Publications

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As transistors start to approach fundamental limits and Moore's law slows down, new devices and architectures are needed to enable continued performance gains. New approaches based on RRAM (resistive random access memory) or memristor crossbars can enable the processing of large amounts of data[1, 2]. One of the most promising applications for RRAM crossbars is brain inspired or neuromorphic computing[3, 4].

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We lay the foundation for a benchmarking methodology for assessing current and future quantum computers. We pose and begin addressing fundamental questions about how to fairly compare computational devices at vastly different stages of technological maturity. We critically evaluate and offer our own contributions to current quantum benchmarking efforts, in particular those involving adiabatic quantum computation and the Adiabatic Quantum Optimizers produced by D-Wave Systems, Inc. We find that the performance of D-Wave's Adiabatic Quantum Optimizers scales roughly on par with classical approaches for some hard combinatorial optimization problems; however, architectural limitations of D-Wave devices present a significant hurdle in evaluating real-world applications. In addition to identifying and isolating such limitations, we develop algorithmic tools for circumventing these limitations on future D-Wave devices, assuming they continue to grow and mature at an exponential rate for the next several years.

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Graphs can be used to model risk management in various systems. Particularly, Caskurlu et al. in [7] have considered a system, which has threats, vulnerabilities and assets, and which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. One can either restricting the permissions of the users, or encapsulating the system assets. The pointed out two strategies correspond to deleting minimum number of elements corresponding to vulnerabilities and assets, such that the flow between threats and assets is reduced below the predefined threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. It is well-known that the Vertex Cover problem is in P on bipartite graphs, however; the computational complexity of the Partial Vertex Cover problem on bipartite graphs has remained open. In this paper, we establish that the Partial Vertex Cover problem is NP-hard on bipartite graphs, which was also recently independently demonstrated [N. Apollonio and B. Simeone, Discrete Appl. Math., 165 (2014), pp. 37–48; G. Joret and A. Vetta, preprint, arXiv:1211.4853v1 [cs.DS], 2012]. We then identify interesting special cases of bipartite graphs, for which the Partial Vertex Cover problem, the closely related Budgeted Maximum Coverage problem, and their weighted extensions can be solved in polynomial time. We also present an 8/9-approximation algorithm for the Budgeted Maximum Coverage problem in the class of bipartite graphs. We show that this matches and resolves the integrality gap of the natural LP relaxation of the problem and improves upon a recent 4/5-approximation.

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In k-hypergraph matching, we are given a collection of sets of size at most k, each with an associated weight, and we seek a maximumweight subcollection whose sets are pairwise disjoint. More generally, in k-hypergraph b-matching, instead of disjointness we require that every element appears in at most b sets of the subcollection. Our main result is a linear-programming based (k - 1 + 1/k)-approximation algorithm for k-hypergraph b-matching. This settles the integrality gap when k is one more than a prime power, since it matches a previously-known lower bound. When the hypergraph is bipartite, we are able to improve the approximation ratio to k - 1, which is also best possible relative to the natural LP. These results are obtained using a more careful application of the iterated packing method. Using the bipartite algorithmic integrality gap upper bound, we show that for the family of combinatorial auctions in which anyone can win at most t items, there is a truthful-in-expectation polynomial-time auction that t-approximately maximizes social welfare. We also show that our results directly imply new approximations for a generalization of the recently introduced bounded-color matching problem. We also consider the generalization of b-matching to demand matching, where edges have nonuniform demand values. The best known approximation algorithm for this problem has ratio 2k on k-hypergraphs. We give a new algorithm, based on local ratio, that obtains the same approximation ratio in a much simpler way.

We study several natural instances of the geometric hitting set problem for input consisting of sets of line segments (and rays, lines) having a small number of distinct slopes. These problems model path monitoring (e.g., on road networks) using the fewest sensors (the “hitting points”). We give approximation algorithms for cases including (i) lines of 3 slopes in the plane, (ii) vertical lines and horizontal segments, (iii) pairs of horizontal/vertical segments. We give hardness and hardness of approximation results for these problems. We prove that the hitting set problem for vertical lines and horizontal rays is polynomially solvable.

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While collection capabilities have yielded an ever-increasing volume of aerial imagery, analytic techniques for identifying patterns in and extracting relevant information from this data have seriously lagged. The vast majority of imagery is never examined, due to a combination of the limited bandwidth of human analysts and limitations of existing analysis tools. In this report, we describe an alternative, novel approach to both encoding and analyzing aerial imagery, using the concept of a geospatial semantic graph. The advantages of our approach are twofold. First, intuitive templates can be easily specified in terms of the domain language in which an analyst converses. These templates can be used to automatically and efficiently search large graph databases, for specific patterns of interest. Second, unsupervised machine learning techniques can be applied to automatically identify patterns in the graph databases, exposing recurring motifs in imagery. We illustrate our approach using real-world data for Anne Arundel County, Maryland, and compare the performance of our approach to that of an expert human analyst.

This report summarizes the first year’s effort on the Enceladus project, under which Sandia was asked to evaluate the potential advantages of adiabatic quantum computing for analyzing large data sets in the near future, 5-to-10 years from now. We were not specifically evaluating the machine being sold by D-Wave Systems, Inc; we were asked to anticipate what future adiabatic quantum computers might be able to achieve. While realizing that the greatest potential anticipated from quantum computation is still far into the future, a special purpose quantum computing capability, Adiabatic Quantum Optimization (AQO), is under active development and is maturing relatively rapidly; indeed, D-Wave Systems Inc. already offers an AQO device based on superconducting flux qubits. The AQO architecture solves a particular class of problem, namely unconstrained quadratic Boolean optimization. Problems in this class include many interesting and important instances. Because of this, further investigation is warranted into the range of applicability of this class of problem for addressing challenges of analyzing big data sets and the effectiveness of AQO devices to perform specific analyses on big data. Further, it is of interest to also consider the potential effectiveness of anticipated special purpose adiabatic quantum computers (AQCs), in general, for accelerating the analysis of big data sets. The objective of the present investigation is an evaluation of the potential of AQC to benefit analysis of big data problems in the next five to ten years, with our main focus being on AQO because of its relative maturity. We are not specifically assessing the efficacy of the D-Wave computing systems, though we do hope to perform some experimental calculations on that device in the sequel to this project, at least to provide some data to compare with our theoretical estimates.

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