For any Boolean function f:{0,1}ⁿ → {0,1} with a complexity measure having value k ≪ n, is it possible to restrict the function f to Θ(k) variables while keeping the complexity preserved at Θ(k)? Instantiation of this question for the measure of circuit complexity of the Boolean function was shown to be related to circuit lower bounds (Buresh-Oppenheim and Santhanam, 2006). Variants of the above question were also shown to have connections to the log-rank conjecture in communication complexity (Hrubeš, 2024) and lower bounds in proof complexity (Razborov, 2016). In the context of communication and query complexity, this question was recently studied by Göös, Newman, Riazanov and Sokolov (2024). They showed, among other results, that query complexity cannot be condensed losslessly. In this work, we show that there exists a Boolean function f such that any restriction of f to O(ℳ(f)) variables has ℳ(⋅)-complexity at most Õ(ℳ(f)^{2/3}), where ℳ is one of block sensitivity (bs), fractional block sensitivity (fbs), certificate complexity (𝖢), deterministic query complexity (𝖣), zero-error randomized query complexity (𝖱₀), and AND (and OR)-decision tree query complexity. This improves upon the results of Göös, Newman, Riazanov, and Sokolov (2024) for 𝖣 and 𝖱₀, and in particular answers their open question about the condensation of block sensitivity. We 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(ℳ(f)) variables such that its ℳ(⋅)-complexity is at least Ω(ℳ(f)^{1/2}), where ℳ ∈ {bs,fbs,𝖢,UC_{min},UC₁,UC,𝖣,deg̃,λ}. In addition, we show lossy condensation for randomized and quantum query complexity with a slightly smaller exponent.

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: - We define 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 VP over min-plus semirings. If we complement super-logarithmically many values, then VNP ≠ VP. - 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. - 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 VNP.

We explore the torus polynomial approximation based approach towards a long-standing question: whether AND can be computed by CC⁰ circuits - the class of constant-depth polynomial size circuits containing MOD_m gates for some natural number m. Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) introduced torus polynomial approximations as an approach for proving lower bounds against ACC⁰ - a class containing CC⁰ where the circuits are also allowed 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, proved by Krishan and Vishwanathan (ITCS 2026), we prove size lower bounds for symmetric CC⁰-circuits computing AND. More precisely, we prove that any depth h symmetric CC⁰ circuit requires 2^Ω̃(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 CC⁰ 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 CC⁰ 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∘MOD_m∘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 Thérien (Information and Computation, 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.

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.

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 in [1] in the context of decision trees). We define the rank-bounded rotation distance between two given full binary trees T 1 and T 2 (with n internal nodes) of rank at most r = max{rank( T 1 ), rank( T 2 )}, 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 (full 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 R ( T 1 , T 2 ). That is, when the given full binary trees are skew trees (we call this variant the 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 R ( T 1 , T 2 ) ≤ n 2 . We show the following upper bound: for any two full binary trees T 1 and T 2 of rank r 1 and r 2 respectively, we have that: d R ( T 1 , T 2 ) ≤ n 2 (1 + (2 n + 1)( r 1 + r 2 – 2)) where r = max{ r 1 , r 2 }. This bound is asymptotically tight for r = 1. En route to our proof of the above theorems, we associate full binary trees to permutations and relate the rotation operation on trees to transpositions in the corresponding permutations. We give exact combinatorial characterizations of permutations that correspond to full binary trees and full skew binary trees under this association. We also precisely characterize the transpositions that correspond to the set of rotations in full binary trees. We also study bi-variate polynomials associated with binary trees (introduced by [2]), and show characterizations and algorithms for computing rotation distances for the case of full skew trees using them.

We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation of a function f is the characteristic vector of the minimum sized set of negated variables needed in any DeMorgan circuit (circuits where negations appear only at the leaves) computing f. We prove trade-off results between the depth and the weight/structure of the orientation vectors in any circuit C computing the CLIQUE function on an n vertex graph. We prove that if C is of depth d and each gate computes a Boolean function with orientation of weight at most w (in terms of the inputs to C), then d × w must be Ω( n). In particular, if the weights are o ( n log k n ) , then C must be of depth ω(log k n). We prove a barrier for our general technique. However, using specific properties of the CLIQUE function (used in Amano Maruoka (2005)) and the Karchmer–Wigderson framework (Karchmer Wigderson (1988)), we go beyond the limitations and obtain lower bounds when the weight restrictions are less stringent. We then study the depth lower bounds when the structure of the orientation vector is restricted. Asymptotic improvements to our results (in the restricted setting) separates NP from NC. As our main tool, we generalize Karchmer–Wigderson games (Karchmer Wigderson (1988)) for monotone functions to work for non-monotone circuits parametrized by the weight/structure of the orientation. We also prove structural results about orientation and prove connections between number of negations and weight of orientations required to compute a function.

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.

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 labelled 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: 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. We use L–REACH as a lens to study structural complexity. In this direction we show that there is a language A in TC 0 for which A–DAGREACH is NP-complete. Using this we show that P vs NP question is equivalent to P vs DAGREACH −1 (P) 1 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.

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.

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.

A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall (STACS 2009), we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai et al. (SODA 2011). Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations ρ and τ of a group H over $ \mathbb{Z}_p^d $ , p a prime, determine if there exists an automorphism : H → H, such that the induced representation ρ𝜙 = ρ ◦ 𝜙 and τ are equivalent, in time poly(|H|, p d ).

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.

In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. An ABP is given by a layered directed acyclic graph with source $s$ and sink $t$, whose edges are labeled by variables taken from the set $\{x_1, x_2, \ldots, x_n\}$ or field constants. It computes the sum of weights of all paths from $s$ to $t$, where the weight of a path is defined as the product of edge-labels on the path. Given a permutation $\pi$ of the $n$ variables, for a $\pi$-ordered ABP ($\pi$-OABP), for any directed path $p$ from $s$ to $t$, a variable can appear at most once on $p$, and the order in which variables appear on $p$ must respect $\pi$. One can think of OABPs as being the arithmetic analogue of ordered binary decision diagrams (OBDDs). We say an ABP $A$ is of read $r$, if any variable appears at most $r$ times in $A$. Our main result pertains to the polynomial identity testing problem, i.e. the problem of deciding whether a given $n$-variate polynomial is identical to the zero polynomial or not. We prove that over any field $\F$, and in the black-box model, i.e. given only query access to the polynomial, read $r$ $\pi$-OABP computable polynomials can be tested in $\DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}]$. In case $\F$ is a finite field, the above time bound holds provided the identity testing algorithm is allowed to make queries to extension fields of $\F$. To establish this result, we combine some basic tools from algebraic geometry with ideas from derandomization in the Boolean domain. Our next set of results investigates the computational limitations of OABPs. It is shown that any OABP computing the determinant or permanent requires size $\Omega(2^n/n)$ and read $\Omega(2^n/n^2)$. We give a multilinear polynomial $p$ in $2n+1$ variables over some specifically selected field $\mathbb{G}$, such that any OABP computing $p$ must read some variable at least $2^n$ times. We prove a strict separation for the computational power of read $(r-1)$ and read $r$ OABPs. Namely, we show that the elementary symmetric polynomial of degree $r$ in $n$ variables can be computed by a size $O(rn)$ read $r$ OABP, but not by a read $(r-1)$ OABP, for any $0 < 2r-1 \leq n$. Finally, we give an example of a polynomial $p$ and two variables orders $\pi \neq \pi'$, such that $p$ can be computed by a read-once $\pi$-OABP, but where any $\pi'$-OABP computing $p$ must read some variable at least $2^n$ times.

The rigidity of a matrix $A$ for target rank $r$ is the minimum number of entries of $A$ that must be changed to ensure that the rank of the altered matrix is at most $r$. Since its introduction by Valiant \cite{Val77}, rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all $\nbyn$ matrices over an infinite field have a rigidity of $(n-r)^2$. It is a long-standing open question to construct infinite families of \emph{explicit} matrices even with superlinear rigidity when $r=\Omega(n)$. In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., $(n-r)^2$, rigidity. The entries of an $\nbyn$ matrix in this family are distinct primitive roots of unity of orders roughly \SL{$\exp(n^4 \log n)$}. To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most $k$ is exactly $n^2 - (n-r)^2 +k$. Finally, we use elimination theory to examine whether the rigidity function is semicontinuous.