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Register a new userApproximating the Permanent of a Random Matrix with Vanishing Mean
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We show an algorithm for computing the permanent of a random matrix with vanishing mean in quasi-polynomial time. Among special cases are the Gaussian, and biased-Bernoulli random matrices with mean 1/lnln(n)^{1/8}. In addition, we can compute the permanent of a random matrix with mean 1/poly(ln(n)) in time 2^{O(n^{eps})} for any small constant eps>0. Our algorithm counters the intuition that the permanent is hard because of the sign problem - namely the interference between entries of a matrix with different signs. A major open question then remains whether one can provide an efficient algorithm for random matrices of mean 1/poly(n), whose conjectured #P-hardness is one of the baseline assumptions of the BosonSampling paradigm.
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We present a randomized approximation scheme for the permanent of a matrix with nonnegative entries. Our scheme extends a recursive rejection sampling method of Huber and Law (SODA 2008) by replacing the upper bound for the permanent with a linear combination of the subproblem bounds at a moderately large depth of the recursion tree. This method, we call deep rejection sampling, is empirically shown to outperform the basic, depth-zero variant, as well as a related method by Kuck et al. (NeurIPS 2019). We analyze the expected running time of the scheme on random $(0, 1)$-matrices where each entry is independently $1$ with probability $p$. Our bound is superior to a previous one for $p$ less than $1/5$, matching another bound that was known to hold when every row and column has density exactly $p$.
We report a cluster of results regarding the difficulty of finding approximate ground states to typical instances of the quantum satisfiability problem $k$-QSAT on large random graphs. As an approximation strategy, we optimize the solution space over `classical product states, which in turn introduces a novel autonomous classical optimization problem, PSAT, over a space of continuous degrees of freedom rather than discrete bits. Our central results are: (i) The derivation of a set of bounds and approximations in various limits of the problem, several of which we believe may be amenable to a rigorous treatment. (ii) A demonstration that an approximation based on a greedy algorithm borrowed from the study of frustrated magnetism performs well over a wide range in parameter space, and its performance reflects structure of the solution space of random $k$-QSAT. Simulated annealing exhibits metastability in similar `hard regions of parameter space. (iii) A generalization of belief propagation algorithms introduced for classical problems to the case of continuous spins. This yields both approximate solutions, as well as insights into the free energy `landscape of the approximation problem, including a so-called dynamical transition near the satisfiability threshold. Taken together, these results allow us to elucidate the phase diagram of random $k$-QSAT in a two-dimensional energy-density--clause-density space.
We study the problem of allocating $m$ items to $n$ agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a $1/e$ approximation factor of the objective. Our main technical contribution is an extension of Gurvitss lower bound on the coefficient of the square-free monomial of a degree $m$-homogeneous stable polynomial on $m$ variables to all homogeneous polynomials. We use this extension to analyze the expected welfare of the allocation returned by our randomized rounding algorithm.
This paper considers a variant of the online paging problem, where the online algorithm has access to multiple predictors, each producing a sequence of predictions for the page arrival times. The predictors may have occasional prediction errors and it is assumed that at least one of them makes a sublinear number of prediction errors in total. Our main result states that this assumption suffices for the design of a randomized online algorithm whose time-average regret with respect to the optimal offline algorithm tends to zero as the time tends to infinity. This holds (with different regret bounds) for both the full information access model, where in each round, the online algorithm gets the predictions of all predictors, and the bandit access model, where in each round, the online algorithm queries a single predictor. While online algorithms that exploit inaccurate predictions have been a topic of growing interest in the last few years, to the best of our knowledge, this is the first paper that studies this topic in the context of multiple predictors for an online problem with unbounded request sequences. Moreover, to the best of our knowledge, this is also the first paper that aims for (and achieves) online algorithms with a vanishing regret for a classic online problem under reasonable assumptions.
Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of Phase Estimation algorithms performs when we apply it to solve the Eigen-Problem of Hermitian matrices. The result is a completely new, efficient and stable, parallel algorithm to compute an approximate spectral decomposition of any Hermitian matrix. The algorithm can be implemented by Boolean circuits in $O(log^2 n)$ parallel time with a total cost of $O(n^{omega+1})$ Boolean operations. This Boolean complexity matches the best known rigorous $O(log^2 n)$ parallel time algorithms, but unlike those algorithms our algorithm is (logarithmically) stable, so further improvements may lead to practical implementations. All previous efficient and rigorous approaches to solve the Eigen-Problem use randomization to avoid bad condition as we do too. Our algorithm makes further use of randomization in a completely new way, taking random powers of a unitary matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian perturbation and a random polynomial power are sufficient to ensure almost pairwise independence of the phases $(mod (2pi))$ is the main technical contribution of this work. This randomization enables us, given a Hermitian matrix with well separated eigenvalues, to sample a random eigenvalue and produce an approximate eigenvector in $O(log^2 n)$ parallel time and $O(n^omega)$ Boolean complexity. We conjecture that further improvements of our method can provide a stable solution to the full approximate spectral decomposition problem with complexity similar to the complexity (up to a logarithmic factor) of sampling a single eigenvector.
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