arXiv subjects

Jayalal Sarma

Publications and source records attributed to Jayalal Sarma.

At least 19 recordsLinked to original sources

arXiv2026

On $CC^0$ Lower Bounds for AND via Torus Polynomials

We explore a torus polynomial approximation based approach towards a long-standing question: whether $AND$ can be computed by $CC^0$ circuits - the class of constant-depth polynomial size circuits containing $MOD_m$ gates for some $m$. Bhrushundi et al. (ITCS 2019) introduced torus polynomial approximations as an approach for proving lower bounds against $ACC^0$ - a class containing $CC^0$ with circuits comprising $AND$, $OR$ and $NOT$ gates. We show how lower bounds for torus polynomials approximating $AND$ can be used to make progress on this question. Using lower bounds on the degree of symmetric torus polynomials approximating $AND$ from Krishan and Vishwanathan (ITCS 2026), we prove size lower bounds for symmetric $CC^0$-circuits computing $AND$. More precisely, we prove that any depth $h$ symmetric $CC^0$ circuit requires $2^{\widetildeΩ(n^{1/O(h)})}$ size to compute $AND$. A key ingredient in our proof is an argument that we can construct symmetric torus polynomials to approximate symmetric $CC^0$ circuits. Our construction exhibits an explicit correspondence between the symmetry of the circuit and that of the polynomial. Using this, we also establish lower bounds for weaker notions of circuit symmetry. Lower bounds for symmetric $CC^0$ circuits were also independently established by Pago (ICALP 2026) using different techniques. In the asymmetric regime, we establish degree upper bounds for depth three circuits of the form $MOD_p \circ MOD_m \circ AND_{O(1)}$ where $m=pq$ is a semiprime. This circuit class is a special case of the constant degree hypothesis, introduced by Barrington, Straubing and Therien (Inf. and Comp., 1990), where $m$ could be an arbitrary composite number. We argue that improved lower bounds for asymmetric torus polynomials approximating $AND$ imply size lower bounds for semiprime $m$ and hence progress on the constant-degree hypothesis.

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arXiv2026

On the Reachability Problem on Monoid-Labelled Undirected Graphs

The labelled reachability problem for undirected graphs with edges labelled by elements of a monoid $M$ (more generally, groupoids or magmas) captures the classes $\sf{L}$ and $\sf{NL}$. Given a graph $G(V, E)$ labelled by $ϕ~\colon E \to M$, $s,t \in V$ and an accepting subset $F \subseteq M$, the problem asks to test whether there is a walk $P$ from $s$ to $t$ in $G$ where $ϕ(P) \in F$. Ramaswamy et al. (2019) studied the variant where the accepting element is part of the input for aperiodic monoids and groups. Motivated by the success in designing space-bounded algorithms for the undirected graph reachability problem, we study the labelled reachability problem when the accepting set is also fixed. This reveals finer complexity bounds and dichotomies for the problem based on the monoid and the accepting set. Previous results imply that the problem is in $\sf{L}$ for any finite accepting subset when $M$ is a group or belongs to $\sf{DA}$. We prove the following (for finite monoids): 1) For any monoid $M$, the problem is in $\sf{L}$ when the accepting element is the identity of $M$. If the accepting element is an idempotent, under suitable constraints, the problem is $\sf{NL}$-hard. 2) For any commutative monoid $M$, the problem is in $\sf{L}$ for all $F \subseteq M$. 3) For any $\mathcal{L}(\mathcal{R})$-commutative union-of-groups (UoG) monoid $M$, the problem is in $\sf{L}$ for all $F\subseteq M$. We show deterministic logspace algorithms for UoG monoids that are neither $\mathcal{L}$-commutative nor $\mathcal{R}$-commutative, under certain constraints. 4) For the monoids $\sf{BA_2}$ and $\sf{U}$, we show a dichotomy: for all $F \subseteq M$, the problem is either $\sf{NL}$-complete or in $\sf{L}$. Our results exploit the connection between Green's relations in the UoG monoids and the properties of the product graph (a graph introduced by Ramaswamy et al. (2019)).

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arXiv2026

Bounds for Hardness Condensation in the Query Model

For any Boolean function $f:\{0,1\}^n \to \{0,1\}$ with a complexity measure having value $k \ll n$, is it possible to restrict the function $f$ to $Θ(k)$ variables while keeping the complexity preserved at $Θ(k)$? This question, in the context of query complexity, was recently studied by G{ö}{ö}s, Newman, Riazanov and Sokolov (STOC 2024). They showed, among other results, that query complexity can not be condensed losslessly. They asked if complexity measures like block sensitivity or unambiguous certificate complexity can be condensed losslessly? In this work, we show that decision tree measures like block sensitivity and certificate complexity, cannot be condensed losslessly. That is, there exists a Boolean function $f$ such that any restriction of $f$ to $O(\mathcal{M}(f))$ variables has $\mathcal{M}(\cdot)$-complexity at most $\tilde{O}(\mathcal{M}(f)^{2/3})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{D}\}$. This also improves upon a result of G{ö}{ö}s, Newman, Riazanov and Sokolov (STOC 2024). We also complement the negative results on lossless condensation with positive results about lossy condensation. In particular, we show that for every Boolean function $f$ there exists a restriction of $f$ to $O(\mathcal{M}(f))$ variables such that its $\mathcal{M}(\cdot)$-complexity is at least $Ω(\mathcal{M}(f)^{1/2})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{UC}_{min},\mathsf{UC}_1,\mathsf{UC},\mathsf{D},\widetilde{\mathsf{deg}},λ\}$. We also show a slightly weaker positive result for randomized and quantum query complexity.

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arXiv2026

A Hierarchy of Tinhofer Graphs: Separations and Membership Testing

Color refinement is an important technique that works very well in practice for the graph isomorphism problem. Tinhofer graphs are the class of graphs for which refinement together with individualization correctly tests graph isomorphism against every other graph, irrespective of the choices of vertices made during individualization. Motivated by the fact that Tinhofer graphs form a natural boundary for efficient graph isomorphism tests based on color refinement, in this paper, we introduce a hierarchy of graph classes within the class of Tinhofer graphs. We call a graph $G$ $k$-Tinhofer if, after $k$ rounds of individualization and refinement, the resulting colored graphs remain isomorphic for every graph $H \cong G$, irrespective of the choices of vertices made during individualization. Arvind et al. (2017) studied a hierarchy of graph classes motivated by color refinement - discrete, amenable, Tinhofer, and refinable graphs. We show that the $k$-Tinhofer hierarchy lies between the class of all graphs and Tinhofer graphs, with refinable graphs coinciding with the first level of the hierarchy. We obtain two characterizations of $k$-Tinhofer graphs: an algebraic characterization in terms of orbit partitions induced by pointwise stabilizers of automorphism groups, and a combinatorial characterization in terms of individualization-refinement trees and quotient graphs. For every fixed integer $k \ge 0$, there exist vertex-colored graphs that are $k$-Tinhofer but not $(k + 1)$-Tinhofer. For every fixed integer $k \ge 0$, the problem of deciding whether a given $k$-Tinhofer graph is ($k + 1$)-Tinhofer is $P$-hard under uniform $\mathsf{AC^0}$ many-one reductions. We show that testing isomorphism between an $(n - k)$-Tinhofer graph $G$ and an arbitrary graph $H$ is fixed-parameter tractable with respect to the parameter $k$.

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arXiv2026

VP, VNP and Algebraic Branching Programs over Min-Plus Semirings

Arithmetic circuit complexity studies the complexity of computing polynomials using only arithmetic operations such as addition, multiplication, subtraction, and division. Polynomials over rings of integers model counting problems. Similarly, polynomials over semirings such as tropical semirings model optimization problems. Circuits over semirings then model so called pure algorithms, algorithms that only use the operations in the semiring. In this paper, we do a complexity-theoretic study of the power and limitations of circuits (which represent dynamic programs) over semirings: i) We define $\mathsf{VNP}$ over min-plus semirings, which can faithfully represent problems such as computing min-weight perfect matchings and min-weight Hamiltonian cycles where we have efficiently verifiable certificates. Unlike over rings, we complement the values in the certificate for free as complementation is impossible over min-plus semirings. We prove a dichotomy theorem that states that if we only complement logarithmically many values, this class is same as $\mathsf{VP}$ over min-plus semirings. If we complement super-logarithmically many values, then $\mathsf{VNP} \neq \mathsf{VP}$. ii) We consider constant-width ABPs (which are also called incremental dynamic programs that are restricted to use only a constant number of registers) and show that even simple problems like computing the min-weight $2$-edge-matching is impossible with width $2$ (or $2$ registers). However, with width $3$ (or $3$ registers), such programs can compute everything. More generally, we show that constant-depth formulas are efficiently simulated by constant-width ABPs. iii) We show that an exponential hypercube sum (min in the semiring) over even provably weak models such as width-$2$ ABPs and products of linear forms are the same as $\mathsf{VNP}$.

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arXiv2026

On Condensation of Block Sensitivity, Certificate Complexity and the $\mathsf{AND}$ (and $\mathsf{OR}$) Decision Tree Complexity

Given an $n$-bit Boolean function with a complexity measure (such as block sensitivity, query complexity, etc.) $M(f) = k$, the hardness condensation question asks whether $f$ can be restricted to $O(k)$ variables such that the complexity measure is $Ω(k)$? In this work, we study the condensability of block sensitivity, certificate complexity, AND (and OR) query complexity and Fourier sparsity. We show that block sensitivity does not condense under restrictions, unlike sensitivity: there exists a Boolean function $f$ with query complexity $k$ such that any restriction of $f$ to $O(k)$ variables has block sensitivity $O(k^{\frac{2}{3}})$. This answers an open question in Göös, Newman, Riazanov, and Sokolov (2024) in the negative. The same function yields an analogous incondensable result for certificate complexity. We further show that $\mathsf{AND}$(and $\mathsf{OR}$) decision trees are also incondensable.

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arXiv2025

Range Avoidance in Boolean Circuits via Turan-type Bounds

Given a circuit $C : \{0,1\}^n \to \{0,1\}^m$ from a circuit class $F$, with $m > n$, finding a $y \in \{0,1\}^m$ such that $\forall x \in \{0,1\}^n$, $C(x) \ne y$, is the range avoidance problem (denoted by $F$-$avoid$). Deterministic polynomial time algorithms (even with access to $NP$ oracles) solving this problem is known to imply explicit constructions of various pseudorandom objects like hard Boolean functions, linear codes, PRGs etc. Deterministic polynomial time algorithms are known for $NC^0_2$-$avoid$ when $m > n$, and for $NC^0_3$-$avoid$ when $m \ge \frac{n^2}{\log n}$, where $NC^0_k$ 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 $NC^0_3$-$avoid$ when $m = n+O\left(n^{2/3}\right)$ is at least as hard as explicit construction of rigid matrices. In this paper, we propose a new approach to solving 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 a $k$-uniform hypergraph $H$ which does not have a sub-hypergraph isomorphic to $H'$? We use our approach to show (using known Turan-type bounds) that there is a constant $c$ such that $mon$-$NC^0_3$-$avoid$ can be solved in deterministic polynomial time when $m > cn^2$. To improve the stretch constraint to linear, we show a new Turan-type theorem for a hypergraph structure (which we call the the loose $chi$-cycles) and use it to show that $mon$-$NC^0_3$-$avoid$ can be solved in deterministic polynomial time when $m > n$, thus improving the known bounds of $NC^0_3$-avoid for the case of monotone circuits.

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arXiv2025

Sensitivity and Query Complexity under Uncertainty

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.

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arXiv2025

Hazard-free Decision Trees

Decision trees are one of the most fundamental computational models for computing Boolean functions $f : \{0, 1\}^n \mapsto \{0, 1\}$. It is well-known that the depth and size of decision trees are closely related to time and number of processors respectively for computing functions in the CREW-PRAM model. For a given $f$, a fundamental goal is to minimize the depth and/or the size of the decision tree computing it. In this paper, we extend the decision tree model to the world of hazard-free computation. 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 input bits. This setting naturally gives rise to ternary decision trees computing functions, which we call hazard-free decision trees. We prove various lower and upper bounds on the depth and size of hazard-free decision trees and compare them to their Boolean counterparts. We prove optimal separations and relate hazard-free decision tree parameters to well-known Boolean function parameters. We show that the analogues of sensitivity, block sensitivity, and certificate complexity for hazard-free functions are all polynomially equivalent to each other and to hazard-free decision tree depth. i.e., we prove the sensitivity theorem in the hazard-free model. We then prove that hazard-free sensitivity satisfies an interesting structural property that is known to hold in the Boolean world. Hazard-free functions with small hazard-free sensitivity are completely determined by their values in any Hamming ball of small radius in $\{0, u, 1\}^n$.

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arXiv2024

Almost-catalytic Computation

Designing algorithms for space bounded models with restoration requirements on the space used by the algorithm is an important challenge posed about the catalytic computation model introduced by Buhrman et al. (2014). Motivated by the scenarios where we do not need to restore unless is useful, we define $ACL(A)$ to be the class of languages that can be accepted by almost-catalytic Turing machines with respect to $A$ (which we call the catalytic set), that uses at most $c\log n$ work space and $n^c$ catalytic space. We show that if there are almost-catalytic algorithms for a problem with catalytic set as $A \subseteq Σ^*$ and its complement respectively, then the problem can be solved by a ZPP algorithm. Using this, we derive that to design catalytic algorithms, it suffices to design almost-catalytic algorithms where the catalytic set is the set of strings of odd weight ($PARITY$). Towards this, we consider two complexity measures of the set $A$ which are maximized for $PARITY$ - random projection complexity (${\cal R}(A)$) and the subcube partition complexity (${\cal P}(A)$). By making use of error-correcting codes, we show that for all $k \ge 1$, there is a language $A_k \subseteq Σ^*$ such that $DSPACE(n^k) \subseteq ACL(A_k)$ where for every $m \ge 1$, $\mathcal{R}(A_k \cap \{0,1\}^m) \ge \frac{m}{4}$ and $\mathcal{P}(A_k \cap \{0,1\}^m)=2^{m/4}$. This contrasts the catalytic machine model where it is unclear if it can accept all languages in $DSPACE(\log^{1+ε} n)$ for any $ε> 0$. Improving the partition complexity of the catalytic set $A$ further, we show that for all $k \ge 1$, there is a $A_k \subseteq \{0,1\}^*$ such that $\mathsf{DSPACE}(\log^k n) \subseteq ACL(A_k)$ where for every $m \ge 1$, $\mathcal{R}(A_k \cap \{0,1\}^m) \ge \frac{m}{4}$ and $\mathcal{P}(A_k \cap \{0,1\}^m)=2^{m/4+Ω(\log m)}$.

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arXiv2024

On Rotation Distance of Rank Bounded Trees

Computing the rotation distance between two binary trees with $n$ internal nodes efficiently (in $poly(n)$ time) is a long standing open question in the study of height balancing in tree data structures. In this paper, we initiate the study of this problem bounding the rank of the trees given at the input (defined by Ehrenfeucht and Haussler (1989) in the context of decision trees). We define the rank-bounded rotation distance between two given binary trees $T_1$ and $T_2$ (with $n$ internal nodes) of rank at most $r$, denoted by $d_r(T_1,T_2)$, as the length of the shortest sequence of rotations that transforms $T_1$ to $T_2$ with the restriction that the intermediate trees must be of rank at most $r$. We show that the rotation distance problem reduces in polynomial time to the rank bounded rotation distance problem. This motivates the study of the problem in the combinatorial and algorithmic frontiers. Observing that trees with rank $1$ coincide exactly with skew trees (binary trees where every internal node has at least one leaf as a child), we show the following results in this frontier : We present an $O(n^2)$ time algorithm for computing $d_1(T_1,T_2)$. That is, when the given trees are skew trees (we call this variant as skew rotation distance problem) - where the intermediate trees are restricted to be skew as well. In particular, our techniques imply that for any two skew trees $d(T_1,T_2) \le n^2$. We show the following upper bound : for any two trees $T_1$ and $T_2$ of rank at most $r_1$ and $r_2$ respectively, we have that: $d_r(T_1,T_2) \le n^2 (1+(2n+1)(r_1+r_2-2))$ where $r = max\{r_1,r_2\}$. This bound is asymptotically tight for $r=1$. En route our proof of the above theorems, we associate binary trees to permutations and bivariate polynomials, and prove several characterizations in the case of skew trees.

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arXiv2022

Power of Decision Trees with Monotone Queries

In this paper, we initiate study of the computational power of adaptive and non-adaptive monotone decision trees - decision trees where each query is a monotone function on the input bits. In the most general setting, the monotone decision tree height (or size) can be viewed as a measure of non-monotonicity of a given Boolean function. We also study the restriction of the model by restricting (in terms of circuit complexity) the monotone functions that can be queried at each node. This naturally leads to complexity classes of the form DT(mon-C) for any circuit complexity class C, where the height of the tree is O(log n), and the query functions can be computed by monotone circuits in the class C. In the above context, we prove the following characterizations and bounds. For any Boolean function f, we show that the minimum monotone decision tree height can be exactly characterized (both in the adaptive and non-adaptive versions of the model) in terms of its alternation (alt(f) is defined as the maximum number of times that the function value changes, in any chain in the Boolean lattice). We also characterize the non-adaptive decision tree height with a natural generalization of certification complexity of a function. Similarly, we determine the complexity of non-deterministic and randomized variants of monotone decision trees in terms of alt(f). We show that DT(mon-C) = C when C contains monotone circuits for the threshold functions (for e.g., if C = TC0). For C = AC0, we are able to show that any function in AC0 can be computed by a sub-linear height monotone decision tree with queries having monotone AC0 circuits. To understand the logarithmic height case in case of AC0 i.e., DT(mon-AC0), we show that functions in DT(mon-AC0) have AC0 circuits with few negation gates.

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arXiv2020

Sensitivity, Affine Transforms and Quantum Communication Complexity

$\newcommand{\F}{\mathbb{F}}$We study the Boolean function parameters sensitivity ($s$), block sensitivity ($bs$), and alternation ($alt$) under specially designed affine transforms. For a function $f:\F_2^n\to \{0,1\}$, and $A=Mx+b$ for $M \in \F_2^{n\times n}$ and $b\in \F_2^n$, the result of the transformation $g$ is defined as $\forall x\in\F_2^n, g(x)=f(Mx+b)$. We study alternation under linear shifts ($M$ is the identity matrix) called the shift invariant alternation (denoted by $salt(f)$). We exhibit an explicit family of functions for which $salt(f)$ is $2^{Ω(s(f))}$. We show an affine transform $A$, such that the corresponding function $g$ satisfies $bs(f,0^n) \le s(g)$, using which we proving that for $F(x,y)=f(x\land y)$, the bounded error quantum communication complexity of $F$ with prior entanglement, $Q^*_{1/3}(F)=Ω(\sqrt{bs(f,0^n)})$. Our proof builds on ideas from Sherstov (2010) where we use specific properties of the above affine transformation. We show, * For a prime $p$ and $0<ε<1$, any $f$ with $deg_p(f)\le(1-ε)\log n$ must satisfy $Q^*_{1/3}(F) = Ω(\frac{n^{ε/2}}{\log n})$. Here, $deg_p(f)$ denotes the degree of the multilinear polynomial of $f$ over $\F_p$. * For any $f$ such that there exists primes $p$ and $q$ with $deg_q(f) \ge Ω(deg_p(f)^δ)$ for $δ> 2$, the deterministic communication complexity - $D(F)$ and $Q^*_{1/3}(F)$ are polynomially related. In particular, this holds when $deg_p(f) = O(1)$. Thus, for this class of functions, this answers an open question (see Buhrman and deWolf (2001)) about the relation between the two measures. We construct linear transformation $A$, such that $g$ satisfies, $alt(f) \le 2s(g)+1$. Using this, we exhibit a family of Boolean functions that rule out a potential approach to settle the XOR Log-Rank conjecture via a proof of Sensitivity conjecture [Hao Huang (2019)].

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arXiv2020

New Bounds for Energy Complexity of Boolean Functions

$\newcommand{\EC}{\mathsf{EC}}\newcommand{\KW}{\mathsf{KW}}\newcommand{\DT}{\mathsf{DT}}\newcommand{\psens}{\mathsf{psens}} \newcommand{\calB}{\cal B} $ For a Boolean function $f:\{0,1\}^n \to \{0,1\}$ computed by a circuit $C$ over a finite basis $\mathcal{B}$, the energy complexity of $C$ (denoted by $\EC_{\calB}(C)$) is the maximum over all inputs $\{0,1\}^n$ the numbers of gates of the circuit $C$ (excluding the inputs) that output a one. Energy Complexity of a Boolean function over a finite basis $\calB$ denoted by $\EC_\calB(f):= \min_C \EC_{\calB}(C)$ where $C$ is a circuit over $\calB$ computing $f$. We study the case when $\calB = \{\land_2, \lor_2, \lnot\}$, the standard Boolean basis. It is known that any Boolean function can be computed by a circuit (with potentially large size) with an energy of at most $3n(1+ε(n))$ for a small $ ε(n)$(which we observe is improvable to $3n-1$). We show several new results and connections between energy complexity and other well-studied parameters of Boolean functions. * For all Boolean functions $f$, $\EC(f) \le O(\DT(f)^3)$ where $\DT(f)$ is the optimal decision tree depth of $f$. * We define a parameter \textit{positive sensitivity} (denoted by $\psens$), a quantity that is smaller than sensitivity and defined in a similar way, and show that for any Boolean circuit $C$ computing a Boolean function $f$, $ \EC(C) \ge \psens(f)/3$. * For a monotone function $f$, we show that $\EC(f) = Ω(\KW^+(f))$ where $\KW^+(f)$ is the cost of monotone Karchmer-Wigderson game of $f$. * Restricting the above notion of energy complexity to Boolean formulas, we show $\EC(F) = Ω\left (\sqrt{L(F)}-depth(F)\right )$ where $L(F)$ is the size and $depth(F)$ is the depth of a formula $F$.

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arXiv2019

Alternation, Sparsity and Sensitivity : Bounds and Exponential Gaps

$\newcommand{\sp}{\mathsf{sparsity}}\newcommand{\s}{\mathsf{s}}\newcommand{\al}{\mathsf{alt}}$ The well-known Sensitivity Conjecture states that for any Boolean function $f$, block sensitivity of $f$ is at most polynomial in sensitivity of $f$ (denoted by $\s(f)$). The XOR Log-Rank Conjecture states that for any $n$ bit Boolean function, $f$ the communication complexity of a related function $f^{\oplus}$ on $2n$ bits, (defined as $f^{\oplus}(x,y)=f(x\oplus y)$) is at most polynomial in logarithm of the sparsity of $f$ (denoted by $\sp(f)$). A recent result of Lin and Zhang (2017) implies that to confirm the above conjectures it suffices to upper bound alternation of $f$ (denoted $\al(f)$) for all Boolean functions $f$ by polynomial in $\s(f)$ and logarithm of $\sp(f)$, respectively. In this context, we show the following : * There exists a family of Boolean functions for which $\al(f)$ is at least exponential in $\s(f)$ and $\al(f)$ is at least exponential in $\log \sp(f)$. En route to the proof, we also show an exponential gap between $\al(f)$ and the decision tree complexity of $f$, which might be of independent interest. * As our main result, we show that, despite the above gap between $\al(f)$ and $\log \sp(f)$, the XOR Log-Rank Conjecture is true for functions with the alternation upper bounded by $poly(\log n)$. It is easy to observe that the Sensitivity Conjecture is also true for this class of functions. * The starting point for the above result is the observation (derived from Lin and Zhang (2017)) that for any Boolean function $f$ and $m \ge 2$, $deg(f)\le \al(f)deg_2(f)deg_m(f)$ where $deg(f)$, $deg_2(f)$ and $deg_m(f)$ are the degrees of $f$ over $\mathbb{R}$, $\mathbb{F}_2$ and $\mathbb{Z}_m$ respectively. We also show three further applications of this observation.

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arXiv2017

Comparator Circuits over Finite Bounded Posets

Comparator circuit model was originally introduced by Mayr and Subramanian (1992) (and further studied by Cook, Filmus and Le (2012)) to capture problems which are not known to be P-complete but still not known to admit efficient parallel algorithms. The class CC is the complexity class of problems many-one logspace reducible to the Comparator Circuit Value Problem and we know that NL is contained in CC which is inturn contained in P. Cook, Filmus and Le (2012) showed that CC is also the class of languages decided by polynomial size comparator circuits. We study generalizations of the comparator circuit model that work over fixed finite bounded posets. We observe that there are universal comparator circuits even over arbitrary fixed finite bounded posets. Building on this, we show that general (resp. skew) comparator circuits of polynomial size over fixed finite distributive lattices characterizes CC (resp. L). Complementing this, we show that general comparator circuits of polynomial size over arbitrary fixed finite lattices exactly characterizes P even when the comparator circuit is skew. In addition, we show a characterization of the class NP by a family of polynomial sized comparator circuits over fixed {\em finite bounded posets}. These results generalize the results by Cook, Filmus and Le (2012) regarding the power of comparator circuits. As an aside, we consider generalizations of Boolean formulae over arbitrary lattices. We show that Spira's theorem (1971) can be extended to this setting as well and show that polynomial sized Boolean formulae over finite fixed lattices capture exactly NC^1.

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arXiv2017

Characterization and Lower Bounds for Branching Program Size using Projective Dimension

We study projective dimension, a graph parameter (denoted by pd$(G)$ for a graph $G$), introduced by (Pudlák, Rödl 1992), who showed that proving lower bounds for pd$(G_f)$ for bipartite graphs $G_f$ associated with a Boolean function $f$ imply size lower bounds for branching programs computing $f$. Despite several attempts (Pudlák, Rödl 1992 ; Babai, Rónyai, Ganapathy 2000), proving super-linear lower bounds for projective dimension of explicit families of graphs has remained elusive. We show that there exist a Boolean function $f$ (on $n$ bits) for which the gap between the projective dimension and size of the optimal branching program computing $f$ (denoted by bpsize$(f)$), is $2^{Ω(n)}$. Motivated by the argument in (Pudlák, Rödl 1992), we define two variants of projective dimension - projective dimension with intersection dimension 1 (denoted by upd$(G)$) and bitwise decomposable projective dimension (denoted by bitpdim$(G)$). As our main result, we show that there is an explicit family of graphs on $N = 2^n$ vertices such that the projective dimension is $O(\sqrt{n})$, the projective dimension with intersection dimension $1$ is $Ω(n)$ and the bitwise decomposable projective dimension is $Ω(\frac{n^{1.5}}{\log n})$. We also show that there exist a Boolean function $f$ (on $n$ bits) for which the gap between upd$(G_f)$ and bpsize$(f)$ is $2^{Ω(n)}$. 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 constant $c>0$ and for any function $f$, $\textrm{bitpdim}(G_f)/6 \le \textrm{bpsize}(f) \le (\textrm{bitpdim}(G_f))^c$. We also study two other variants of projective dimension and show that they are exactly equal to well-studied graph parameters - bipartite clique cover number and bipartite partition number respectively.

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arXiv2017

On the Complexity of L-reachability

We initiate a complexity theoretic study of the language based graph reachability problem (L-REACH) : Fix a language L. Given a graph whose edges are labeled with alphabet symbols of the language L and two special vertices s and t, test if there is path P from s to t in the graph such that the concatenation of the symbols seen from s to t in the path P forms a string in the language L. We study variants of this problem with different graph classes and different language classes and obtain complexity theoretic characterizations for all of them. Our main results are the following: 1.Restricting the language using formal language theory we show that the complexity of L-REACH increases with the power of the formal language class. We show that there is a regular language for which the L-REACH is NL-complete even for undirected graphs. In the case of linear languages, the complexity of L-REACH does not go beyond the complexity of L itself. Further, there is a deterministic context-free language L for which L-DAGREACH is LogCFL-complete. 2.We use L-REACH as a lens to study structural complexity. In this direction we show that there is a language A in TC0 for which A-DAGREACH is NP-complete. Using this we show that P vs NP question is equivalent to P vs DAGREACH-1(P) question. This leads to the intriguing possibility that by proving DAGREACH-1(P) is contained in some subclass of P, we can prove an upward translation of separation of complexity classes. Note that we do not know a way to upward translate the separation of complexity classes.

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