The MAX BISECTION problem seeks a maximum-size cut that evenly divides the vertices of a given undirected graph. An open problem raised by Austrin, Benabbas, and Georgiou [SODA'13, TALG'16] is whether MAX BISECTION can be approximated as well as MAX CUT, i.e., to within \(\alpha_{\mathrm{GW}} \approx 0.8785672\ldots\), which is the approximation ratio achieved by the celebrated Goemans-Williamson algorithm for MAX CUT, which is best possible assuming the Unique Games Conjecture (UGC). They conjectured that the answer is yes.

In this paper, we continue the study of robust satisfiability of promise CSPs (PCSPs), initiated in (Brakensiek, Guruswami, Sandeep, STOC 2023), and obtain the following results:

The input to the Multiway Cut problem is a weighted undirected graph, with nonnegative edge weights, and k designated terminals. The goal is to partition the vertices of the graph into k parts, each containing exactly one of the terminals, such that the sum of weights of the edges connecting vertices in different parts of the partition is minimized. The problem is APX-hard for k≥3. The currently best known approximation algorithm for the problem for arbitrary k, obtained by Sharma and Vondrák [STOC 2014] more than a decade ago, has an approximation ratio of 1.2965. We present an algorithm with an improved approximation ratio of 1.2787. Also, for small values of k ≥ 4 we obtain the first improvements in 25 years over the currently best approximation ratios obtained by Karger, Klein, Stein, Thorup, and Young [STOC 1999]. (For k=3 an optimal approximation algorithm is known.)

Abstract. Assuming the unique games conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the max cut problem is [Formula: see text], obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. The current best approximation algorithm for max di-cut, i.e., the max cut problem in directed graphs, achieves a ratio of about 0.87401, leaving open the question of whether max di-cut can be approximated as well as max cut. We obtain a slightly improved algorithm for max di-cut and a new UGC-hardness for it, showing that [Formula: see text], where [Formula: see text] is the best approximation ratio that can be obtained in polynomial time for max di-cut under UGC. Our new upper bound shows that max di-cut cannot be approximated as well as max cut, which separates max di-cut from max cut and resolves a question raised by Feige and Goemans. A natural generalization of max di-cut is the max [Formula: see text]-and problem in which each constraint is of the form [Formula: see text], where [Formula: see text] and [Formula: see text] are literals, i.e., variables or their negations (in max di-cut each constraint is of the form [Formula: see text] where [Formula: see text] and [Formula: see text] are variables). Austrin separated max [Formula: see text]-and from max cut by showing that [Formula: see text] and conjectured that max [Formula: see text]-and and max di-cut have the same approximation ratio. Our new lower bound on max di-cut refutes this conjecture, completing the separation of the three problems max [Formula: see text]-and, max di-cut, and max cut. We also obtain a new lower bound for max [Formula: see text]-and, showing that [Formula: see text]. Our upper bound on max di-cut is achieved via a simple, analytical proof. The new lower bounds on max di-cut and max [Formula: see text]-and, i.e., the new approximation algorithms, use experimentally discovered distributions of rounding functions which are then verified via computer-assisted proofs. Code for the project is available at https://github.com/jbrakensiek/max-dicut .

We prove a hardness of sampling result for the anti-ferromagnetic Ising model on random graphs of average degree d for large constant d, proving that when the normalized inverse temperature satisfies β > 1 (asymptotically corresponding to the condensation threshold), then w.h.p. over the random graph there is no stable sampling algorithm that can output a sample close in W₂ distance to the Gibbs measure. The results also apply to a fixed-magnetization version of the model, showing that there are no stable sampling algorithms for low but positive temperature max and min bisection distributions. These results show a gap in the tractability of search and sampling problems: while there are efficient algorithms to find near optimizers, stable sampling algorithms cannot access the Gibbs distribution concentrated on such solutions. Our techniques involve extensions of the interpolation technique relating behavior of the mean field Sherrington-Kirkpatrick model to behavior of Ising models on random graphs of average degree d for large d. While previous interpolation arguments compared the free energies of the two models, our argument compares the average energies and average overlaps in the two models.

No abstract available.

Non-Gaussian Component Analysis (NGCA) is the statistical task of finding a non-Gaussian direction in a high-dimensional dataset. Specifically, given i.i.d. samples from a distribution$P_{v}^{A}$on$\mathbb{R}^{n}$that behaves like a known distribution$A$in a hidden direction$v$and like a standard Gaussian in the orthogonal complement, the goal is to approximate the hidden direction. The standard formulation posits that the first$k$- moments of$A$match those of the standard Gaussian and the$k$-th moment differs. Under mild assumptions, this problem has sample complexity$O(n)$. On the other hand, all known efficient algorithms require$\Omega(n^{k/2})$samples. Prior work developed sharp Statistical Query and low-degree testing lower bounds suggesting an information-computation tradeoff for this problem. Here we study the complexity of NGCA in the Sum-of-Squares (SoS) framework. Our main contribution is the first super-constant degree SoS lower bound for NGCA. Specifically, we show that if the non-Gaussian distribution$A$matches the first$(k-1)$moments of$\mathrm{N}(\mathrm{O},\ 1)$and satisfies other mild conditions, then with fewer than$n^{(1-\varepsilon)k/2}$many samples from the normal distribution, with high probability, degree$(\log n)^{\frac{1}{2}-o_{n}(1)}\mathbf{SoS}$fails to refute the existence of such a direction$v$. Our result significantly strengthens prior work by establishing a super-polynomial information-computation tradeoff against a broader family of algorithms. As corollaries, we obtain SoS lower bounds for several problems in robust statistics and the learning of mixture models. Our SoS lower bound proof introduces a novel technique’ that we believe may be of broader interest, and a number of refinements over existing methods. As in previous work, we use the framework of [Barak et al. FOCS 2016], where we express the moment matrix$M$as a sum of graph matrices, find a factorization$M\approx LQL^{T}$using minimum vertex separators, and show that with high probability$Q$is positive semidefinite (PSD) while the errors are small. Our technical innovations involve the following. First, instead of the minimum weight separator used in prior work, we crucially make use of the minimum square separator. Second, proving that$Q$is PSD poses significant challenges due to an intrinsic reason. In all prior work, the major part of$Q$was always a constant term, meaning a matrix whose entries are constant functions of the input. Here, however, even after removing a small error term,$Q$remains a nontrivial linear combination of non-constant, equally dominating terms. We develop an algebraic method to address this difficulty, which may have wider applications. Specifically, we model the multiplications between the “important” graph matrices by an R.-algebra, construct a representation of this algebra, and use it to analyze$Q$. Via this approach, we show that the PSDness of$Q$boils down to the multiplicative identities of Hermite polynomials.

In this paper, we investigate the total coefficient size of Nullstellensatz proofs. We show that Nullstellensatz proofs of the pigeonhole principle on $n$ pigeons require total coefficient size $2^{Ω(n)}$ and that there exist Nullstellensatz proofs of the ordering principle on $n$ elements with total coefficient size $2^n - n$.

We prove that for every D ∈ N, and large enough constant d ∈ N, with high probability over the choice of G ∼ G(n,d/n), the Erdos-Renyi random graph distribution, the canonical degree 2D Sum-of-Squares relaxation fails to certify that the largest independent set in G is of size o(n/√d D4). In particular, degree D sum-of-squares strengthening can reduce the integrality gap of the classical theta SDP relaxation by at most a O(D4) factor. This is the first lower bound for >4-degree Sum-of-Squares (SoS) relaxation for any problems on ultra sparse random graphs (i.e. average degree of an absolute constant). Such ultra-sparse graphs were a known barrier for previous methods and explicitly identified as a major open direction. Indeed, the only other example of an SoS lower bound on ultra-sparse random graphs was a degree-4 lower bound for Max-Cut. Our main technical result is a new method to obtain spectral norm estimates on graph matrices (a class of low-degree matrix-valued polynomials in G(n,d/n)) that are accurate to within an absolute constant factor. All prior works lose log n factors that trivialize any lower bound on o(logn)-degree random graphs. We combine these new bounds with several upgrades on the machinery for analyzing lower-bound witnesses constructed by pseudo-calibration so that our analysis does not lose any ω(1)-factors that would trivialize our results. In addition to other SoS lower bounds, we believe that our methods for establishing spectral norm estimates on graph matrices will be useful in the analyses of numerical algorithms on average-case inputs.

Given independent standard Gaussian points $v_1, \ldots, v_n$ in dimension $d$, for what values of $(n, d)$ does there exist with high probability an origin-symmetric ellipsoid that simultaneously passes through all of the points? This basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. Based on strong numerical evidence, Saunderson, Parrilo, and Willsky [Proc. of Conference on Decision and Control, pp. 6031-6036, 2013] conjectured that the ellipsoid fitting problem transitions from feasible to infeasible as the number of points $n$ increases, with a sharp threshold at $n \sim d^2/4$. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = d^2/\mathrm{polylog}(d)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix and a careful analysis of its Neumann expansion via the theory of graph matrices.

Assuming the Unique Games Conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the MAX CUT problem is $\alpha_{\text {CUT}} \simeq 0.87856$, obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. Currently, the best approximation algorithm for MAX DI-CUT, i.e., the MAX CUT problem in directed graphs, achieves a ratio of about 0.87401, leaving open the question whether MAX DI-CUT can be approximated as well as MAX CUT. We obtain a slightly improved algorithm for MAX DI-CUT and a new UG-Chardness result for it, showing that $0.87446 \leq \alpha_{\text {DI-CUT}} \leq 0.87461$, where $\alpha_{\text {DI-CUT}}$ is the best approximation ratio that can be obtained in polynomial time for MAX DI-CUT under UGC. The new upper bound separates MAX DI-CUT from MAX CUT, i.e., shows that MAX DI-CUT cannot be approximated as well as MAX CUT, resolving a question raised by Feige and Goemans. A natural generalization of MAX DI-CUT is the MAX 2-AND problem in which each constraint is of the form $z_{1} \wedge {z_{2}}$, where $z_{1}$ and ${z_{2}}$ are literals, i.e., variables or their negations. (In MAX DI-CUT each constraint is of the form $\bar{x}_{1} \wedge {x_{2}}$, where $x_{1}$ and ${x_{2}}$ are variables.) Austrin separated MAX 2-AND from MAX CUT by showing that $\alpha_{2 \mathrm{AND}} \leq 0.87435$ and conjectured that MAX 2-AND and MAX DI-CUT have the same approximation ratio. Our new lower bound on MAX DI-CUT refutes this conjecture, completing the separation of the three problems MAX 2-AND, MAX DI-CUT and MAX CUT. We also obtain a new lower bound for MAX 2-AND showing that $0.87414 \leq \alpha_{2 \text {AND}} \leq 0.87435$. Our upper bound on MAXDI-CUT is achieved via a simple analytical proof. The new lower bounds on MAX DI-CUT and MAX 2-AND, i.e., the new approximation algorithms, use experimentally-discovered distributions of rounding functions which are then verified via computer-assisted proofs.11Code for the project: https://github.com/jbrakensiek/max-dicut

We prove that unary Sherali-Adams requires proofs of size $n^{\Omega(d)}$ to rule out the existence of an $n^{\Theta(1)}$-clique in Erdős-Rényi random graphs whose maximum clique is of size $d \leq 2 \log n$. This lower bound is tight up to the multiplicative constant in the exponent. We obtain this result by introducing a technique inspired by pseudo-calibration which may be of independent interest. The technique involves defining a measure on monomials that precisely captures the contribution of a monomial to a refutation. This measure intuitively captures progress and should have further applications in proof complexity.

Given a graph and an integer k, Densest k-Subgraph is the algorithmic task of finding the subgraph on k vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an O(n1/4 + )-factor approximation (for any > 0) due to Bhaskara et al. [STOC ’10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an O(n1/4 − )-factor approximation. In the average case, Densest k-Subgraph is a prototypical noisy inference task which is conjectured to exhibit a statistical-computational gap.

Cohen, Peri and Ta-Shma [Gil Cohen et al., 2021] considered the following question: Assume the vertices of an expander graph are labelled by ± 1. What "test" functions f : {±1}^t → {±1} can or cannot distinguish t independent samples from those obtained by a random walk? [Gil Cohen et al., 2021] considered only balanced labellings, and proved that for all symmetric functions the distinguishability goes down to zero with the spectral gap λ of the expander G. In addition, [Gil Cohen et al., 2021] show that functions computable by AC⁰ circuits are fooled by expanders with vanishing spectral expansion. We continue the study of this question. We generalize the result to all labelling, not merely balanced ones. We also improve the upper bound on the error of symmetric functions. More importantly, we give a matching lower bound and show a symmetric function with distinguishability going down to zero with λ but not with t. Moreover, we prove a lower bound on the error of functions in AC⁰ in particular, we prove that a random walk on expanders with constant spectral gap does not fool AC⁰.

In this paper, we consider low-degree polynomials of inner products between a collection of random vectors. We give an almost orthogonal basis for this vector space of polynomials when the random vectors are Gaussian, spherical, or Boolean. In all three cases, our basis admits an interesting combinatorial description based on the topology of the underlying graph of inner products. We also analyze the expected value of the product of two polynomials in our basis. In all three cases, we show that this expected value can be expressed in terms of collections of matchings on the underlying graph of inner products. In the Gaussian and Boolean cases, we show that this expected value is always non-negative. In the spherical case, we show that this expected value can be negative but we conjecture that if the underlying graph of inner products is planar then this expected value will always be non-negative.

Principal Components Analysis (PCA) is a dimension-reduction technique widely used in machine learning and statistics. However, due to the dependence of the principal components on all the dimensions, the components are notoriously hard to interpret. Therefore, a variant known as sparse PCA is often preferred. Sparse PCA learns principal components of the data but enforces that such components must be sparse. This has applications in diverse fields such as computational biology and image processing. To learn sparse principal components, it's well known that standard PCA will not work, especially in high dimensions, and therefore algorithms for sparse PCA are often studied as a separate endeavor. Various algorithms have been proposed for Sparse PCA over the years, but given how fundamental it is for applications in science, the limits of efficient algorithms are only partially understood. In this work, we study the limits of the powerful Sum of Squares (SoS) family of algorithms for Sparse PCA. SoS algorithms have recently revolutionized robust statistics, leading to breakthrough algorithms for long-standing open problems in machine learning, such as optimally learning mixtures of gaussians, robust clustering, robust regression, etc. Moreover, it is believed to be the optimal robust algorithm for many statistical problems. Therefore, for sparse PCA, it's plausible that it can beat simpler algorithms such as diagonal thresholding that have been traditionally used. In this work, we show that this is not the case, by exhibiting strong tradeoffs between the number of samples required, the sparsity and the ambient dimension, for which SoS algorithms, even if allowed sub-exponential time, will fail to optimally recover the component. Our results are complemented by known algorithms in literature, thereby painting an almost complete picture of the behavior of efficient algorithms for sparse PCA. Since SoS algorithms encapsulate many algorithmic techniques such as spectral or statistical query algorithms, this solidifies the message that known algorithms are optimal for sparse PCA. Moreover, our techniques are strong enough to obtain similar tradeoffs for Tensor PCA, another important higher order variant of PCA with applications in topic modeling, video processing, etc.

The Sum-of-Squares (SoS) hierarchy of semidefinite programs is a powerful algorithmic paradigm which captures state-of-the-art algorithmic guarantees for a wide array of problems. In the average case setting, SoS lower bounds provide strong evidence of algorithmic hardness or information-computation gaps. Prior to this work, SoS lower bounds have been obtained for problems in the “dense” input regime, where the input is a collection of independent Rademacher or Gaussian random variables, while the sparse regime has remained out of reach. We make the first progress in this direction by obtaining strong SoS lower bounds for the problem of Independent Set on sparse random graphs. We prove that with high probability over an Erdós-Rénvi random graph$G\sim G_{n_{J}\frac{d}{u}}$with average degree$d > \log^{2}n$, degree-Dsos SoS fails to refute the existence of an independent set of size$k=\displaystyle \Omega(\frac{n}{\sqrt{d}(\log n)(\mathrm{D}_{\mathrm{S}\mathrm{o}\mathrm{S}})^{c_{0}}})$in$G$(where$c_{0}$is an absolute constant), whereas the true size of the largest independent set in$G$is$O(\displaystyle \frac{n\log d}{d})$. Our proof involves several significant extensions of the techniques used for proving SoS lower bounds in the dense setting. Previous lower bounds are based on the pseudo-calibration heuristic of Barak et al. [FOCS 2016] which produces a candidate SoS solution using a planted distribution indistinguishable from the input distribution via low-degree tests. In the sparse case the natural planted distribution does admit low-degree distinguishers, and we show how to adapt the pseudo-calibration heuristic to overcome this. Another notorious technical challenge for the sparse regime is the quest for matrix norm bounds. In this paper, we obtain new norm bounds for graph matrices in the sparse setting. While in the dense setting the norms of graph matrices are characterized by the size of the minimum vertex separator of the corresponding graph, this turns not to be the case for sparse graph matrices. Another contribution of our work is developing a new combinatorial understanding of structures needed to understand the norms of sparse graph matrices.

We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in n variables with roots placed in points k-1 and k. Functions of this type have played a central role in deepening the understanding of the performance of the SoS method for various unconstrained Boolean hypercube optimization problems, including the Max Cut problem. Recently, Lee, Prakash, de Wolf, and Yuen proved a lower bound on the SoS rank for SQFs of Ω(√{k(n-k)}) and conjectured the lower bound of Ω(n) by similarity to a polynomial representation of the n-bit OR function. Leveraging recent developments on Chebyshev polynomials, we refute the Lee-Prakash-de Wolf-Yuen conjecture and prove that the SoS rank for SQFs is at most O(√{nk}log(n)). We connect this result to two constrained Boolean hypercube optimization problems. First, we provide a degree O(√n) SoS certificate that matches the known SoS rank lower bound for an instance of Min Knapsack, a problem that was intensively studied in the literature. Second, we study an instance of the Set Cover problem for which Bienstock and Zuckerberg conjectured an SoS rank lower bound of n/4. We refute the Bienstock-Zuckerberg conjecture and provide a degree O(√nlog(n)) SoS certificate for this problem.

Abstract. MAX NAE-SAT is a natural optimization problem, closely related to its better-known relative MAX SAT. The approximability status of MAX NAE-SAT is almost completely understood if all clauses have the same size [Formula: see text] for some [Formula: see text]. We refer to this problem as MAX NAE-[Formula: see text]-SAT. For [Formula: see text], it is a slight extension of the celebrated MAX CUT problem. For [Formula: see text], it is related to the MAX CUT problem in graphs that can be fractionally covered by triangles. For [Formula: see text], it is known that an approximation ratio of [Formula: see text], obtained by choosing a random assignment, is optimal, assuming [Formula: see text]. For every [Formula: see text], an approximation ratio of at least [Formula: see text] can be obtained for MAX NAE-[Formula: see text]-SAT. There was some hope, therefore, that there is also a [Formula: see text]-approximation algorithm for MAX NAE-SAT, where clauses of all sizes are allowed simultaneously. Our main result is that there is no [Formula: see text]-approximation algorithm for MAX NAE-SAT, assuming the Unique Games Conjecture (UGC). In fact, even for almost satisfiable instances of MAX NAE-[Formula: see text]-SAT (i.e., MAX NAE-SAT where all clauses have size 3 or 5), the best approximation ratio that can be achieved, assuming UGC, is at most [Formula: see text]. Using calculus of variations, we extend the analysis of O’Donnell and Wu for MAX CUT to MAX NAE-[Formula: see text]-SAT. We obtain an optimal algorithm, assuming UGC, for MAX NAE-[Formula: see text]-SAT, slightly improving on previous algorithms. The approximation ratio of the new algorithm is about 0.9089. This gives a full understanding of MAX NAE-[Formula: see text]-SAT for every [Formula: see text]. Interestingly, the rounding function used by this optimal algorithm is the solution of an integral equation. We complement our theoretical results with some experimental results. We describe an approximation algorithm for almost satisfiable instances of MAX NAE-[Formula: see text]-SAT with a conjectured approximation ratio of 0.8728, and an approximation algorithm for almost satisfiable instances of MAX NAE-SAT with a conjectured approximation ratio of 0.8698. We further conjecture that these are essentially the best approximation ratios that can be achieved for these problems, assuming the UGC. Somewhat surprisingly, the rounding functions used by these approximation algorithms are nonmonotone step functions that assume only the values [Formula: see text].

Given a predicate P: {-1, 1}^k → {-1, 1}, let CSP(P) be the set of constraint satisfaction problems whose constraints are of the form P. We say that P is approximable if given a nearly satisfiable instance of CSP(P), there exists a probabilistic polynomial time algorithm that does better than a random assignment. Otherwise, we say that P is approximation resistant. In this paper, we analyze presidential type predicates, which are balanced linear threshold functions where all of the variables except the first variable (the president) have the same weight. We show that almost all presidential type predicates P are approximable. More precisely, we prove the following result: for any δ₀ > 0, there exists a k₀ such that if k ≥ k₀, δ ∈ (δ₀,1 - 2/k], and {δ}k + k - 1 is an odd integer then the presidential type predicate P(x) = sign({δ}k{x₁} + ∑_{i = 2}^{k} {x_i}) is approximable. To prove this, we construct a rounding scheme that makes use of biases and pairwise biases. We also give evidence that using pairwise biases is necessary for such rounding schemes.

In this paper, we analyze the sum of squares hierarchy (SOS) on the ordering principle on n elements (which has N = Θ(n²) variables). We prove that degree O(√nlog(n)) SOS can prove the ordering principle. We then show that this upper bound is essentially tight by proving that for any ε > 0, SOS requires degree Ω(n^(1/2 - ε)) to prove the ordering principle.

The Sum-of-Squares (SoS) hierarchy is a semi-definite programming meta-algorithm that captures state-of-the-art polynomial time guarantees for many optimization problems such as Max- k-CSPs and Tensor PCA. On the flip side, a SoS lower bound provides evidence of hardness, which is particularly relevant to average-case problems for which NP-hardness may not be available. In this paper, we consider the following average case problem, which we call the Planted Affine Planes (PAP) problem: Given m random vectors d1, ..., dmin Rn, can we prove that there is no vector v ∈ IRnsuch that for all u ∈ [m], 〈v, du〉2= 1? In other words, can we prove that m random vectors are not all contained in two parallel hyperplanes at equal distance from the origin? We prove that for m ≤ n3/2-ε, with high probability, degree- nΩ(ε)SoS fails to refute the existence of such a vector v. When the vectors d1, ..., dmare chosen from the multivariate normal distribution, the PAP problem is equivalent to the problem of proving that a random n-dimensional subspace of Rmdoes not contain a boolean vector. As shown by Mohanty-Raghavendra-Xu [STOC 2020], a lower bound for this problem implies a lower bound for the problem of certifying energy upper bounds on the Sherrington-Kirkpatrick Hamiltonian, and so our lower bound implies a degree- nΩ(ε)SoS lower bound for the certification version of the Sherrington-Kirkpatrick problem.

In this paper, we develop machinery which makes it much easier to prove sum of squares lower bounds when the problem is symmetric under permutations of $[1,n]$ and the unsatisfiability of our problem comes from integrality arguments, i.e. arguments that an expression must be an integer. Roughly speaking, to prove SOS lower bounds with our machinery it is sufficient to verify that the answer to the following three questions is yes: 1. Are there natural pseudo-expectation values for the problem? 2. Are these pseudo-expectation values rational functions of the problem parameters? 3. Are there sufficiently many values of the parameters for which these pseudo-expectation values correspond to the actual expected values over a distribution of solutions which is the uniform distribution over permutations of a single solution? We demonstrate our machinery on three problems, the knapsack problem analyzed by Grigoriev, the MOD 2 principle (which says that the complete graph $K_n$ has no perfect matching when $n$ is odd), and the following Turan type problem: Minimize the number of triangles in a graph $G$ with a given edge density. For knapsack, we recover Grigoriev's lower bound exactly. For the MOD 2 principle, we tighten Grigoriev's linear degree sum of squares lower bound, making it exact. Finally, for the triangle problem, we prove a sum of squares lower bound for finding the minimum triangle density. This lower bound is completely new and gives a simple example where constant degree sum of squares methods have a constant factor error in estimating graph densities.

In this paper, we show that there exists a balanced linear threshold function (LTF) which is unique games hard to approximate, refuting a conjecture of Austrin, Benabbas, and Magen. We also show that the almost monarchy predicate P(x) = sign((k−4)x1 + ∑i=2kxi) is approximable for sufficiently large k.

The problem of finding large cliques in random graphs and its “planted” variant, where one wants to recover a clique of size ω > log ( n ) added to an Erdős-Rényi graph G ∼ G ( n ,1/2), have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size ω = Ω (√ n ). By contrast, information theoretically, one can recover planted cliques so long as ω > log ( n ). In this work, we continue the investigation of algorithms from the Sum of Squares hierarchy for solving the planted clique problem begun by Meka, Potechin, and Wigderson [2] and Deshpande and Montanari [25]. Our main result is that degree four SoS does not recover the planted clique unless ω > √ n / polylog n , improving on the bound ω > n 1/3 due to Reference [25]. An argument of Kelner shows that the this result cannot be proved using the same certificate as prior works. Rather, our proof involves constructing and analyzing a new certificate that yields the nearly tight lower bound by “correcting” the certificate of References [2, 25, 27].

In this paper, we show that while almost all functions require exponential size branching programs to compute, for all functions $f$ there is a branching program computing a doubly exponential number of copies of $f$ which has linear size per copy of $f$. This result disproves a conjecture about non-uniform catalytic computation, rules out a certain type of bottleneck argument for proving non-monotone space lower bounds, and can be thought of as a constructive analogue of Razborov's result that submodular complexity measures have maximum value $O(n)$.

We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in $\mathbb R^n$ from $r⋅\tilde O(n^1.5)$ randomly observed entries of the tensor. This bound improves over the previous best one of $r⋅\tilde O(n^2)$ by reduction to exact matrix completion. Our bound also matches the best known results for the easier problem of approximate tensor completion (Barak & Moitra, 2015). Our algorithm and analysis extends seminal results for exact matrix completion (Candes & Recht, 2009) to the tensor setting via the sum-of-squares method. The main technical challenge is to show that a small number of randomly chosen monomials are enough to construct a degree-3 polynomial with precisely planted orthogonal global optima over the sphere and that this fact can be certified within the sum-of-squares proof system.

We study planted problems-finding hidden structures in random noisy inputs-through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new algorithms for planted problems, often achieving the best known polynomial-time guarantees in terms of accuracy of recovered solutions and robustness to noise. One theme in recent work is the design of spectral algorithms which match the guarantees of SoS algorithms for planted problems. Classical spectral algorithms are often unable to accomplish this: the twist in these new spectral algorithms is the use of spectral structure of matrices whose entries are low-degree polynomials of the input variables. We prove that for a wide class of planted problems, including refuting random constraint satisfaction problems, tensor and sparse PCA, densest-ksubgraph, community detection in stochastic block models, planted clique, and others, eigenvalues of degree-d matrix polynomials are as powerful as SoS semidefinite programs of degree d. For such problems it is therefore always possible to match the guarantees of SoS without solving a large semidefinite program. Using related ideas on SoS algorithms and lowdegree matrix polynomials (and inspired by recent work on SoS and the planted clique problem [BHK+16]), we prove a new SoS lower bound for the tensor PCA problem.

The Sum Of Squares hierarchy is one of the most powerful tools we know of for solving combinatorial optimization problems. However, its performance is only partially understood. Improving our understanding of the sum of squares hierarchy is a major open problem in computational complexity theory. A key component of analyzing the sum of squares hierarchy is understanding the behavior of certain matrices whose entries are random but not independent. For these matrices, there is a random input graph and each entry of the matrix is a low degree function of the edges of this input graph. Moreoever, these matrices are generally invariant (as a function of the input graph) when we permute the vertices of the input graph. In this paper, we bound the norms of all such matrices up to a polylogarithmic factor.

We prove that with high probability over the choice of a random graph G from the Erdös-Rényi distribution G(n,1/2), the nO(d)-time degree d Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least n1/2-c(d/log n)1/2for some constant c > 0. This yields a nearly tight n1/2-o(1)bound on the value of this program for any degree d = o(log n). Moreover we introduce a new framework that we call pseudo-calibration to construct Sum-of-Squares lower bounds. This framework is inspired by taking a computational analogue of Bayesian probability theory. It yields a general recipe for constructing good pseudo-distributions (i.e., dual certificates for the Sum-of-Squares semidefinite program), and sheds further light on the ways in which this hierarchy differs from others.

The problem of finding large cliques in random graphs and its “planted” variant, where one wants to recover a clique of size ω ≫ log (n) added to an Erdős-Rényi graph , have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size . By contrast, information theoretically, one can recover planted cliques so long as ω ≫ log (n). In this work, we continue the investigation of algorithms from the sum of squares hierarchy for solving the planted clique problem begun by Meka, Potechin, and Wigderson [MPW15] and Deshpande and Montanari [DM15b]. Our main results improve upon both these previous works by showing: 1. Degree four SoS does not recover the planted clique unless , improving upon the bound ω ≫ n1/3 due to [DM15b]. 2. For , degree 2d SoS does not recover the planted clique unless ω ≫ n1/(d+1)/(2d polylog n), improving upon the bound due to [MPW15]. Our proof for the second result is based on a fine spectral analysis of the certificate used in the prior works [MPW15, DM15b, FK03] by decomposing it along an appropriately chosen basis. Along the way, we develop combinatorial tools to analyze the spectrum of random matrices with dependent entries and to understand the symmetries in the eigenspaces of the set symmetric matrices inspired by work of Grigoriev [Gri01a] An argument of Kelner shows that the first result cannot be proved using the same certificate. Rather, our proof involves constructing and analyzing a new certificate that yields the nearly tight lower bound by “correcting” the certificate of [MPW15, DM15b, FK03]

Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size k is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for k = Θ(√n). In this paper we study the complexity of the planted clique problem under algorithms from the Sum-Of-Squares hierarchy. We prove the first average case lower bound for this model: for almost all graphs in G(n,1/2), r rounds of the SOS hierarchy cannot find a planted k-clique unless k ≥ (√n/log n)1/rCr. Thus, for any constant number of rounds planted cliques of size no(1) cannot be found by this powerful class of algorithms. This is shown via an integrability gap for the natural formulation of maximum clique problem on random graphs for SOS and Lasserre hierarchies, which in turn follow from degree lower bounds for the Positivestellensatz proof system.

We prove tight size bounds on monotone switching networks for the k-clique problem, and for an explicit monotone problem by analyzing the generation problem with a pyramid structure of height h. This gives alternative proofs of the separations of m-NC from m-P and of m-NCi from m-NCi+1, different from Raz-McKenzie (Combinatorica '99). The enumerative-combinatorial and Fourier analytic techniques in this work are very different from a large body of work on circuit depth lower bounds, and may be of independent interest.

We prove that any monotone switching network solving directed connectivity on N vertices must have size NΩ(log N).