Fast Sampling and Counting k -SAT Solutions in the Local Lemma Regime

We give new algorithms based on Markov chains to sample and approximately count satisfying assignments to k -uniform CNF formulas where each variable appears at most d times. For any k and d satisfying kd < n o(1) and k ≥ 20 log k + 20 log d + 60, the new sampling algorithm runs in close to linear time, and the counting algorithm runs in close to quadratic time. Our approach is inspired by Moitra (JACM, 2019), which remarkably utilizes the Lovász local lemma in approximate counting. Our main technical contribution is to use the local lemma to bypass the connectivity barrier in traditional Markov chain approaches, which makes the well-developed MCMC method applicable on disconnected state spaces such as SAT solutions. The benefit of our approach is to avoid the enumeration of local structures and obtain fixed polynomial running times, even if k = ω (1) or d = ω (1).

A Partition-Based Group Testing Algorithm for Estimating the Number of Infected Individuals

The dangers of COVID-19 remain ever-present worldwide. The asymptomatic nature of COVID-19 obfuscates the signs policy makers look for when deciding to reopen public areas or further quarantine. In much of the world, testing resources are often scarce, creating a need for testing potentially infected individuals that prioritizes efficiency. This report presents an advancement to Beigel and Kasif's Approximate Counting Algorithm (ACA). ACA estimates the infection rate with a number of tests that is logarithmic in the population size. Our newer version of the algorithm provides an extra level of efficiency: each subject is tested exactly once. A simulation of the algorithm, created for and presented as part of this paper, can be used to find a linear regression of the results with R^2 > 0.999. This allows stakeholders and members of the biomedical community to estimate infection rates for varying population sizes and ranges of infection rates.

Approximate Counting of k -Paths: Simpler, Deterministic, and in Polynomial Space

Recently, Brand et al. [STOC 2018] gave a randomized mathcal O(4 k m ε -2 -time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ε based on exterior algebra. Prior to our work, this has been the state-of-the-art. In this article, we revisit the algorithm by Alon and Gutner [IWPEC 2009, TALG 2010], and obtain the following results: • We present a deterministic 4 k + O (√ k (log k +log 2 ε -1 )) m -time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. • Additionally, we present a randomized 4 k +mathcal O(log k (log k +logε -1 )) m -time polynomial-space algorithm. Our algorithm is simple—we only make elementary use of the probabilistic method. Here, n and m are the number of vertices and the number of edges, respectively. Additionally, our approach extends to approximate counting of other patterns of small size (such as q -dimensional p -matchings).

Chapter 26. Approximate Model Counting

Model counting, or counting solutions of a set of constraints, is a fundamental problem in Computer Science with diverse applications. Since exact counting is computationally hard (#P complete), approximate counting techniques have received much attention over the past few decades. In this chapter, we focus on counting models of propositional formulas, and discuss in detail universal-hashing based approximate counting, which has emerged as the predominant paradigm for state-of-the-art approximate model counters. These counters are randomized algorithms that exploit properties of universal hash functions to provide rigorous approximation guarantees, while piggybacking on impressive advances in propositional satisfiability solving to scale up to problem instances with a million variables. We elaborate on various choices in designing such approximate counters and the implications of these choices. We also discuss variants of approximate model counting, such as DNF counting and weighted counting.

Fine-Grained Reductions from Approximate Counting to Decision

In this article, we introduce a general framework for fine-grained reductions of approximate counting problems to their decision versions. (Thus, we use an oracle that decides whether any witness exists to multiplicatively approximate the number of witnesses with minimal overhead.) This mirrors a foundational result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the polynomial-time setting, and a similar result of Müller (IWPEC 2006) in the FPT setting. Using our framework, we obtain such reductions for some of the most important problems in fine-grained complexity: the Orthogonal Vectors problem, 3SUM, and the Negative-Weight Triangle problem (which is closely related to All-Pairs Shortest Path). While all these problems have simple algorithms over which it is conjectured that no polynomial improvement is possible, our reductions would remain interesting even if these conjectures were proved; they have only polylogarithmic overhead and can therefore be applied to subpolynomial improvements such as the n 3 / exp(Θ (√ log n ))-time algorithm for the Negative-Weight Triangle problem due to Williams (STOC 2014). Our framework is also general enough to apply to versions of the problems for which more efficient algorithms are known. For example, the Orthogonal Vectors problem over GF( m ) d for constant m can be solved in time n · poly ( d ) by a result of Williams and Yu (SODA 2014); our result implies that we can approximately count the number of orthogonal pairs with essentially the same running time. We also provide a fine-grained reduction from approximate #SAT to SAT. Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for some 1 < c < 2 and all k there is an O ( c n )-time algorithm for k -SAT. Then we prove that for all k , there is an O (( c + o (1)) n )-time algorithm for approximate # k -SAT. In particular, our result implies that the Exponential Time Hypothesis (ETH) is equivalent to the seemingly weaker statement that there is no algorithm to approximate #3-SAT to within a factor of 1+ɛ in time 2 o ( n )/ ɛ 2 (taking ɛ > 0 as part of the input).

Near-optimal ground state preparation

Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.

Approximately counting bases of bicircular matroids

We give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.