The framework of algebraic natural proofs was independently introduced in the works of Forbes, Shpilka and Volk (2018), and Grochow, Kumar, Saks and Saraf (2017), to study the efficacy of commonly used techniques for proving lower bounds in algebraic complexity. We use the known connections between algebraic hardness and pseudorandomness to shed some more light on the question relating to this framework, as follows. The first set of results is obtained by algebraizing the well-known method of generating hardness from non-trivial hitting sets, and by generalizing the result of Heintz and Schnorr (1980) that proves the existence of hitting sets for . The conditional hardness of equations for uses the fact that pseudorandomness against a class can be extracted from a polynomial that is (sufficiently) hard for that class (Kabanets and Impagliazzo, 2004).
Consider the expected query complexity of computing the k-fold direct product f^⊗ k of a function f to error ε with respect to a distribution μ^k. One strategy is to sequentially compute each of the k copies to error ε/k with respect to μ and apply the union bound. We prove a strong direct sum theorem showing that this naive strategy is essentially optimal. In particular, computing a direct product necessitates a blowup in both query complexity and error. Strong direct sum theorems contrast with results that only show a blowup in query complexity or error but not both. There has been a long line of such results for distributional query complexity, dating back to (Impagliazzo, Raz, Wigderson 1994) and (Nisan, Rudich, Saks 1994), but a strong direct sum theorem had been elusive. A key idea in our work is the first use of the Hardcore Theorem (Impagliazzo 1995) in the context of query complexity. We prove a new "resilience lemma" that accompanies it, showing that the hardcore of f^⊗ k is likely to remain dense under arbitrary partitions of the input space.
Classical results of Brent, Kuck, and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula. Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas. This improves on previous results in the regime when d is small (i.e. d = s^o(1) ). In particular, for the setting of d = O(log s), along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas. This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this “low-degree” and “low-depth” setting. We also show that these results cannot be improved in the monotone setting, even for commutative formulas.
The quality of enumeration algorithms is often measured by their delay, that is, the maximal time spent between the output of two distinct solutions. If the goal is to enumerate $t$ distinct solutions for any given $t$, then another relevant measure is the maximal time needed to output $t$ solutions divided by $t$, a notion we call the amortized delay of the algorithm, since it can be seen as the amortized complexity of the problem of enumerating $t$ elements in the set. In this paper, we study the relation between these two notions of delay, showing different schemes allowing one to transform an algorithm with polynomial amortized delay for which one has a blackbox access into an algorithm with polynomial delay. We complement our results by providing several lower bounds and impossibility theorems in the blackbox model.
We prove tight lower bounds for the following variant of the counting problem considered by Aaronson et al. The task is to distinguish whether an input set $x\subseteq [n]$ has size either $k$ or $k'=(1+\epsilon)k$. We assume the algorithm has access to * the membership oracle, which, for each $i\in [n]$, can answer whether $i\in x$, or not; and * the uniform superposition $|\psi_x\rangle = \sum_{i\in x} |i\rangle/\sqrt{|x|}$ over the elements of $x$. Moreover, we consider three different ways how the algorithm can access this state: ** the algorithm can have copies of the state $|\psi_x\rangle$; ** the algorithm can execute the reflecting oracle which reflects about the state $|\psi_x\rangle$; ** the algorithm can execute the state-generating oracle (or its inverse) which performs the transformation $|0\rangle\mapsto |\psi_x\rangle$. Without the second type of resources (related to $|\psi_x\rangle$), the problem is well-understood, see Brassard et al. The study of the problem with the second type of resources was recently initiated by Aaronson et al. We completely resolve the problem for all values of $1/k \le \epsilon\le 1$, giving tight trade-offs between all types of resources available to the algorithm. Thus, we close the main open problems from Aaronson et al. The lower bounds are proven using variants of the adversary bound by Belovs and employing analysis closely related to the Johnson association scheme.
We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate TC0-Frege. This extends the line of results of Krajicek and Pudlak (Information and Computation, 1998), Bonet, Pitassi, and Raz (FOCS, 1997), and Bonet, Domingo, Gavalda, Maciel, and Pitassi (Computational Complexity, 2004), who showed that Extended Frege, TC0-Frege and AC(0)-Frege, respectively, cannot be weakly automated by classical algorithms if either the RSA cryptosystem or the Diffie-Hellman key exchange protocol are secure. To the best of our knowledge, this is the first interaction between quantum computation and propositional proof search.
We relate various complexity measures like sensitivity, block sensitivity, certificate complexity for multi-output functions to the query complexities of such functions. Using these relations, we show that the deterministic query complexity of total search problems is at most the third power of its pseudo-deterministic query complexity. Previously, a fourth-power relation was shown by Goldreich, Goldwasser and Ron (ITCS'13). Using our proof along with a decision-tree manipulation technique, we give a simple and self-contained proof that the SearchCNF problem on random k -CNF has pseudo-deterministic query complexity Ω(n^1/3) ; a lower bound of Ω(√(n)) is known, due to Goldwasser, Impagliazzo, Pitassi, and Santhanam (CCC'21), but via a significantly more complex proof. We improve the known separation between pseudo-deterministic and randomized decision tree size for total search problems in two ways: (1) We exhibit an exp(Ω(n^1/4)) separation for the SearchCNF relation for random k -CNFs. This seems to be the first exponential lower bound on the pseudo-deterministic size complexity of SearchCNF associated with random k -CNFs. (2) We exhibit an exp(Ω(n)) separation for the ApproxHW relation. The previous best known separation for any relation was exp(Ω(n^1/2)) . We also separate pseudo-determinism from randomness in AND and CONJ decision trees, and determinism from pseudo-determinism in Parity decision trees. Finally, for a hypercube colouring problem, that was introduced by Goldwasswer et al. to analyze the pseudo-deterministic complexity of a complete problem in TFNPdt , we prove that either the monotone block-sensitivity or the anti-monotone block sensitivity is Ω(n^1/3) ; Goldwasser et al. showed an Ω(n^1/2) bound for general block-sensitivity.
The complexity class quantum statistical zero-knowledge(QSZK) captures computational difficulties of the time-bounded quantum state testing problem with respect to the trace distance, deciding whether T(rho(0), rho(1)) is at least alpha or at most beta, known as the Quan-tum State Distinguishability Problem (QSDP) introduced by Watrous(FOCS 2002). However,QSDP [alpha, beta] is in QSZK only within the con-stant polarizing regime, where alpha and beta are constants satisfying alpha(2 )> beta (rather than alpha > beta), similar to its classical counterpart shown by Sahaiand Vadhan (JACM 2003)due to the polarization lemma (error reduction for SDP). Recently, Berman, Degwekar, Rothblum, and Vasudevan(TCC 2019)extended the SZK containment of SDP beyond the polarizing regime via the time-bounded distribution testing problems with respect to the triangular discrimination and the Jensen-Shannon divergence. Our work introduces properquantum analogs for these problems by defining quantum counterparts for triangular discrimination. We investigate whether the quantum analogs behave similarly to their classical counterparts and examine the limitations of existing approaches to polarization regarding quantum distances. These new QSZK-complete problems improve QSZK containments of QSDP beyond the polariz-ing regime and establish a simple QSZK-hardness for the quantum entropy difference problem (QEDP)defined by Ben-Aroya, Schwartz, and Ta-Shma (ToC 2010). Furthermore, we prove that QSDP with some exponentially small errors is in PP, while the same problem without error is in NQP.
We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to Fürer's question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison with the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.
A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settings blackbox derandomization, i.e., derandomization through pseudorandom generators, has been shown equivalent to lower bounds for decision problems against circuits. Recently, Chen and Tell (FOCS'21) established nearequivalences in the BPP setting between whitebox derandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function f on the given instance. Follow-up works managed to obtain full equivalences in the BPP setting by exploiting a compression property of classical pseudorandom generator constructions. In particular, Chen, Tell, and Williams (FOCS'23) showed that derandomization of BPP is equivalent to constructive lower bounds against algorithms that go through a compression phase. In this paper, we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness-to-derandomization direction, consider a length-preserving function f computable by a nondeterministic algorithm that runs in time n^a . We show that if every Arthur-Merlin protocol that runs in time n^c for c=O(log^2 a) can only compute f correctly on finitely many inputs, then AM is in NP. We also obtain equivalences between constructive lower bounds against Arthur-Merlin protocols that go through a compression phase and derandomization of AM via targeted generators. Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs of proximity for nondeterministic computations. As a by-product of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM, an issue raised by Goldreich (LNCS, 2011). By-products in the average-case setting include the first uniform hardness vs. randomness trade-offs for AM, as well as an unconditional mild derandomization result for AM.
Littlewood-Richardson, Kronecker and plethysm coefficients are fundamental multiplicities of interest in Representation Theory and Algebraic Combinatorics. Determining a combinatorial interpretation for the Kronecker and plethysm coefficients is a major open problem and prompts the consideration of their computational complexity. Recently, it was shown that they behave relatively well with respect to quantum computation, and for some large families there are polynomial-time quantum algorithms (Larocca and Havlicek in Quantum algorithms for representation-theoretic multiplicities, 2024. arXiv:2407.17649 ) (also Bravyi et al. in PRX Quantum 5(1):010329, 2024). In this paper, we show that for many of those cases the Kronecker and plethysm coefficients can also be computed in polynomial time via classical algorithms, thereby refuting some of the conjectures in Larocca and Havlicek (2024). This vastly limits the cases in which the desired superpolynomial quantum speedup could be achieved.
We propose a framework of algorithm vs. hardness for all Max-CSPs and demonstrate it for a large class of predicates. This framework extends the work of Raghavendra [STOC, 2008], who showed a similar result for almost satisfiable Max-CSPs. Our framework is based on a new hybrid approximation algorithm, which uses a combination of the Gaussian elimination technique (i.e., solving a system of linear equations over an Abelian group) and the semidefinite programming relaxation. We complement our algorithm with a matching dictator vs. quasirandom test that has perfect completeness. The analysis of our dictator vs. quasirandom test is based on a novel invariance principle, which we call the mixed invariance principle. Our mixed invariance principle is an extension of the invariance principle of Mossel, O'Donnell and Oleszkiewicz [Annals of Mathematics, 2010] which plays a crucial role in Raghavendra's work. The mixed invariance principle allows one to relate 3-wise correlations over discrete probability spaces with expectations over spaces that are a mixture of Guassian spaces and Abelian groups, and may be of independent interest.
Lifting theorems are used to transfer lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure A for some function f , we compose f with a carefully chosen gadget function g and get essentially the same lower bound on a complexity measure B for the lifted function f ♢ g . Lifting theorems have applications in many different areas, such as circuit complexity, communication complexity, proof complexity, etc. One of the main questions in the context of lifting is how to choose a suitable gadget g . Generally, to get better results, i.e., to minimize the losses when transferring lower bounds, we need the gadget to be of a constant size (number of inputs). Unfortunately, in many settings we know lifting results only for gadgets of size that grows with the size of f , and it is unclear whether they can be improved to constant-size gadgets. This motivates us to identify the properties of gadgets that make lifting possible. In this paper, we systematically study the question: ‘For which gadgets does the lifting result hold?’ in the following four settings: lifting from decision tree depth to decision tree size, lifting from conjunction DAG width to conjunction DAG size, lifting from decision tree depth to parity decision tree depth and size, and lifting from block sensitivity to deterministic and randomized communication complexities. In all the cases, we prove the complete classification of gadgets by exposing the properties of gadgets that make lifting results hold. The structure of the results shows that there are no intermediate cases—for every gadget, there is either a polynomial lifting or no lifting at all. As a byproduct of our studies, we prove the log-rank conjecture for the class of functions that can be represented as f♢ OR ♢ XOR for some function f .
Given a non-negative real matrix M of non-negative rank at least r, can we witness this fact by a small submatrix of M? While Moitra (SIAM J. Comput. 2013) proved that this cannot be achieved exactly, we show that such a witnessing is possible approximately: An m× n matrix of non-negative rank r always contains a submatrix with at most r3 rows and columns with non-negative rank at least Ω(r/log nlog m) . A similar result is proved for the 1-partition number of a Boolean matrix and, consequently, also for its two-player deterministic communication complexity. Tightness of the latter estimate is closely related to the log-rank conjecture of Lovász and Saks.
We obtain new catalytic algorithms for space-bounded derandomization. In the catalytic computation model introduced by (Buhrman, Cleve, Koucký, Loff, and Speelman STOC 2013), we are given a small worktape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. We prove that BPSPACE[S] ⊆ CSPACE[S,S^2] where BPSPACE[S] corresponds to randomized space S computation, and CSPACE[S,C] corresponds to catalytic algorithms that use O(S) bits of workspace and O(C) bits of catalytic space. Previously, only BPSPACE[S]⊆ CSPACE[S,2^O(S)] was known. In fact, we prove a general tradeoff, that for every α∈ [1,1.5] , BPSPACE[S] ⊆ CSPACE[S^α,S^3-α]. We do not use the algebraic techniques of prior work on catalytic computation. Instead, we develop an algorithm that branches based on if the catalytic tape is conditionally random, and instantiate this primitive in a recursive framework. Our result gives an alternate proof of the best known time-space tradeoff for BPSPACE[S] , due to (Cai, Chakaravarthy, and van Melkebeek, Theory Comput. Sys. 2006). As a final application, we extend our results to solve search problems in CSPACE[S,S^2] . As far as we are aware, this constitutes the first study of search problems in the catalytic computing model.
The classic TQBF problem can be viewed as a game in which two players alternate turns assigning truth values to a CNF formula's variables in a prescribed order, and the winner is determined by whether the CNF gets satisfied. The complexity of deciding which player has a winning strategy in this game is well-understood: it is NL-complete for 2-CNFs and PSPACE-complete for 3-CNFs. We continue the study of the unordered variant of this game, in which each turn consists of picking any remaining variable and assigning it a truth value. The complexity of deciding who can win on a given CNF is less well-understood; prior work by the authors showed it is in L for 2-CNFs and PSPACE-complete for 5-CNFs. We conjecture it may be efficiently solvable on 3-CNFs, and we make progress in this direction by proving the problem is in P, indeed in L, for 3-CNFs with a certain restriction, namely that each width-3 clause has at least one variable that appears in no other clause. Another (incomparable) restriction of this problem was previously shown to be tractable by Kutz.
We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give algorithms with complexity n^p + 1 for n × n by n × n^p matrix multiplication. In fact, we prove a precise numerical barrier for this method. Our barrier improves the previously known barriers, both in the numerical sense, as well as in its generality. In particular, we prove that any lower bound on the dual exponent of matrix multiplication α via the big Coppersmith-Winograd tensors cannot exceed 0.6218 .
We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate TC^0 -Frege. This extends the line of results of Krajííček and Pudlík(Information and Computation, 1998), Bonet, Pitassi, and Raz (SIAM Journal on Computing, 2000),and Bonet, Domingo, Gavaldá, Maciel, and Pitassi (Computational Complexity, 2004), who showed that ExtendedFrege, TC^0 -Frege and AC^0 -Frege, respectively, cannot be weakly automated by classical algorithms if either the RSA cryptosystem or the Diffie-Hellman key exchange protocol are secure. To the best of our knowledge, this is the first interaction between quantum computation and propositional proof search.
This paper explores a fine-grained version of the Watrous conjecture, including the randomized and quantum algorithms with success probabilities arbitrarily close to $1/2$. Our contributions include the following: i) An analysis of the optimal success probability of quantum and randomized query algorithms of two fundamental partial symmetric Boolean functions given a fixed number of queries. We prove that for any quantum algorithm computing these two functions using $T$ queries, there exist randomized algorithms using $\mathsf{poly}(T)$ queries that achieve the same success probability as the quantum algorithm, even if the success probability is arbitrarily close to 1/2. ii) We establish that for any total symmetric Boolean function $f$, if a quantum algorithm uses $T$ queries to compute $f$ with success probability $1/2+\beta$, then there exists a randomized algorithm using $O(T^2)$ queries to compute $f$ with success probability $1/2+\Omega(\delta\beta^2)$ on a $1-\delta$ fraction of inputs, where $\beta,\delta$ can be arbitrarily small positive values. As a corollary, we prove a randomized version of Aaronson-Ambainis Conjecture for total symmetric Boolean functions in the regime where the success probability of algorithms can be arbitrarily close to 1/2. iii) We present polynomial equivalences for several fundamental complexity measures of partial symmetric Boolean functions. Specifically, we first prove that for certain partial symmetric Boolean functions, quantum query complexity is at most quadratic in approximate degree for any error arbitrarily close to 1/2. Next, we show exact quantum query complexity is at most quadratic in degree. Additionally, we give the tight bounds of several complexity measures, indicating their polynomial equivalence.
