From Spin Glasses to Hard Satisfiable Formulas

Jia, Haixia; Moore, Cristopher; Selman, Bart

doi:10.1007/11527695_16

Cited by 21 publications

(22 citation statements)

References 16 publications

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“…The special structure of XORSAT gives rise to a global algorithm when the instance is satisfiable, allowing for a solution in polynomial time using Gaussian elimination [24]. However, when unsatisfiable, occurring for ρ > 4 ⋅ 0 918 ≈ 3 7, this same structure makes these problems very difficult for local search solvers [22,25]. In addition, the choice of instances out of XORSAT clauses makes them particularly difficult also for algorithms based on message passing [26].…”

Section: Resultsmentioning

confidence: 99%

Evidence of Exponential Speed‐Up in the Solution of Hard Optimization Problems

et al. 2018

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Optimization problems pervade essentially every scientific discipline and industry. A common form requires identifying a solution satisfying the maximum number among a set of many conflicting constraints. Often, these problems are particularly difficult to solve, requiring resources that grow exponentially with the size of the problem. Over the past decades, research has focused on developing heuristic approaches that attempt to find an approximation to the solution. However, despite numerous research efforts, in many cases even approximations to the optimal solution are hard to find, as the computational time for further refining a candidate solution also grows exponentially with input size. In this paper, we show a noncombinatorial approach to hard optimization problems that achieves an exponential speed-up and finds better approximations than the current state of the art. First, we map the optimization problem into a Boolean circuit made of specially designed, self-organizing logic gates, which can be built with (nonquantum) electronic elements with memory. The equilibrium points of the circuit represent the approximation to the problem at hand. Then, we solve its associated nonlinear ordinary differential equations numerically, towards the equilibrium points. We demonstrate this exponential gain by comparing a sequential MATLAB implementation of our solver with the winners of the 2016 Max-SAT competition on a variety of hard optimization instances. We show empirical evidence that our solver scales linearly with the size of the problem, both in time and memory, and argue that this property derives from the collective behavior of the simulated physical circuit. Our approach can be applied to other types of optimization problems, and the results presented here have far-reaching consequences in many fields.

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Section: Resultsmentioning

confidence: 99%

Evidence of Exponential Speed‐Up in the Solution of Hard Optimization Problems

et al. 2018

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“…We choose XORSAT because it is a prototypical problem in both physics and theoretical computer science. Even though XORSAT can be solved in polynomial time with Gaussian elimination, it nevertheless has evaded efficient solution with any local algorithm, including variants of the Davis-Putnam algorithm [22], message-passing methods [23], stochastic search [24], simulated annealing [20], and quantum adiabatic annealing [10].…”

Section: A the Xorsat Problemmentioning

confidence: 99%

Obstacles to quantum annealing in a planar embedding of XORSAT

et al. 2019

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We introduce a planar embedding of the k-regular k-XORSAT problem, in which solutions are encoded in the ground state of a classical statistical mechanics model of reversible logic gates arranged on a square grid and acting on bits that represent the Boolean variables of the problem. The special feature of this embedding is that the resulting model lacks a finite-temperature phase transition, thus bypassing the first-order thermodynamic transition known to occur in the random graph representation of XORSAT. In spite of this attractive feature, the thermal relaxation into the ground state displays remarkably slow glassy behavior. The question addressed in this paper is whether this planar embedding can afford an efficient path to solution of k-regular k-XORSAT via quantum adiabatic annealing. We first show that our model bypasses an avoided level crossing and consequent exponentially small gap in the limit of small transverse fields. We then present quantum Monte Carlo results for our embedding of the k-regular k-XORSAT that strongly support a picture in which second-order and firstorder transitions develop at a finite transverse field for k = 2 and k = 3, respectively. This translates into power-law and exponential dependences in the scaling of energy gaps with system size, corresponding to timesto-solution which are, respectively, polynomial and exponential in the number of variables. We conclude that neither classical nor quantum annealing can efficiently solve our reformulation of XORSAT, even though the original problem can be solved in polynomial time by Gaussian elimination. arXiv:1904.01860v2 [cond-mat.stat-mech]

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“…For the fault diagnosis problems studied here, the locally-structured approach also performs better, and we pursue improvements to the locally-structured algorithm of [3]. Figure 2: Comparison of the D-Wave quantum annealing hardware performance in solving XOR-3-SAT problems [21], using global (blue) and locally-structured (red) modelling strategies. Each XOR-3-SAT instance is randomly generated subject to having a unique solution and a clause-to-variable ratio of 1.0.…”

Section: Approaches To Embeddingmentioning

confidence: 99%

Mapping Constrained Optimization Problems to Quantum Annealing with Application to Fault Diagnosis

et al. 2016

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Current quantum annealing (QA) hardware suffers from practical limitations such as finite temperature, sparse connectivity, small qubit numbers, and control error. We propose new algorithms for mapping boolean constraint satisfaction problems (CSPs) onto QA hardware mitigating these limitations. In particular we develop a new embedding algorithm for mapping a CSP onto a hardware Ising model with a fixed sparse set of interactions, and propose two new decomposition algorithms for solving problems too large to map directly into hardware.The mapping technique is locally-structured, as hardware compatible Ising models are generated for each problem constraint, and variables appearing in different constraints are chained together using ferromagnetic couplings. In contrast, global embedding techniques generate a hardware independent Ising model for all the constraints, and then use a minor-embedding algorithm to generate a hardware compatible Ising model. We give an example of a class of CSPs for which the scaling performance of D-Wave's QA hardware using the local mapping technique is significantly better than global embedding.We validate the approach by applying D-Wave's hardware to circuit-based fault-diagnosis. For circuits that embed directly, we find that the hardware is typically able to find all solutions from a min-fault diagnosis set of size N using 1000N samples, using an annealing rate that is 25 times faster than a leading SAT-based sampling method. Further, we apply decomposition algorithms to find min-cardinality faults for circuits that are up to 5 times larger than can be solved directly on current hardware. arXiv:1603.03111v1 [quant-ph]

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From Spin Glasses to Hard Satisfiable Formulas

Cited by 21 publications

References 16 publications

Evidence of Exponential Speed‐Up in the Solution of Hard Optimization Problems

Evidence of Exponential Speed‐Up in the Solution of Hard Optimization Problems

Obstacles to quantum annealing in a planar embedding of XORSAT

Mapping Constrained Optimization Problems to Quantum Annealing with Application to Fault Diagnosis

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