A recent paper of Braverman, Cohen, and Garg (STOC'18) introduced the concept of a weighted pseudorandom generator (WPRG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with (potentially negative) real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of WPRGs for ordered branching programs whose seed length has a better dependence on the error parameter s than the classic PRG construction of Nisan (STOC'90 and Combinatorica 1992). In this article we give an explicit construction of WPRGs that achieve parameters that are impossible to achieve by a PRG. In particular, we construct a WPRG for ordered permutation branching programs of unbounded width with a single accept state that has seed length O(log(3/2) n) for error parameter epsilon = 1/poly(n), where n is the input length. In contrast, recent work of Hoza et al. (ITCS'21) shows that any PRG for this model requires seed length Omega(log(2) =) to achieve error epsilon= 1/poly(=). As a corollary, we obtain explicit WPRGs with seed length O(log(3/2 )=) and error s = 1/poly(=) for ordered permutation branching programs of width F = poly(=) with an arbitrary number of accept states. Previously, seed length o(log(2 )=) was only known when both the width and the reciprocal of the error are subpolynomial, i. e., omega = n(0(1)) and epsilon = 1/n(0(1)) (Braverman, Rao, Raz, Yehudayoff, FOCS'10 and SICOMP 2014). The starting point for our work was the recent family of space-efficient algorithms for estimating random-walk probabilities in directed graphs by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford, and Vadhan (FOCS'20), which are based on spectral graph theory and space-efficient Laplacian solvers. We interpret these algorithms as giving WPRGs with large seed length, which we then derandomize to obtain our results. We also note that this approach gives a simpler proof of the original result of Braverman, Cohen, and Garg, as independently discovered by Cohen, Doron, Renard, Sberlo, and Ta-Shma (CCC'21).
The Stabbing Planes proof system [8] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations mod2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [25] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.
We propose a combinatorial hypothesis regarding a subspace vs. subspace agreement test, and prove that if correct it leads to a proof of the 2-to-1 Games Conjecture, albeit with imperfect completeness. This paper presents the second installment in a line of work by various subsets of the authors (with additional contributions by Barak, Kothari, and Steurer (ITCS'19)), which led to a proof of the 2-to-2 Games Conjecture.
During the last decade, an active line of research in proof complexity has been into the space complexity of proofs and how space is related to other measures. By now these aspects of resolution are fairly well understood, but many open problems remain for the related but stronger polynomial calculus (PC/PCR) proof system. For instance, the space complexity of many standard "benchmark formulas" is still open, as well as the relation of space to size and degree in PC/PCR. We prove that if a formula requires large resolution width, then making XOR substitution yields a formula requiring large PCR space, providing some circumstantial evidence that degree might be a lower bound for space. More importantly, this immediately yields formulas that are very hard for space but very easy for size, exhibiting a size-space separation similar to what is known for resolution. Using related ideas, we show that if a graph has good expansion and in addition its edge set can be partitioned into short cycles, then the Tseitin formula over this graph requires large PCR space. In particular, Tseitin formulas over random 4-regular graphs almost surely require space at least $\Omega{\sqrt{n}}$. Our proofs use techniques recently introduced in [Bonacina-Galesi '13]. Our final contribution, however, is to show that these techniques provably cannot yield non-constant space lower bounds for the functional pigeonhole principle, delineating the limitations of this framework and suggesting that we are still far from characterizing PC/PCR space.
This paper studies expansion properties of the (generalized) Johnson Graph. For natural numbers t < ` < k, the nodes of the graph are sets of size ` in a universe of size k. Two sets are connected if their intersection is of size t. The Johnson graph arises often in combinatorics and theoretical computer science: it represents a “slice” of the noisy hypercube, and it is the graph that underlies direct product tests as well as a candidate hard unique game. We prove that any small set of vertices in the graph either has near perfect edge expansion or is not pseudorandom. Here “not pseudorandom” means that the set becomes denser when conditioning on containing a small set of elements. In other words, we show that slices of the noisy hypercube – while not small set expanders like the noisy hypercube – only have non-expanding small sets of a certain simple structure. This paper is related to a recent line of work establishing the 2-to-2 Theorem in PCP. The result was motivated, in part, by [7] which hypothesized and made partial progress on similar result for the Grassmann graph. In turn, our result for the Johnson graphs served as a crucial step towards the full result for the Grassmann graphs completed subsequently in [20].
We give a corrected proof that if PP subset of BQP/qpoly (probabilistic polynomial time can be efficiently simulated by quantum circuits with quantum advice), then the Counting Hierarchy collapses, as originally claimed by Aaronson (CCC'06). This recovers the related unconditional claim that PP does not have circuits of any fixed-polynomial size nk even with quantum advice. Our result is based on proving that YQP & lowast;, an oblivious version of QMA boolean AND coQMA, is contained in APP, a PP-low subclass of PP with an arbitrarily small but nonzero promise gap.
We present a candidate reduction from the 3-Lin problem to the 2-to-2 Games problem and present a combinatorial hypothesis about Grassmann graphs which, if correct, is sufficient to show the soundness of the reduction in a certain non-standard sense. A reduction that is sound in this non-standard sense implies that it is NP-hard to distinguish whether an n -vertex graph has an independent set of size ( 1- 1/√2 ) n - o ( n ) or whether every independent set has size o ( n ), and consequently, that it is NP-hard to approximate the Vertex Cover problem within a factor √2- o (1).
We introduce a new type of seeded extractors we dub seed-protecting extractors. Informally, a seeded extractor is seed protecting against a class C of functions, mappings seeds to seeds, if the seed Y remains close to uniform even after observing the output Ext(X, A(Y)) for every choice of A is an element of C (or, more generally, observing the outputs corresponding to several adversaries from C). The results of this paper are structural. We establish what we believe to be surprising relations, in fact, equivalences between seed-protecting extractors and each of the well-studied strengthenings of seeded extractors: strong extractors, nonmalleable extractors (albeit only against permutations), and two-source extractors, where each case is classified by a suitable class C. Our work motivates the study of non-malleable extractors against permutations and puts forth a novel approach for their construction. Indeed, the existing machinery developed for constructing non-malleable extractors focuses on the output and so is aimed towards breaking correlations. Instead, our work suggests developing techniques for protecting the seed.
A fundamental fact about bounded-degree graph expanders is that three notions of expansion -- vertex expansion, edge expansion, and spectral expansion -- are all equivalent. In this paper, we study to what extent such a statement is true for linear-algebraic notions of expansion. There are two well-studied notions of linear-algebraic expansion, namely dimension expansion (defined in analogy to graph vertex expansion) and quantum expansion (defined in analogy to graph spectral expansion). Lubotzky and Zelmanov proved that the latter implies the former. We prove that the converse is false: there are dimension expanders which are not quantum expanders. Moreover, this asymmetry is explained by the fact that there are two distinct linear-algebraic analogues of graph edge expansion. The first of these is quantum edge expansion, which was introduced by Hastings, and which he proved to be equivalent to quantum expansion. We introduce a new notion, termed dimension edge expansion, which we prove is equivalent to dimension expansion and which is implied by quantum edge expansion. Thus, the separation above is implied by a finer one: dimension edge expansion is strictly weaker than quantum edge expansion. This new notion also leads to a new, more modular proof of the Lubotzky--Zelmanov result that quantum expanders are dimension expanders.
The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings (Quantum Information and Computation, 2014) -- which posits the existence of a local Hamiltonian with a super-constant circuit lower bound on the complexity of all low-energy states -- identifies a fundamental obstacle to the resolution of the quantum PCP conjecture. In this work, we provide new techniques, based on entropic and local indistinguishability arguments, that prove circuit lower bounds for all the low-energy states of local Hamiltonians arising from quantum error-correcting codes. For local Hamiltonians arising from nearly linear-rate and polynomial-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy $o(n)$ (which can be viewed as an almost linear NLTS theorem). Such codes are known to exist and are not necessarily locally-testable, a property previously suspected to be essential for the NLTS conjecture. Curiously, such codes can also be constructed on a two-dimensional lattice, showing that low-depth states cannot accurately approximate the ground-energy in physically relevant systems.
The Lovasz Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast. Besides conditions for existence of and fast convergence to desirable objects, one may naturally ask further questions regarding properties of these algorithms. For instance, are they parallelizable?, how many solutions can they output?, what is the expected weight of a solution?, etc. These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs.
We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under simultaneous row and column permutations of the matrix. We establish unconditional exponential lower bounds on the size of any symmetric circuit for computing the permanent. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.
We write down an explicit sequence of tensors in $C^m\otimes C^m\otimes C^m$, for all $m$ sufficiently large, having border rank at least $2.02m$, overcoming a longstanding barrier. We obtain our lower bounds via the border substitution method.
The spectral gap of a Markov chain can be bounded by the spectral gaps of constituent "restriction" chains and a "projection" chain, and the strength of such a bound is the content of various decomposition theorems. In this paper, we introduce a new parameter that allows us to improve upon these bounds. We further define a notion of orthogonality between the restriction chains and "complementary" restriction chains. This leads to a new Complementary Decomposition theorem, which does not require analyzing the projection chain. For $\epsilon$-orthogonal chains, this theorem may be iterated $O(1/\epsilon)$ times while only giving away a constant multiplicative factor on the overall spectral gap. As an application, we provide a $1/n$-orthogonal decomposition of the nearest neighbor Markov chain over $k$-class biased monotone permutations on [$n$], as long as the number of particles in each class is at least $C\log n$. This allows us to apply the Complementary Decomposition theorem iteratively $n$ times to prove the first polynomial bound on the spectral gap when $k$ is as large as $\Theta(n/\log n)$. The previous best known bound assumed $k$ was at most a constant.
We show that if a k-CNF requires width w to refute in resolution, then it requires space square root of √ω to refute in polynomial calculus, where the space of a polynomial calculus refutation is the number of monomials that must be kept in memory when working through the proof. This is the first analogue, in polynomial calculus, of Atserias and Dalmau's result lower-bounding clause space in resolution by resolution width. As a by-product of our new approach to space lower bounds we give a simple proof of Bonacina's recent result that total space in resolution (the total number of variable occurrences that must be kept in memory) is lower-bounded by the width squared. As corollaries of the main result we obtain some new lower bounds on the PCR space needed to refute specific formulas, as well as partial answers to some open problems about relations between space, size, and degree for polynomial calculus.
Let $A \in \{0,1\}^{n \times n}$ be a matrix with $z$ zeroes and $u$ ones and $x$ be an $n$-dimensional vector of formal variables over a semigroup $(S, \circ)$. How many semigroup operations are required to compute the linear operator $Ax$? As we observe in this paper, this problem contains as a special case the well-known range queries problem and has a rich variety of applications in such areas as graph algorithms, functional programming, circuit complexity, and others. It is easy to compute $Ax$ using $O(u)$ semigroup operations. The main question studied in this paper is: can $Ax$ be computed using $O(z)$ semigroup operations? We prove that in general this is not possible: there exists a matrix $A \in \{0,1\}^{n \times n}$ with exactly two zeroes in every row (hence $z=2n$) whose complexity is $\Theta(n\alpha(n))$ where $\alpha(n)$ is the inverse Ackermann function. However, for the case when the semigroup is commutative, we give a constructive proof of an $O(z)$ upper bound. This implies that in commutative settings, complements of sparse matrices can be processed as efficiently as sparse matrices (though the corresponding algorithms are more involved). Note that this covers the cases of Boolean and tropical semirings that have numerous applications, e.g., in graph theory. As a simple application of the presented linear-size construction, we show how to multiply two $n\times n$ matrices over an arbitrary semiring in $O(n^2)$ time if one of these matrices is a 0/1-matrix with $O(n)$ zeroes (i.e., a complement of a sparse matrix).
For f : {0, 1}(n) -> {0, 1} and S subset of {1, 2, . . , , n}, let J(s)(+) (f) be the probability that, for G uniform from {0, 1}(n), there is some y is an element of {0, 1}(n) with f(y) = 1 and x equivalent to y outside S. We are interested in estimating, for given mu(f) (:= E(f)) and <, the least possible value of maxJ(s)(+) (f) : |S| = m}. A theorem of Kahn, Kalai, and Linial (KKL) gave some understanding of this issue and led to several stronger conjectures. Here we disprove a pair of conjectures from the late 80s, as follows. (1) The KKL Theorem implies that there is a fixed alpha > 0 so that if mu(f) approximate to 1/2, and c > 0, then there is a set S of size at most alpha cn with J(s)(+) (f) >= 1 - n(-c). We show that for every delta > 0 there is an f with mu(f) approximate to 1/2 and J(s)(+) (f) <= 1 - n(-c) for every S of size (1/2 - delta)n, where C = C-delta. This disproves a conjecture of Benny Chor from 1989. (2) We also show that for fixed delta > 0 there are c, alpha > 0 and Boolean functions f such that mu(f) > exp[-n(1-c)] and J(s)(+) (f) <= exp[-n(alpha)] for each S of size (1/2 - delta)n. This disproves a conjecture of the third author from the late 80s.
Given a rooted tree and a ranking of its leaves, what is the minimum number of inversions of the leaves that can be attained by ordering the tree? This variation of the well-known problem of counting inversions in arrays originated in mathematical psychology. It has the evaluation of the Mann-Whitney statistic for detecting differences between distributions as a special case. We study the complexity of the problem in the comparison-query model, the standard model for problems like sorting, selection, and heap construction. The complexity depends heavily on the shape of the tree: for trees of unit depth, the problem is trivial; for many other shapes, we establish lower bounds close to the strongest known in the model, namely the lower bound of log2(n!) for sorting n items. For trees with n leaves we show, in increasing order of closeness to the sorting lower bound:
