Abstract In an influential paper, Erdős and Selfridge introduced the Maker-Breaker game played on a hypergraph, or equivalently, on a monotone CNF. The players take turns assigning values to variables of their choosing, and Breaker’s goal is to satisfy the CNF, while Maker’s goal is to falsify it. The Erdős–Selfridge Theorem says that the least number of clauses in any monotone CNF with k literals per clause where Maker has a winning strategy is $$\varvec{\Theta (2^k)}$$ Θ ( 2 k ) . We study the analogous question when the CNF is not necessarily monotone. We prove bounds of $$\varvec{\Theta (\sqrt{2}\,^k)}$$ Θ ( 2 k ) when Maker plays last, and $$\varvec{\Omega (1.5^k)}$$ Ω ( 1 . 5 k ) and $$\varvec{O(r^k)}$$ O ( r k ) when Breaker plays last, where $$\varvec{r=(1+\sqrt{5})/2\approx 1.618}$$ r = ( 1 + 5 ) / 2 ≈ 1.618 is the golden ratio.

Abstract 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 -complete for 2-CNFs and -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 for 2-CNFs and -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 , indeed in , 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.

In the model of fault tolerant decision trees introduced by Kenyon and Yao, there is a known upper bound E on the total number of queries that may be faulty (i.e., get the wrong bit). We consider this computational problem: Given as input the truth table of a function f: {0,1}ⁿ → {0,1} and a value of E, find the minimum possible height (worst-case number of queries) of any decision tree that computes f while tolerating up to E many faults. We design an algorithm for this problem that runs in time Õ(binom(n+E,E)⋅(2E+3)ⁿ), which is polynomial in the size of the truth table when E is a constant. This generalizes a standard algorithm for the non-fault tolerant setting.

We provide a complete picture of the extent to which amplification of success probability is possible for randomized algorithms having access to one NP oracle query, in the settings of two-sided, onesided, and zero-sided error. We generalize this picture to amplifying one-query algorithms with q-query algorithms, and we show our inclusions are tight for relativizing techniques.

In a STOC 1976 paper, Schaefer proved that it is PSPACE-complete to determine the winner of the so-called Maker-Breaker game on a given set system, even when every set has size at most 11. Since then, there has been no improvement on this result. We prove that the game remains PSPACE-complete even when every set has size 6.

We investigate the power of randomness in two-party communication complexity. In particular, we study the model where the parties can make a constant number of queries to a function that has an efficient one-sided-error randomized protocol. The complexity classes defined by this model comprise the Randomized Boolean Hierarchy, which is analogous to the Boolean Hierarchy but defined with one-sidederror randomness instead of nondeterminism. Our techniques connect the Nondeterministic and Randomized Boolean Hierarchies, and we provide a complete picture of the relationships among complexity classes within and across these two hierarchies. In particular, we prove that the Randomized Boolean Hierarchy does not collapse, and we prove a query-to-communication lifting theorem for all levels of the Nondeterministic Boolean Hierarchy and use it to resolve an open problem stated in the paper by Halstenberg and Reischuk (CCC 1988) which initiated the study of this hierarchy.

Suppose we have randomized decision trees for an outer function f and an inner function g. The natural approach for obtaining a randomized decision tree for the composed function (f∘ gⁿ)(x¹,…,xⁿ) = f(g(x¹),…,g(xⁿ)) involves amplifying the success probability of the decision tree for g, so that a union bound can be used to bound the error probability over all the coordinates. The amplification introduces a logarithmic factor cost overhead. We study the question: When is this log factor necessary? We show that when the outer function is parity or majority, the log factor can be necessary, even for models that are more powerful than plain randomized decision trees. Our results are related to, but qualitatively strengthen in various ways, known results about decision trees with noisy inputs.

Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta<1, this requires Omega(n) communication even to get within statistical distance 1-beta^n of the target distribution. Previously, Ambainis, Schulman, Ta-Shma, Vazirani, and Wigderson (FOCS 1998) proved that Omega(sqrt{n}) communication is required to get within some constant statistical distance epsilon>0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}.

The complexity class ZPP^{NP[1]} (corresponding to zero-error randomized algorithms with access to one NP oracle query) is known to have a number of curious properties. We further explore this class in the settings of time complexity, query complexity, and communication complexity. - For starters, we provide a new characterization: ZPP^{NP[1]} equals the restriction of BPP^{NP[1]} where the algorithm is only allowed to err when it forgoes the opportunity to make an NP oracle query. - Using the above characterization, we prove a query-to-communication lifting theorem, which translates any ZPP^{NP[1]} decision tree lower bound for a function f into a ZPP^{NP[1]} communication lower bound for a two-party version of f. - As an application, we use the above lifting theorem to prove that the ZPP^{NP[1]} communication lower bound technique introduced by Göös, Pitassi, and Watson (ICALP 2016) is not tight. We also provide a "primal" characterization of this lower bound technique as a complexity class.

We study problems in randomized communication complexity when the protocol is only required to attain some small advantage over purely random guessing, i.e., it produces the correct output with probability at least epsilon greater than one over the codomain size of the function. Previously, Braverman and Moitra (STOC 2013) showed that the set-intersection function requires Theta(epsilon n) communication to achieve advantage epsilon. Building on this, we prove the same bound for several variants of set-intersection: (1) the classic "tribes" function obtained by composing with And (provided 1/epsilon is at most the width of the And), and (2) the variant where the sets are uniquely intersecting and the goal is to determine partial information about (say, certain bits of the index of) the intersecting coordinate.

We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between $${\mathsf{P}}$$ and $${\mathsf{PSPACE}}$$ , short of proving lower bounds against classes for which no explicit lower bounds were already known. Our article also serves as an up-to-date survey on the state of structural communication complexity. Among our new results we show that $${\mathsf{MA} \not\subseteq \mathsf{ZPP}^{\mathsf{NP}[1]}}$$ , that is, Merlin–Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one $${\mathsf{NP}}$$ query. Here the class $$\mathsf{ZPP}^{\mathsf{NP}[1]}$$ has the property that generalizing it in the slightest ways would make it contain $${\mathsf{AM} \cap \mathsf{coAM}}$$ , for which it is notoriously open to prove any explicit lower bounds. We also prove that $${\mathsf{US} \not\subseteq \mathsf{ZPP}^{\mathsf{NP}[1]}}$$ , where $${\mathsf{US}}$$ is the class whose canonically complete problem is the variant of set-disjointness where yes-instances are uniquely intersecting. We also prove that $${\mathsf{US} \not\subseteq \mathsf{coDP}}$$ , where $${\mathsf{DP}}$$ is the class of differences of two $${\mathsf{NP}}$$ sets. Finally, we explore an intriguing open issue: Are rank-1 matrices inherently more powerful than rectangles in communication complexity? We prove a new separation concerning $${\mathsf{PP}}$$ that sheds light on this issue and strengthens some previously known separations.

We show that deterministic communication complexity can be superlogarithmic in the partition number of the associated communication matrix. We also obtain near-optimal deterministic lower bounds for the Clique vs. Independent Set problem, which in particular yields new lower bounds for the log-rank conjecture. All of these results follow from a simple adaptation of a communication-to-query simulation theorem of Raz and McKenzie [ Combinatorica, 19 (1999), pp. 403--435] together with lower bounds for the analogous query complexity questions.

We prove that the P^NP-type query complexity (alternatively, decision list width) of any boolean function f is quadratically related to the P^NP-type communication complexity of a lifted version of f. As an application, we show that a certain "product" lower bound method of Impagliazzo and Williams (CCC 2010) fails to capture P^NP communication complexity up to polynomial factors, which answers a question of Papakonstantinou, Scheder, and Song (CCC 2014).

We show that randomized communication complexity can be superlogarithmic in the partition number of the associated communication matrix, and we obtain near-optimal randomized lower bounds for the Clique vs. Independent Set problem. These results strengthen the deterministic lower bounds obtained in prior work (Goos, Pitassi, and Watson, FOCS 2015). One of our main technical contributions states that information complexity when the cost is measured with respect to only 1-inputs (or only 0-inputs) is essentially equivalent to information complexity with respect to all inputs.

We exhibit an $n$-node graph whose independent set polytope requires extended formulations of size exponential in $\Omega(n/\log n)$. Previously, no explicit examples of $n$-dimensional $0/1$-polytopes were known with extension complexity larger than exponential in $\Theta(\sqrt{n})$. Our construction is inspired by a relatively little-known connection between extended formulations and (monotone) circuit depth.

We develop a new method to prove communication lower bounds for composed functions of the form $f\circ g^n$, where $f$ is any boolean function on $n$ inputs and $g$ is a sufficiently “hard” two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of $f \circ g^n$ can be simulated by a nonnegative combination of juntas. This is a new formalization for the intuition that each low-communication randomized protocol can only “query” a few inputs of $f$ as encoded by the gadget $g$. Consequently, we characterize the communication complexity of $f\circ g^n$ in all known one-sided (i.e., not closed under complement) zero-communication models by a corresponding query complexity measure of $f$. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work. We show that $\mathsf{SBP}^{\sf cc}$ (a class characterized by corruption) is not closed under intersection. An immediate corollary is that $\mathsf{MA}^{\sf cc} \neq \mathsf{SBP}^{\sf cc}$. These results answer questions of Klauck [Proceedings of the 18th Conference on Computational Complexity (CCC), IEEE Computer Society, Los Alamitos, CA, 2003, pp. 118--134] and Böhler, Glasser, and Meister [J. Comput. System Sci., 72 (2006), pp. 1043--1076]. We also show that the approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. [Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP), Springer, Berlin, 2014, pp. 701--712] for partial matrices. In subsequent work, our structure theorem has been applied to resolve the communication complexity of the clique versus independent set problem.

We show that deterministic communication complexity can be super logarithmic in the partition number of the associated communication matrix. We also obtain near-optimal deterministic lower bounds for the Clique vs. Independent Set problem, which in particular yields new lower bounds for the log-rank conjecture. All these results follow from a simple adaptation of a communication-to-query simulation theorem of Raz and McKenzie (Combinatorica 1999) together with lower bounds for the analogous query complexity questions.

We study whether information complexity can be used to attack the long-standing open problem of proving lower bounds against Arthur{Merlin (AM) communication protocols. Our starting point is to show that|in contrast to plain randomized communication complexity|every boolean function admits an AM communication protocol where on each yes- input, the distribution of Merlin's proof leaks no information about the input and moreover, this proof is unique for each outcome of Arthur's randomness. We posit that these two properties of zero information leakage and unambiguity on yes-inputs are interesting in their own right and worthy of investigation as new avenues toward AM.

We develop a new method to prove communication lower bounds for composed functions of the form f o gn where f is any boolean function on n inputs and g is a sufficiently "hard" two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of f o gn can be simulated by a nonnegative combination of juntas. This is the strongest yet formalization for the intuition that each low-communication randomized protocol can only "query" few inputs of f as encoded by the gadget g. Consequently, we characterize the communication complexity of f o gn in all known one-sided zero-communication models by a corresponding query complexity measure of f. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work: We show that SBPcc (a class characterized by corruption) is not closed under intersection. An immediate corollary is that MAcc ≠ SBPcc. These results answer questions of Klauck (CCC 2003) and Bohler et al. (JCSS 2006). We also show that approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. (ICALP) for partial matrices.

We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least alpha(n) on yes-inputs and at most beta(n) on no-inputs, for some functions alpha(n)>beta(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions alpha and beta, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where alpha=1/2+epsilon(n) and beta=1/2-epsilon(n). The following contributions play a crucial role in our characterization and are interesting in their own right. (1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a "restricted" class SBP (which lies between MA and AM) and an "unrestricted" class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP. (2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003). (3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness.

We consider the problems of deciding whether the joint distribution sampled by a given circuit satisfies certain statistical properties such as being i.i.d., being exchangeable, being pairwise independent, having two coordinates with identical marginals, having two uncorrelated coordinates, and many other variants. We give a proof that simultaneously shows all these problems are C_{=P}-complete, by showing that the following promise problem (which is a restriction of all the above problems) is C_{=P}-complete: Given a circuit, distinguish the case where the output distribution is uniform and the case where every pair of coordinates is neither uncorrelated nor identically distributed. This completeness result holds even for samplers that are depth-3 circuits. We also consider circuits that are d-local, in the sense that each output bit depends on at most d input bits. We give linear-time algorithms for deciding whether a 2-local sampler's joint distribution is fully independent, and whether it is exchangeable. We also show that for general circuits, certain approximation versions of the problems of deciding full independence and exchangeability are SZK-complete. We also introduce a bounded-error version of C_{=P}, which we call BC_{=P}, and we investigate its structural properties.

We show that “a little more time gives a lot more power to sampling algorithms.” We prove that for every constant $k\ge 2$, every polynomial time bound $t$, and every polynomially small $\epsilon$, there exists a family of distributions on $k$ elements that can be sampled exactly in polynomial time but cannot be sampled within statistical distance $1-1/k-\epsilon$ in time $t$. This implies the following general time hierarchy for sampling distributions on arbitrary-size domains such as $\{0,1\}^n$: For every polynomial time bound $t$ and every constant $\epsilon>0$, there exists a family of distributions that can be sampled exactly in polynomial time but cannot be sampled within statistical distance $1-\epsilon$ in time $t$. Our proof involves reducing the problem to a communication problem over a certain type of noisy channel. To solve the latter problem we use a type of list-decodable code for a setting where there is no bound on the number of errors but each error gives more information than an erasure. This type of code can be constructed using certain known traditional list-decodable codes, but we give a new construction that is elementary, self-contained, and tailored to this setting.

We show that "a little more time gives a lot more power to sampling algorithms." We prove that for every constant k ≥ 2, every polynomial time bound t, and every polynomially small ε, there exists a family of distributions on k elements that can be sampled exactly in polynomial time but cannot be sampled within statistical distance 1-1/k-ε in time t. This implies the following general time hierarchy for sampling distributions on arbitrary-size domains such as {0,1}n: For every polynomial time bound t and every constant ε>0, there exists a family of distributions that can be sampled exactly in polynomial time but cannot be sampled within statistical distance 1-ε in time t. Our proof involves reducing the problem to a communication problem over a certain type of noisy channel. To solve the latter problem we use a type of list-decodable code for a setting where there is no bound on the number of errors but each error gives more information than an erasure. This type of code can be constructed using certain known traditional list-decodable codes, but we give a new construction that is elementary, self-contained, and tailored to this setting.

We prove a lower bound on the amount of nonuniform advice needed by black-box reductions for the Dense Model Theorem of Green, Tao, and Ziegler, and of Reingold, Trevisan, Tulsiani, and Vadhan. The latter theorem roughly says that for every distribution D that is delta-dense in a distribution that is epsilon'-indistinguishable from uniform, there exists a "dense model" for D, that is, a distribution that is delta-dense in the uniform distribution and is epsilon-indistinguishable from D. This epsilon-indistinguishability is with respect to an arbitrary small class of functions F. For the natural case where epsilon' >= Omega(epsilon delta) and epsilon >= delta^{O(1)}, our lower bound implies that Omega(sqrt{(1/epsilon)log(1/delta)} log|F|) advice bits are necessary. There is only a polynomial gap between our lower bound and the best upper bound for this case (due to Zhang), which is O((1/epsilon^2)log(1/delta) log|F|). Our lower bound can be viewed as an analog of list size lower bounds for list-decoding of error-correcting codes, but for "dense model decoding" instead. Our proof introduces some new techniques which may be of independent interest, including an analysis of a majority of majorities of p-biased bits. The latter analysis uses an extremely tight lower bound on the tail of the binomial distribution, which we could not find in the literature.

We define a combinatorial checkerboard to be a function f:{1, ⋯, m}d→ {1, -1} of the form f(u1, ⋯, ud) = Πi=1dfi(ui) for some functions fi:{1, ⋯, m} → {1, -1}. This is a variant of combinatorial rectangles, which can be defined in the same way but using {0, 1} instead of {1, -1}. We consider the problem of constructing explicit pseudorandom generators for combinatorial checkerboards. This is a generalization of small-bias generators, which correspond to the case m=2. We construct a pseudorandom generator that ϵ-fools all combinatorial checkerboards with seed length O(log m + log d·log log d + log3/21/ϵ). Previous work by Impagliazzo, Nisan, and Wigderson implies a pseudorandom generator with seed length O(log m + log2d + log d·log 1/ϵ). Our seed length is better except when 1/ϵ ≥ dω(log d).