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1. Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs

   Afonso S Bandeira, Jess Banks (October 2021, Proceedings of Machine Learning Research)

   null (Ed.)

   Abstract We study the problem of efficiently refuting the k-colorability of a graph, or equivalently, certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular graphs, suggesting that there is no polynomial-time algorithm that improves upon a classical spectral algorithm. Our evidence takes the form of a "computationally-quiet planting": we construct a distribution of d-regular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishable by a large class of algorithms. We generalize our results to the more general problem of certifying an upper bound on the maximum k-cut. This quiet planting is achieved by minimizing the effect of the planted structure (e.g. colorings or cuts) on the graph spectrum. Specifically, the planted structure corresponds exactly to eigenvectors of the adjacency matrix. This avoids the pushout effect of random matrix theory, and delays the point at which the planting becomes visible in the spectrum or local statistics. To illustrate this further, we give similar results for a Gaussian analogue of this problem: a quiet version of the spiked model, where we plant an eigenspace rather than adding a generic low-rank perturbation. Our evidence for computational hardness of distinguishing two distributions is based on three different heuristics: stability of belief propagation, the local statistics hierarchy, and the low-degree likelihood ratio. Of independent interest, our results include general-purpose bounds on the low-degree likelihood ratio for multi-spiked matrix models, and an improved low-degree analysis of the stochastic block model. 

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2. A Polynomial Degree Bound on Equations for Non-rigid Matrices and Small Linear Circuits

   https://doi.org/10.1145/3543685  

   Volk, Ben Lee; Kumar, Mrinal (June 2022, ACM Transactions on Computation Theory)

   We show that there is an equation of degree at most poly( n ) for the (Zariski closure of the) set of the non-rigid matrices: That is, we show that for every large enough field 𝔽, there is a non-zero n 2 -variate polynomial P ε 𝔽[ x 1, 1 , ..., x n, n ] of degree at most poly( n ) such that every matrix M that can be written as a sum of a matrix of rank at most n /100 and a matrix of sparsity at most n 2 /100 satisfies P(M) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer, and Landsberg [ 9 ] and improves the best upper bound known for this problem down from exp ( n 2 ) [ 9 , 12 ] to poly( n ). We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n 2 /200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [ 21 ] to construct low-degree “universal” maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low-degree annihilating polynomial completes the proof. As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [ 11 ]. 

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3. Efficient Algorithms for Cardinality Estimation and Conjunctive Query Evaluation With Simple Degree Constraints

   https://doi.org/10.1145/3725233  

   Im, Sungjin; Moseley, Benjamin; Ngo, Hung; Pruhs, Kirk (June 2025, Proceedings of the ACM on Management of Data)

   Cardinality estimation and conjunctive query evaluation are two of the most fundamental problems in database query processing. Recent work proposed, studied, and implemented a robust and practical information-theoretic cardinality estimation framework. In this framework, the estimator is the cardinality upper bound of a conjunctive query subject to ''degree-constraints'', which model a rich set of input data statistics. For general degree constraints, computing this bound is computationally hard. Researchers have naturally sought efficiently computable relaxed upper bounds that are as tight as possible. The polymatroid bound is the tightest among those relaxed upper bounds. While it is an open question whether the polymatroid bound can be computed in polynomial-time in general, it is known to be computable in polynomial-time for some classes of degree constraints. Our focus is on a common class of degree constraints called simple degree constraints. Researchers had not previously determined how to compute the polymatroid bound in polynomial time for this class of constraints. Our first main result is a polynomial time algorithm to compute the polymatroid bound given simple degree constraints. Our second main result is a polynomial-time algorithm to compute a ''proof sequence'' establishing this bound. This proof sequence can then be incorporated in the PANDA-framework to give a faster algorithm to evaluate a conjunctive query. In addition, we show computational limitations to extending our results to broader classes of degree constraints. Finally, our technique leads naturally to a new relaxed upper bound called theflow bound,which is computationally tractable. 

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4. Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning

   Brand; Kalev, Amir; Li, Tongyang; Lin, Cedric Yen-Yu; Svore, Krysta M.; Wu, Xiaodi (July 2019, Leibniz international proceedings in informatics)

   We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest. 

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5. Faster Linear Algebra for Distance Matrices

   Indyk, Piotr; Silwal, Sandeep (January 2022, Conference on Neural Information Processing Systems)

   The distance matrix of a dataset X of n points with respect to a distance function f represents all pairwise distances between points in X induced by f. Due to their wide applicability, distance matrices and related families of matrices have been the focus of many recent algorithmic works. We continue this line of research and take a broad view of algorithm design for distance matrices with the goal of designing fast algorithms, which are specifically tailored for distance matrices, for fundamental linear algebraic primitives. Our results include efficient algorithms for computing matrix-vector products for a wide class of distance matrices, such as the l1 metric for which we get a linear runtime, as well as a quadratic lower bound for any algorithm which computes a matrix-vector product for the l_infty case. Our upper bound results have many further downstream applications, including the fastest algorithm for computing a relative error low-rank approximation for the distance matrix induced by l1 and l2 functions and the fastest algorithm for computing an additive error lowrank approximation for the l2 metric, in addition to applications for fast matrix multiplication among others. We also give algorithms for constructing distance matrices and show that one can construct an approximate l2 distance matrix in time faster than the bound implied by the Johnson-Lindenstrauss lemma. 

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