Resolution over parities (Res(⊕)) is a proof system introduced by Itsykson and Sokolov [MFCS ’14] as a stepping stone towards proving AC0[2]-Frege lower bounds. A recent line of work has established lower bounds against depth-restricted Res(⊕) refutations. Prior to this work, the state of the art was exponential lower bounds against depth O(N logN) Res(⊕) proved by Efremenko and Itsykson [CCC ’25], where N is the number of variables in the CNF. In this work we prove exponential lower bounds against depth O(N2−є) Res(⊕) refutations. The lifted Tseitin formula we consider has O(N) clauses of width 6, which lets the allowed depth be almost quadratic not only in the number of variables, but also in the CNF size. We also prove depth-restricted lower bounds for variants of the bit pigeonhole principle (BPHP), including an exponential lower bound for depth O(n2−є) Res(⊕) refutations of BPHP with n+1 pigeons and n holes.

Exact and point-wise approximating representations of Boolean functions by real polynomials have been of great interest in the theory of computing. We focus on the study of sparsity of such representations. Our results include the following: First, we show that for every total Boolean function, its exact and approximate sparsity in the De Morgan basis are polynomially related to each other in the log scale, ignoring poly-log(n) factors. This answers an open question posed by Knop, Lovett, McGuire and Yuan (STOC 2021). It builds on and is analogous to the seminal result of Nisan and Szegedy (Computational Complexity 1994) who proved the same for degree and approximate degree. Second, we consider more powerful representations using generalized monomials, where each monomial is an indicator of a sub-cube. There are 3n such monomials, where n is the number of variables. We prove that even for these representations, the sparsity and approximate sparsity of total Boolean functions remain polynomially related to each other in the log scale, ignoring poly-log(n) factors. Third, we show that for every total Boolean function f, the log of its De Morgan sparsity characterizes up to polynomial loss and ignoring poly-log(n) factors, the quantum and classical 2-party bounded-error communication complexity of f ∘ EQ4, where EQ4 is Equality of two 2-bit strings, one held by Alice and the other by Bob. As a consequence, we show that bounded-error quantum protocols cannot exhibit super-polynomial cost advantage over their classical counterparts, for computing such functions.

Probabilistic polynomials over commutative rings offer a powerful way of representing Boolean functions. Although many degree lower bounds for such representations have been proved, sparsity lower bounds (counting the number of monomials in the polynomials) have not been so common. Sparsity upper bounds are of great interest for potential algorithmic applications, since sparse probabilistic polynomials are the key technical tool behind the best known algorithms for many core problems, including dense All-Pairs Shortest Paths, and the existence of sparser polynomials would lead to breakthrough algorithms for these problems. In this paper, we prove several strong lower bounds on the sparsity of probabilistic and approximate polynomials computing Boolean functions when 0 means "false". Our main result is that the AND of n ORs of c log n variables requires probabilistic polynomials (over any commutative ring which isn't too large) of sparsity n^Ω(log c) to achieve even 1/4 error. The lower bound is tight, and it rules out a large class of polynomial-method approaches for refuting the APSP and SETH conjectures via matrix multiplication. Our other results include: - Every probabilistic polynomial (over a commutative ring) for the disjointness function on two n-bit vectors requires exponential sparsity in order to achieve exponentially low error. - A generic lower bound that any function requiring probabilistic polynomials of degree d must require probabilistic polynomials of sparsity Ω(2^d). - Building on earlier work, we consider the probabilistic rank of Boolean functions which generalizes the notion of sparsity for probabilistic polynomials, and prove separations of probabilistic rank and probabilistic sparsity. Some of our results and lemmas are basis independent. For example, over any basis {a,b} for true and false where a ≠ b, and any commutative ring R, the AND function on n variables has no probabilistic R-polynomial with 2^o(n) sparsity, o(n) degree, and 1/2^o(n) error simultaneously. This AND lower bound is our main technical lemma used in the above lower bounds.

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 $$\text{SearchCNF}$$ problem on random $$\text{k}$$ -CNF has pseudo-deterministic query complexity $$\Omega(n^{1/3})$$ ; a lower bound of $$\Omega(\sqrt{n})$$ 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 $$\text{exp}(\widetilde{\Omega}(n^{1/4}))$$ separation for the $$\text{SearchCNF}$$ relation for random $$k$$ -CNFs. This seems to be the first exponential lower bound on the pseudo-deterministic size complexity of $$\text{SearchCNF}$$ associated with random $$k$$ -CNFs. (2) We exhibit an $${\text{exp}(\Omega(n))}$$ separation for the $$\text{ApproxHW}$$ relation. The previous best known separation for any relation was $${\text{exp}(\Omega(n^{1/2}))}$$ . We also separate pseudo-determinism from randomness in $$\text{AND}$$ and $$\text{CONJ}$$ decision trees, and determinism from pseudo-determinism in $$\text{Parity}$$ 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 $$\text{TFNPdt}$$ , we prove that either the monotone block-sensitivity or the anti-monotone block sensitivity is $${\Omega(n^{1/3})}$$ ; Goldwasser et al. showed an $${\Omega(n^{1/2})}$$ bound for general block-sensitivity.

Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities is an outstanding problem that has received a lot of attention after its introduction by Raz and Tzamaret [Ann. Pure Appl. Log.'08]. Very recently, Efremenko, Garlík and Itsykson [ECCC'23] proved the first exponential lower bounds on the size of ResLin proofs that were additionally restricted to be bottom-regular. We show that there are formulas for which such regular ResLin proofs of unsatisfiability continue to have exponential size even though there exists short proofs of their unsatisfiability in ordinary, non-regular resolution. This is the first super-polynomial separation between the power of general ResLin and and that of regular ResLin for any natural notion of regularity. Our argument, while building upon the work of Efremenko et al., uses additional ideas from the literature on lifting theorems.

We show that the deterministic decision tree complexity of a (partial) function or relation f lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation f∘g as long as the gadget g satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([Göös, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of f, which could be exponentially smaller than its deterministic counterpart when either f is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to f. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., Res(⊕), of the unsatisfiability of closely related constant-width CNF formulas.

A classic result of Nisan [SICOMP ’91] states that the deterministic decision tree *depth* complexity of every total Boolean function is at most the cube of its randomized decision tree *depth* complexity. The question whether randomness helps in significantly reducing the *size* of decision trees appears not to have been addressed. We show that the logarithm of the deterministic decision tree size complexity of every total Boolean function on n input variables is at most the fourth power of the logarithm of its bounded-error randomized decision tree size complexity, ignoring a polylogarithmic factor in the input size.

We prove two results that shed new light on the monotone complexity of the spanning tree polynomial, a classic polynomial in algebraic complexity and beyond. First, we show that the spanning tree polynomials having $n$ variables and defined over constant-degree expander graphs, have monotone arithmetic complexity $2^{Ω(n)}$. This yields the first strongly exponential lower bound on the monotone arithmetic circuit complexity for a polynomial in VP. Before this result, strongly exponential size monotone lower bounds were known only for explicit polynomials in VNP (Gashkov-Sergeev'12, Raz-Yehudayoff'11, Srinivasan'20, Cavalar-Kumar-Rossman'20, Hrubes-Yehudayoff'21). Recently, Hrubes'20 initiated a program to prove lower bounds against general arithmetic circuits by proving $ε$-sensitive lower bounds for monotone arithmetic circuits for a specific range of values for $ε\in (0,1)$. We consider the spanning tree polynomial $ST_{n}$ defined over the complete graph on $n$ vertices and show that the polynomials $F_{n-1,n} - ε\cdot ST_{n}$ and $F_{n-1,n} + ε\cdot ST_{n}$ defined over $n^2$ variables, have monotone circuit complexity $2^{Ω(n)}$ if $ε\geq 2^{-Ω(n)}$ and $F_{n-1,n} = \prod_{i=2}^n (x_{i,1} +\cdots + x_{i,n})$ is the complete set-multilinear polynomial. This provides the first $ε$-sensitive exponential lower bound for a family of polynomials inside VP. En-route, we consider a problem in 2-party, best partition communication complexity of deciding whether two sets of oriented edges distributed among Alice and Bob form a spanning tree or not. We prove that there exists a fixed distribution, under which the problem has low discrepancy with respect to every nearly-balanced partition. This result could be of interest beyond algebraic complexity.

Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}ⁿ → {-1,1} and G ∈ {AND₂, XOR₂}, the bounded-error quantum communication complexity of the composed function f∘G equals O(𝖰(f) log n), where 𝖰(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that 𝖱^{cc}(f∘G) ≤ 2𝖱(f), where 𝖱^{cc} and 𝖱 denote bounded-error communication and query complexity, respectively. Chakraborty et al. (CCC'20) exhibited a total function for which the log n overhead in the BCW simulation is required. This established the somewhat surprising fact that quantum reductions are in some cases inherently more expensive than classical reductions. We improve upon their result in several ways. - We show that the log n overhead is not required when f is symmetric (i.e., depends only on the Hamming weight of its input), generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). Our upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication. - In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. This also provides a different, and arguably simpler, proof of Aaronson and Ambainis’s O(√n) communication upper bound for Set-Disjointness. - In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2 (this corresponds to the unbounded-error model of communication). - We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free.

This issue of SICOMP contains six specially selected papers from STOC 2020, the Fifty-second Annual ACM Symposium on the Theory of Computing, which was held June 22--26, 2020, initially planned at Chicago, Illinois, but due to COVID-19 was an online conference in the end. The papers here were chosen to represent the range and quality of the STOC program. These papers have been revised and extended by their authors and subjected to the standard thorough reviewing process of SICOMP. The program committee for STOC 2020 consisted of an executive committee made up of Nima Anari, Boaz Barak, Sébastien Bubeck, Mark Bun, Arkadev Chattopadhyay, Chandra Chekuri, Julia Chuzhoy, Marek Cygan, Ilias Diakonikolas, Yevgeniy Dodis, Sebastian Forster, Ankit Garg, Nika Haghtalab, Prahladh Harsha, Justin Holmgren, Piotr Indyk, Rahul Jain, Sanjeev Khanna, Dakshita Khurana, Pravesh Kothari, Robert Krauthgamer, Marvin Künnemann, Tengyu Ma, Rafael Oliveira, Merav Parter, Sofya Raskhodnikova, Robert Robere, Dana Ron, Noga Ron-Zewi, Thatchaphol Saranurak, Balasubramanian Sivan, Christian Sohler, Madhur Tulsiani, Omri Weinstein, Christian Wulff-Nilsen, and Henry Yuen. The program chair was Julia Chuzhoy. Included in this issue are the following papers: ``Explicit Near-Ramanujan Graphs of Every Degree" by Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes shows a deterministic poly$(n)$-time algorithm that outputs a $d$-regular graph on $\Theta(n)$ vertices that is $\epsilon$-near-Ramanujan. ``Reducing Path TSP to TSP" by Vera Traub, Jens Vygen, and Rico Zenklusen presents a black-box reduction from the path version of the traveling salesman problem (Path TSP) to the classical tour version (TSP). ``Nearly Optimal Static Las Vegas Succinct Dictionary" by Huacheng Yu obtains a randomized dictionary data structure using ${OPT}+{poly}\lg n+O(\lg^{(\ell)} U)$ bits of space with expected constant query time for the static dictionary problem. ``Separating the Communication Complexity of Truthful and Nontruthful Algorithms for Combinatorial Auctions" by Sepehr Assadi, Hrishikesh Khandeparkar, Raghuvansh Raj Saxena, and S. Matthew Weinberg provides the first separation in the approximation guarantee achievable by truthful and nontruthful combinatorial auctions with polynomial communication. ``Strong Average-Case Circuit Lower Bounds from Nontrivial Derandomization" by Lijie Chen and Hanlin Ren establish a connection between nondeterministic algorithms estimating the acceptance probability of a given circuit and average-case lower bounds for nondeterministic time classes. ``Improved Bounds for Perfect Sampling of $k$-Colorings in Graphs" by Siddharth Bhandari and Sayantan Chakraborty presents a randomized algorithm that takes as input an undirected $n$-vertex graph $G$ with maximum degree $\Delta$ and an integer $k>3\Delta$ and returns a random proper $k$-coloring of $G$. We thank the authors, the STOC 2020 program committee, the STOC 2020 external reviewers, and the SICOMP referees for all of their hard work. Arkadev Chattopadhyay, Marek Cygan, Noga Ron-Zewi, Christian Wulff-Nilsen - Guest editors

We exhibit a natural function $F_n$ on $n$ variables that can be computed by just a linear-size decision list of “Equalities,” but whose sign-rank is $2^{\Omega(n^{1/4})}$. This yields the following two new unconditional complexity class separations. 1. Boolean circuit complexity. The function $F_n$ can be computed by linear-size depth-two threshold formulas when the weights of the threshold gates are unrestricted (${THR} \circ {THR}$), but any ${THR} \circ {MAJ}$ circuit (the weights of the bottom threshold gates are polynomially bounded in $n$) computing $F_n$ requires size $2^{\Omega(n^{1/4})}$. This provides the first separation between the Boolean circuit complexity classes ${THR} \circ {MAJ}$ and ${THR} \circ {THR}$. While Amano and Maruoka [Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science, 2005, pp. 107--118] and Hansen and Podolskii [Proceedings of the 25th Annual IEEE Conference on Computational Complexity, 2010, pp. 270--279] emphasized that superpolynomial separations between the two classes remained a basic open problem, our separation is in fact exponential. In contrast, Goldmann, H\aastad, and Razborov [Comput. Complexity, 2 (1992), pp. 277--300] showed more than twenty-five years ago that functions efficiently computable by ${MAJ} \circ {THR}$ circuits can also be efficiently computed by ${MAJ} \circ {MAJ}$ circuits. In view of this, it was not even clear if ${THR} \circ {THR}$ was significantly more powerful than ${THR} \circ {MAJ}$ until our work, and there was no candidate function identified for the potential separation. 2. Communication complexity. The function $F_n$ (under the natural partition of the inputs) lies in the communication complexity class ${P}^{{MA}}$. Since $F_n$ has large sign-rank, this implies ${P}^{{MA}} \nsubseteq {UPP}$, strongly resolving a recent open problem posed by Göös, Pitassi, and Watson [ Comput. Complexity, 27 (2018), pp. 245--304]. In order to prove our main result, we view $F_n$ as an ${XOR}$ function and develop a technique to lower bound the sign-rank of such functions. This requires novel approximation-theoretic arguments against polynomials of unrestricted degree. Further, our work highlights for the first time the class “decision lists of exact thresholds” as a common frontier for making progress on longstanding open problems in threshold circuits and communication complexity.

The optimal constants are found for Lebesgue norm multilinear inequalities of Holder-Brascamp-Lieb type for arbitrary discrete Abelian groups. Previously a criterion for finiteness of the constants had been established for finitely generated Abelian groups, and the optimal constant had been found in the torsion-free case. The main step here is the analysis of finite groups.

Valiant (1980) showed that general arithmetic circuits with negation can be exponentially more powerful than monotone ones. We give the first improvement to this classical result: we construct a family of polynomials Pn in n variables, each of its monomials has non-negative coefficient, such that Pn can be computed by a polynomial-size depth-three formula but every monotone circuit computing it has size 2Ω(n1/4/log(n)).

Lifting theorems are theorems that relate the query complexity of a function $f:\{0,1\}^{n}\to \{0,1\}$ to the communication complexity of the composed function $f\circ g^{n}$ for some “gadget” $g:\{ 0,1\}^{b}\times \{0,1\}^{b}\to \{0,1\}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget $g$. We prove a new lifting theorem that works for all gadgets $g$ that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: first, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.

Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function f:{-1,1}ⁿ → {-1,1} and •:{-1,1}² → {-1,1} the two-party bounded-error quantum communication complexity of (f ∘ •) is O(Q(f) log n), where Q(f) is the bounded-error quantum query complexity of f. Note that the bounded-error randomized communication complexity of (f ∘ •) is bounded by O(R(f)), where R(f) denotes the bounded-error randomized query complexity of f. Thus, the BCW simulation has an extra O(log n) factor appearing that is absent in classical simulation. A natural question is if this factor can be avoided. Razborov (IZV MATH'03) showed that the bounded-error quantum communication complexity of Set-Disjointness is Ω(√n). The BCW simulation yields an upper bound of O(√n log n). Høyer and de Wolf (STACS'02) showed that this can be reduced to c^(log^* n) for some constant c, and subsequently Aaronson and Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the quantum communication complexity of the Set-Disjointness function (which is NOR_n ∘ ∧) is O(Q(NOR_n)). Perhaps somewhat surprisingly, we show that when • = ⊕, then the extra log n factor in the BCW simulation is unavoidable. In other words, we exhibit a total function F:{-1,1}ⁿ → {-1,1} such that Q^{cc}(F ∘ ⊕) = Θ(Q(F) log n). To the best of our knowledge, it was not even known prior to this work whether there existed a total function F and 2-bit function •, such that Q^{cc}(F ∘ •) = ω(Q(F)).

The canonical problem that gives an exponential separation between deterministic and randomized communication complexity in the classical two-party communication model is "Equality". In this work we show that even allowing access to an "Equality" oracle, deterministic protocols remain exponentially weaker than randomized ones. More precisely, we exhibit a total function on n bits with randomized one-sided communication complexity O(log n), but such that every deterministic protocol with access to "Equality" oracle needs Omega(n) cost to compute it. Additionally we exhibit a natural and strict infinite hierarchy within BPP, starting with the class P^{EQ} at its bottom.

We prove a new query-to-communication lifting for randomized protocols, with inner product as gadget. This allows us to use a much smaller gadget, leading to a more efficient lifting. Prior to this work, such a theorem was known only for deterministic protocols, due to Chattopadhyay et al. [Arkadev Chattopadhyay et al., 2017] and Wu et al. [Xiaodi Wu et al., 2017]. The only query-to-communication lifting result for randomized protocols, due to Göös, Pitassi and Watson [Mika Göös et al., 2017], used the much larger indexing gadget. Our proof also provides a unified treatment of randomized and deterministic lifting. Most existing proofs of deterministic lifting theorems use a measure of information known as thickness. In contrast, Göös, Pitassi and Watson [Mika Göös et al., 2017] used blockwise min-entropy as a measure of information. Our proof uses the blockwise min-entropy framework to prove lifting theorems in both settings in a unified way.

We construct a simple and total XOR function F on 2n variables that has only O(√n) spectral norm, O(n2) approximate rank and O(n2.5) approximate nonnegative rank. We show it has polynomially large randomized bounded-error communication complexity of Ω(√n). This yields the first exponential gap between the logarithm of the approximate rank and randomized communication complexity for total functions. Thus F witnesses a refutation of the Log-Approximate-Rank Conjecture (LARC) which was posed by Lee and Shraibman as a very natural analogue for randomized communication of the still unresolved Log-Rank Conjecture for deterministic communication. The best known previous gap for any total function between the two measures is a recent 4th-power separation by G'o'os, Jayram, Pitassi and Watson.

We generalize the deterministic simulation theorem of Raz & McKenzie (Combinatorica 19(3):403–435, 1999 ), to any gadget which satisfies a certain hitting property. We prove that inner product and gap-Hamming satisfy this property, and as a corollary, we obtain a deterministic simulation theorem for these gadgets, where the gadget’s input size is logarithmic in the input size of the outer function. This yields the first deterministic simulation theorem with a logarithmic gadget size, answering an open question posed by Göös, Pitassi & Watson (in: Proceedings of the 56th FOCS, 2015 ). Our result also implies the previous results for the indexing gadget, with better parameters than was previously known. Moreover, a simulation theorem with logarithmic-sized gadget implies a quadratic separation in the deterministic communication complexity and the logarithm of the 1-partition number, no matter how high the 1-partition number is with respect to the input size—something which is not achievable by previous results of Göös, Pitassi & Watson ( 2015 ).

We exhibit a natural function F, that can be computed by just a linear sized decision list of 'Equalities', but whose sign rank is exponentially large. This yields the following two new unconditional complexity class separations. The first is an exponential separation between the depth-two threshold circuit classes Threshold-of Majority and Threshold-of-Threshold, answering an open question posed by Amano and Maruoka [MFCS '05] and Hansen and Podolskii [CCC '10]. The second separation shows that the communication complexity class P^MA is not contained in UPP, strongly resolving a recent open problem posed by Goos, Pitassi and Watson [ICALP '16]. In order to prove our main result, we view F as an XOR function and develop a technique to lower bound the sign rank of such functions. This requires novel approximation theoretic arguments against polynomials of unrestricted degree. Further, our work highlights for the first time the class 'decision lists of exact thresholds' as a common frontier for making progress on longstanding open problems in Threshold circuits and communication complexity.

We develop a new technique for proving lower bounds in the setting of asymmetric communication, a model that was introduced in the famous works of Miltersen (STOC’94) and Miltersen, Nisan, Safra and Wigderson (STOC’95). At the core of our technique is the first simulation theorem in the asymmetric setting, where Alice gets a p × n matrix x over F2 and Bob gets a vector y ∈ F2n. Alice and Bob need to evaluate f(x· y) for a Boolean function f: {0,1}p → {0,1}. Our simulation theorems show that a deterministic/randomized communication protocol exists for this problem, with cost C· n for Alice and C for Bob, if and only if there exists a deterministic/randomized *parity decision tree* of cost Θ(C) for evaluating f.

We use a technique of “lifting” functions introduced by Krause and Pudlak [Theor. Comput. Sci., 1997], to amplify degree-hardness measures of a function to corresponding monomial-hardness properties of the lifted function. We then show that any symmetric function F projects onto a “lift” of another suitable symmetric function f . These two key results enable us to prove several results on the complexity of symmetric functions in various models, as given below: 1. We provide a characterization of the approximate spectral norm of symmetric functions in terms of the spectrum of the underlying predicate, affirming a conjecture of Ada et al. [APPROX-RANDOM, 2012] which has several consequences. 2. We characterize symmetric functions computable by quasi-polynomial sized Threshold of Parity circuits. 3. We show that the approximate spectral norm of a symmetric function f characterizes the (quantum and classical) bounded error communication complexity of f o XOR. 4. Finally, we characterize the weakly-unbounded error communication complexity of symmetric XOR functions, resolving a weak form of a conjecture by Shi and Zhang [Quantum Information & Computation, 2009]

We prove tight network topology dependent bounds on the round complexity of computing well studied k-party functions such as set disjointness and element distinctness. Unlike the usual case in the CONGEST model in distributed computing, we fix the function and then vary the underlying network topology. This complements the recent such results on total communication that have received some attention. We also present some applications to distributed graph computation problems. Our main contribution is a proof technique that allows us to reduce the problem on a general graph topology to a relevant two-party communication complexity problem. However, unlike many previous works that also used the same high level strategy, we do not reason about a two-party communication problem that is induced by a cut in the graph. To ‘stitch’ back the various lower bounds from the two party communication problems, we use the notion of timed graph that has seen prior use in network coding. Our reductions use some tools from Steiner tree packing and multi-commodity flow problems that have a delay constraint.

We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. [1] and later studied by Beimel et al. [4] for its connection to the famous direct sum question. In this problem, let f: {0, 1}2n → {0,1} be any boolean function. Alice and Bob get k inputs x1,⋯, xk and y1,⋯, yk respectively, with xi, yi ∈ {0, 1}n. They want to output a k-bit vector v, such that there exists one index i for which vi = f(xi,yi). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds. To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson [19]. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola [23], to show that Greater-Than is weakly regular.

Proving superpolylogarithmic lower bounds for dynamic data structures has remained an open problem despite years of research. Pǎtraşcu proposed an exciting approach for breaking this barrier via a two-player communication model in which one player gets private advice at the beginning of the protocol. He gave reductions from the problem of solving an asymmetric version of set-disjointness in his model to a diverse collection of natural dynamic data structure problems in the cell probe model. He also conjectured that, for any hard problem in the standard two-party communication model, the asymmetric version of the problem is hard in his model, provided not too much advice is given. In this paper, we prove several surprising results about his model. We show that there exist Boolean functions requiring linear randomized communication complexity in the two-party model, for which the asymmetric versions in his model have deterministic protocols with exponentially smaller complexity. For set-disjointness, which also requires linear randomized communication complexity in the two-party model, we give a deterministic protocol for the asymmetric version in his model with a quadratic improvement in complexity. These results demonstrate that Pǎtraşcu's conjecture, as stated, is false. In addition, we show that the randomized and deterministic communication complexities of problems in his model differ by no more than a logarithmic multiplicative factor. We also prove lower bounds in some restricted versions of this model for natural functions such as set-disjointness and inner product. All of our upper bounds conform to these restrictions. Moreover, a special case of one of these lower bounds implies a new proof of a strong lower bound on the tradeoff between the query time and the amortized update time of dynamic data structures with nonadaptive query algorithms.

We consider the point-to-point message passing model of communication in which there are $k$ processors with individual private inputs, each $n$-bit long. Each processor is located at the node of an underlying undirected graph and has access to private random coins. An edge of the graph is a private channel of communication between its endpoints. The processors have to compute a given function of all their inputs by communicating along these channels. While this model has been widely used in distributed computing, strong lower bounds on the amount of communication needed to compute simple functions have just begun to appear. In this work, we prove a tight lower bound of $Ω(kn)$ on the communication needed for computing the Tribes function, when the underlying graph is a star of $k+1$ nodes that has $k$ leaves with inputs and a center with no input. Lower bound on this topology easily implies comparable bounds for others. Our lower bounds are obtained by building upon the recent information theoretic techniques of Braverman et.al (FOCS'13) and combining it with the earlier work of Jayram, Kumar and Sivakumar (STOC'03). This approach yields information complexity bounds that is of independent interest.

In the `Number-on-Forehead' (NOF) model of multiparty communication, the input is a k times m boolean matrix A (where k is the number of players) and Player i sees all bits except those in the i-th row, and the players communicate by broadcast in order to evaluate a specified function f at A. We discover new computational power when k exceeds log m. We give a protocol with communication cost poly-logarithmic in m, for block composed functions with limited block width. These are functions of the form f o g where f is a symmetric b-variate function, and g is a (kr)-variate function and (f o g)(A) is defined, for a k times (br) matrix to be f(g(A-1),...,g(A-b)) where A-i is the i-th (k times r) block of A. Our protocol works provided that k > 1+ ln b + (2 to the power of r). Ada et al. (ICALP'2012) previously obtained simultaneous and deterministic efficient protocols for composed functions of block-width one. The new protocol is the first to work for block composed functions with block-width greather than one. Moreover, it is simultaneous, with vanishingly small error probability, if public coin randomness is allowed. The deterministic and zero-error version barely uses interaction.

We consider the communication cost of computing functions when inputs are distributed among the vertices of an undirected graph. The communication is assumed to be point-to-point: a processor sends messages only to its neighbors. The processors in the graph act according to a pre-determined protocol, which can be randomized and may err with some small probability. The communication cost of the protocol is the total number of bits exchanged in the worst case. Extending recent work that assumed that the graph was the complete graph (with unit edge lengths), we develop a methodology for showing lower bounds that are sensitive to the graph topology. In particular, for a broad class of graphs, we obtain a lower bound of the form Ω(k2n), for computing a function of k inputs, each of which is n-bits long and located at a different vertex. Previous works obtained lower bounds of the form Ω(k n). This methodology yields a variety of other results including the following: A tight lower bound (ignoring poly-log factors) for Element Distinctness, settling a question of Phillips, Verbin and Zhang (SODA '12), a distributed XOR lemma, a lower bound for composed functions, settling a question of Phillips et al., new topology-dependent bounds for several natural graph problems considered by Woodruff and Zhang (DISC '13). To obtain these results we use tools from the theory of metric embeddings and represent the topological constraints imposed by the graph as a collection of cuts, each cut providing a setting where our understanding of two-party communication complexity can be effectively deployed.

We present an algorithm which computes the multilinear factors of bivariate lacunary polynomials. It is based on a new Gap theorem which allows to test whether P(X)=∑kj=1 αjXαj(1+X)βjis identically zero in polynomial time. The algorithm we obtain is more elementary than the one by Kaltofen and Koiran (ISSAC'05) since it relies on the valuation of polynomials of the previous form instead of the height of the coefficients. As a result, it can be used to find some linear factors of bivariate lacunary polynomials over a field of large finite characteristic in probabilistic polynomial time.

In 1989 Kushilevitz initiated the study of iinformation-theoretic privacy within the context of communication complexity. Unfortunately, it has been shown that most interesting functions are not privately computable. The unattainability of perfect privacy for many functions motivated the study of approximate privacy. Feigenbaum et al. define notions of worst-case as well as average-case approximate privacy, and present several interesting upper bounds, and some open problems for further study. In this paper, we obtain asymptotically tight bounds on the tradeoffs between both the worst-case and average-case approximate privacy of protocols and their communication cost for Vickrey-auctions. Further, we relate the notion of average-case approximate privacy to other measures based on information cost of protocols. This enables us to prove exponential lower bounds on the subjective approximate privacy of protocols for computing the Intersection function, independent of its communication cost. This proves a conjecture of Feigenbaum et al.

We formulate a new connection between instance compressibility [1]), where the compressor uses circuits from a class C, and correlation with circuits in C. We use this connection to prove the first lower bounds on general probabilistic multi-round instance compression. We show that there is no probabilistic multi-round compression protocol for Parity in which the computationally bounded party uses a non-uniform AC0-circuit and transmits at most n/(log(n))ω(1)bits. This result is tight, and strengthens results of Dubrov and Ishai. We also show that a similar lower bound holds for Majority. We also consider the question of round separation, i.e., whether for each r ≥ 1, there are functions which can be compressed better with r rounds of compression than with r - 1 rounds. We answer this question affirmatively for compression using constant-depth polynomial-size circuits. Finally, we prove the first non-trivial lower bounds for 1-round compressibility of Parity by polynomial size ACC0[p] circuits where p is an odd prime.

Proving superpolylogarithmic lower bounds for dynamic data structures has remained an open problem despite years of research. Recently Pătraşcu proposed an exciting new approach for breaking this barrier via a two player communication model in which one player gets private advice at the beginning of the protocol. He gave reductions from the problem of solving an asymmetric version of set-disjointness in his model to a diverse collection of natural dynamic data structure problems in the cell probe model. He also conjectured that, for any hard problem in the standard two-party communication model, the asymmetric version of the problem is hard in his model, provided not too much advice is given. In this paper, we prove several surprising results about his model. We show that there exist Boolean functions requiring linear randomized communication complexity in the two-party model, for which the asymmetric versions in his model have deterministic protocols with exponentially smaller complexity. For set-disjointness, which also requires linear randomized communication complexity in the two-party model, we give a deterministic protocol for the asymmetric version in his model with a quadratic improvement in complexity. These results demonstrate that Pătraşcu's conjecture, as stated, is false. In addition, we show that the randomized and deterministic communication complexities of problems in his model differ by no more than a logarithmic multiplicative factor. We also prove lower bounds in some restricted versions of this model for natural functions such as set-disjointness and inner product. All of our upper bounds conform to these restrictions.

We consider a system of linear constraints over any finite Abelian group G of the following form: ℓi(x1, ..., xn) ≡ ℓi,1x1+ ⋯ + ℓi,nxn∈ Aifor i=1, ..., N and each Ai⊂ G, ℓi,jis an element of G and xi's are Boolean variables. Our main result shows that the subset of the Boolean cube that satisfies these constraints has exponentially small correlation with the MODqboolean function, when the order of G and q are co-prime numbers. Our work extends the recent result of Chattopadhyay and Wigderson (FOCS'09) who obtain such a correlation bound for linear systems over cyclic groups whose order is a product of two distinct primes or has at most one prime factor. Our result also immediately yields the first exponential bounds on the size of boolean depth-four circuits of the form MAJ ο AND ο ANY ο(1)ο MODmfor computing the MODqfunction, when m, q are co-prime. No superpolynomial lower bounds were known for such circuits for computing any explicit function. This completely solves an open problem posed by Beigel and Maciel (Complexity'97).

We study solution sets to systems of 'generalized' linear equations of the form: ¿i(x1, x2, ···, xn) in ¿ Ai(mod m) where ¿1,..., ¿tare linear forms in n Boolean variables, each Aiis an arbitrary subset of Zm, and m is a composite integer that is a product of two distinct primes, like 6. Our main technical result is that such solution sets have exponentially small correlation, i.e. with the boolean function MODq, when m and q are relatively prime. This bound is independent of the number t of equations. This yields progress on limiting the power of constant-depth circuits with modular gates. We derive the first exponential lower bound on the size of depth-three circuits of type MAJ o AND o MODAm(i.e having a MAJORITY gate at the top, AND/OR gates at the middle layer and generalized MODmgates at the base) computing the function MODq. This settles an open problem of Beigel and Maciel (Complexity'97) for the case of such modulus m. Our technique makes use of the work of Bourgain on estimating exponential sums involving a low-degree polynomial and ideas involving matrix rigidity from the work of Grigoriev and Razborov on arithmetic circuits over finite fields.

We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'number on the forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the degree-discrepancy lemma in the recent work of Sherstov (2007). Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer i.e. circuits of type MAJ o SYMM o ANYO(1)cannot simulate the circuit class AC0in sub-exponential size. Further, even if the fan-in of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC0in quasi-polynomial size. This is in contrast to the classical result of Yao and Beigel-Tarui that shows that such circuits, having only MAJORITY gales, can simulate the class ACC0in quasi-polynomial size when the bottom fan-in is increased to poly-logarithmic size. In the second part, we simplify the arguments in the breakthrough work of Bourgain (2005) for obtaining exponentially small upper bounds on the correlation between the boolean function MODqand functions represented bv polynomials of small degree over Zm, when m,q ges 2 are co-prime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of Bourgain et al. (2005). It is known that such estimates imply that circuits of type MAJ o MODmo ANDisinlogncannot compute the MODqfunction in sub-exponential size. It remains a major open question to determine if such circuits can simulate ACC0in polynomial size when the bottom fan-in is increased to poly-logarithmic size.

Let CCo(n)[m] be the class of circuits that have size o(n) and in which all gates are MOD[m] gates. We show that CC [m] circuits cannot compute MODqin sub-linear size when m, q > 1 are co-prime integers. No non-trivial lower bounds were known before on the size of CC [m] circuits of constant depth for computing MODq. On the other hand, our results show circuits of type MAJ o CCo(n)[m] need exponential size to compute MODq. Using Bourgain's recent breakthrough result on estimates of exponential sums, we extend our bound to the case where small fan-in AND gates are allowed at the bottom of such circuits i.e. circuits of type MAJ o CC[m] o ANDepsiv log n, where epsiv > 0 is a sufficiently small constant. CC [m] circuits of constant depth need superlinear number of wires to compute both the AND and MODqfunctions. To prove this, we show that any circuit computing such functions has a certain connectivity property that is similar to that of superconcentration. We show a superlinear lower bound on the number of edges of such graphs extending results on superconcentrators