improving the known bounds of NC⁰₃-Avoid for the case of monotone circuits. In contrast, we note that efficient algorithms for solving Monotone-NC⁰₆-Avoid, already imply explicit constructions for rigid matrices. - We also generalise our argument to solve the special case of range avoidance for NC⁰_k where each output function computed by the circuit is the majority function on its inputs, where m > n².
Cite as

Neha Kuntewar and Jayalal Sarma. Avoiding Range via Turan-Type Bounds. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 62:1-62:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kuntewar_et_al:LIPIcs.APPROX/RANDOM.2025.62,
  author =	{Kuntewar, Neha and Sarma, Jayalal},
  title =	{{Avoiding Range via Turan-Type Bounds}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{62:1--62:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.62},
  URN =		{urn:nbn:de:0030-drops-244281},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.62},
  annote =	{Keywords: circuit lower bounds, explicit constructions, range avoidance, linear hypergraphs, Tur\'{a}n number of hypergraphs}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.17
Sensitivity and Query Complexity Under Uncertainty

Authors: Deepu Benson, Balagopal Komarath, Nikhil Mande, Sai Soumya Nalli, Jayalal Sarma, and Karteek Sreenivasaiah

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
In this paper, we study the query complexity of Boolean functions in the presence of uncertainty, motivated by parallel computation with an unlimited number of processors where inputs are allowed to be unknown. We allow each query to produce three results: zero, one, or unknown. The output could also be: zero, one, or unknown, with the constraint that we should output "unknown" only when we cannot determine the answer from the revealed input bits. Such an extension of a Boolean function is called its hazard-free extension. - We prove an analogue of Huang’s celebrated sensitivity theorem [Annals of Mathematics, 2019] in our model of query complexity with uncertainty. - We show that the deterministic query complexity of the hazard-free extension of a Boolean function is at most quadratic in its randomized query complexity and quartic in its quantum query complexity, improving upon the best-known bounds in the Boolean world. - We exhibit an exponential gap between the smallest depth (size) of decision trees computing a Boolean function, and those computing its hazard-free extension. - We present general methods to convert decision trees for Boolean functions to those for their hazard-free counterparts, and show optimality of this construction. We also parameterize this result by the maximum number of unknown values in the input. - We show lower bounds on size complexity of decision trees for hazard-free extensions of Boolean functions in terms of the number of prime implicants and prime implicates of the underlying Boolean function.
Cite as

Deepu Benson, Balagopal Komarath, Nikhil Mande, Sai Soumya Nalli, Jayalal Sarma, and Karteek Sreenivasaiah. Sensitivity and Query Complexity Under Uncertainty. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{benson_et_al:LIPIcs.MFCS.2025.17,
  author =	{Benson, Deepu and Komarath, Balagopal and Mande, Nikhil and Nalli, Sai Soumya and Sarma, Jayalal and Sreenivasaiah, Karteek},
  title =	{{Sensitivity and Query Complexity Under Uncertainty}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{17:1--17:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.17},
  URN =		{urn:nbn:de:0030-drops-241240},
  doi =		{10.4230/LIPIcs.MFCS.2025.17},
  annote =	{Keywords: CREW-PRAM, query complexity, decision trees, sensitivity, hazard-free extensions}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2016.37
Characterization and Lower Bounds for Branching Program Size Using Projective Dimension

Authors: Krishnamoorthy Dinesh, Sajin Koroth, and Jayalal Sarma

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

Abstract
We study projective dimension, a graph parameter (denoted by pd(G) for a graph G), introduced by Pudlak and Rodl (1992). For a Boolean function f(on n bits), Pudlak and Rodl associated a bipartite graph G_f and showed that size of the optimal branching program computing f (denoted by bpsize(f)) is at least pd(G_f) (also denoted by pd(f)). Hence, proving lower bounds for pd(f) imply lower bounds for bpsize(f). Despite several attempts (Pudlak and Rodl (1992), Ronyai et.al, (2000)), proving super-linear lower bounds for projective dimension of explicit families of graphs has remained elusive. We observe that there exist a Boolean function f for which the gap between the pd(f) and bpsize(f) is 2^{Omega(n)}. Motivated by the argument in Pudlak and Rodl (1992), we define two variants of projective dimension - projective dimension with intersection dimension 1 (denoted by upd(f)) and {bitwise decomposable projective dimension} (denoted by bpdim(f)). We show the following results: (a) We observe that there exist a Boolean function f for which the gap between upd(f) and bpsize(f) is 2^{Omega(n)}. In contrast, we also show that the bitwise decomposable projective dimension characterizes size of the branching program up to a polynomial factor. That is, there exists a large constant c>0 and for any function f, bpdim(f)/6 <= bpsize(f) <= (bpdim(f))^c. (b) We introduce a new candidate function family f for showing super-polynomial lower bounds for bpdim(f). As our main result, we demonstrate gaps between pd(f) and the above two new measures for f: pd(f) = O(sqrt{n}), upd(f) = Omega(n), bpdim(f) = Omega({n^{1.5}}/{log(n)}). (c) Although not related to branching program lower bounds, we derive exponential lower bounds for two restricted variants of pd(f) and upd(f) respectively by observing that they are exactly equal to well-studied graph parameters - bipartite clique cover number and bipartite partition number respectively.
Cite as

Krishnamoorthy Dinesh, Sajin Koroth, and Jayalal Sarma. Characterization and Lower Bounds for Branching Program Size Using Projective Dimension. 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. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{dinesh_et_al:LIPIcs.FSTTCS.2016.37,
  author =	{Dinesh, Krishnamoorthy and Koroth, Sajin and Sarma, Jayalal},
  title =	{{Characterization and Lower Bounds for Branching Program Size Using Projective Dimension}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{37:1--37:14},
  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.2016.37},
  URN =		{urn:nbn:de:0030-drops-68722},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.37},
  annote =	{Keywords: Projective Dimension, Lower Bounds, Branching Program Size}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2014.597
Polynomial Min/Max-weighted Reachability is in Unambiguous Log-space

Authors: Anant Dhayal, Jayalal Sarma, and Saurabh Sawlani

Published in: LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

Abstract
For a graph G(V,E) and a vertex s in V, a weighting scheme (w : E -> N) is called a min-unique (resp. max-unique) weighting scheme, if for any vertex v of the graph G, there is a unique path of minimum (resp. maximum) weight from s to v. Instead, if the number of paths of minimum (resp. maximum) weight is bounded by n^c for some constant c, then the weighting scheme is called a min-poly (resp. max-poly) weighting scheme. In this paper, we propose an unambiguous non-deterministic log-space (UL) algorithm for the problem of testing reachability in layered directed acyclic graphs (DAGs) augmented with a min-poly weighting scheme. This improves the result due to Reinhardt and Allender [Reinhardt/Allender, SIAM J. Comp., 2000] where a UL algorithm was given for the case when the weighting scheme is min-unique. Our main technique is a triple inductive counting, which generalizes the techniques of [Immermann, Siam J. Comp.,1988; Szelepcsényi, Acta Inf.,1988] and [Reinhardt/Allender, SIAM J. Comp., 2000], combined with a hashing technique due to [Fredman et al.,J. ACM, 1984] (also used in [Garvin et al., Comp. Compl.,2014]). We combine this with a complementary unambiguous verification method, to give the desired UL algorithm. At the other end of the spectrum, we propose a UL algorithm for testing reachability in layered DAGs augmented with max-poly weighting schemes. To achieve this, we first reduce reachability in DAGs to the longest path problem for DAGs with a unique source, such that the reduction also preserves the max-poly property of the graph. Using our techniques, we generalize the double inductive counting method in [Limaye et al., CATS, 2009] where UL algorithms were given for the longest path problem on DAGs with a unique sink and augmented with a max-unique weighting scheme. An important consequence of our results is that, to show NL = UL, it suffices to design log-space computable min-poly (or max-poly) weighting schemes for DAGs.
Cite as

Anant Dhayal, Jayalal Sarma, and Saurabh Sawlani. Polynomial Min/Max-weighted Reachability is in Unambiguous Log-space. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 597-609, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{dhayal_et_al:LIPIcs.FSTTCS.2014.597,
  author =	{Dhayal, Anant and Sarma, Jayalal and Sawlani, Saurabh},
  title =	{{Polynomial Min/Max-weighted Reachability is in Unambiguous Log-space}},
  booktitle =	{34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)},
  pages =	{597--609},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-77-4},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{29},
  editor =	{Raman, Venkatesh and Suresh, S. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.597},
  URN =		{urn:nbn:de:0030-drops-48744},
  doi =		{10.4230/LIPIcs.FSTTCS.2014.597},
  annote =	{Keywords: Reachability Problem, Space Complexity, Unambiguous Algorithms}
}
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