Author

# Xiaorong Sun

Bio: Xiaorong Sun is an academic researcher. The author has contributed to research in topics: Boolean network & Disjunctive normal form. The author has an hindex of 1, co-authored 1 publications receiving 7 citations.

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01 Jan 1992

TL;DR: This thesis presents a new efficient algorithm to recognize threshold Boolean functions, and shows that iterated roof duality applied to a class of quadratic pseudo-Boolean functions which are naturally associated to graphs, provides the exact values of the stability number for the special case of odd $K\sb4$-free graphs.

Abstract: This thesis reports on a series of studies concerning on the one hand, classes of structured Boolean functions, and on the other hand, quadratic pseudo-Boolean optimization. The approach is combinatorial and algorithmic. The central topics revolve around the recognition of structured Boolean functions, their generalizations, and roof duality for quadratic pseudo-Boolean functions.
In the first part of the thesis, we give linear-time combinatorial algorithms for recognizing various generalizations of Horn related formulae (A Boolean function f in disjunctive normal form (DNF), is called Horn if each term involves at most one negative variable).
In the second part, we present a new efficient algorithm to recognize threshold Boolean functions, i.e., functions for which there exists a hyperplane separating their set of true points from their set of false points. We show also a hierarchy of generalizations of regular Boolean functions, which are themselves natural generalizations of threshold functions. For any of these functions, if the set of minimal true points is given, then the set of maximal false points can be found in polynomial time. A new way of representing positive Boolean functions using disjunctive condensed forms (DCFs) is also studied. Several polynomial algorithms whose inputs are DNFs, are generalized to the case when the inputs are DCFs (which are shorter than DNFs).
In the third part, we study the (NP-hard) problem of minimizing quadratic pseudo-Boolean functions, i.e., quadratic real valued polynomials whose variables take only the values 0 and 1. We shall describe a new network flow based algorithm for finding lower bounds of the minimum. This approach gives the same lower bounds as others (such as the roof duality of Hammer, Hansen and Simeone), but provides a faster algorithm to compute the lower bound. The max-flow approach can also quickly identify the optimal values of a subset of variables. Computational results are also reported.
To provide better lower bounds than roof duality, we present a new approach called iterated roof duality. It is shown that iterated roof duality applied to a class of quadratic pseudo-Boolean functions which are naturally associated to graphs, provides the exact values of the stability number for the special case of odd $K\sb4$-free graphs.

7 citations

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01 Jan 2006

TL;DR: It is shown that 100% data reduction is achieved using the proposed preprocessing techniques for MAX-CUT graphs derived from VLSI design, MAX-Clique in c-fat graphsderived from fault diagnosis, and minimum vertex cover problems in random planar graphs of up to 500 000 vertices.

Abstract: We propose several efficient preprocessing techniques for Unconstrained Quadratic Binary Optimization (QUBO), including the direct use of enhanced versions of known basic techniques (e.g., implications derived from first and second order derivatives and from roof–duality), and the integrated use (e.g., by probing and by consensus) of the basic techniques. The application of these techniques is implemented using a natural network flow model of QUBO. The use of the proposed preprocessing techniques provides: (i) a lower bound of the minimum of the objective function, (ii) the values of some of the variables in some or every optimum, (iii) binary relations (equations, inequalities, or non-equalities) between the values of certain pairs of variables in some or every optimum, and (iv) the decomposition (if possible) of the original problem into several smaller pairwise independent QUBO problems. Extensive computational experience showing the efficiency of the proposed techniques is presented. Substantial problem simplifications, improving considerably the existing techniques, are demonstrated on benchmark problems as well as on randomly generated ones. In particular, it is shown that 100% data reduction is achieved using the proposed preprocessing techniques for MAX-CUT graphs derived from VLSI design, MAX-Clique in c-fat graphs derived from fault diagnosis, and minimum vertex cover problems in random planar graphs of up to 500 000 vertices. Acknowledgements: The third author was partially financially supported by the Portuguese FCT and by the FSE in the context of the III Quadro Comunitario de Apoio.

94 citations

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01 Jan 1998

TL;DR: This paper focuses on the development of branch-and-cut algorithms for discrete optimization problems and in polyhedral outer-approximation methods for continuous nonconvex programming problems.

Abstract: Discrete and continuous nonconvex programming problems arise in a host of practical applications in the context of production, location-allocation, distribution, economics and game theory, process design, and engineering design situations. Several recent advances have been made in the development of branch-and-cut algorithms for discrete optimization problems and in polyhedral outer-approximation methods for continuous nonconvex programming problems. At the heart of these approaches is a sequence of linear programming problems that drive the solution process. The success of such algorithms is strongly tied in with the strength or tightness of the linear programming representations employed.

52 citations

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TL;DR: A graph-theoretic approach is provided to provide bounds, which includes as a special case the roof dual bound, and it is shown that these bounds can be computed in O(n^3) time by using network flow techniques.

50 citations

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01 Jan 2008TL;DR: This dissertation investigates the Quadratic Unconstrained Binary Optimization (QUBO) problem and shows that there is a unique maximal set of persistencies for the linearization model for QUBO, and improved the Iterated Roof–Dual bound (IRD) by proposing two combinatorial methods.

Abstract: This dissertation investigates the Quadratic Unconstrained Binary Optimization (QUBO) problem, i.e. the problem of minimizing a quadratic function in binary variables. QUBO is studied at two complementary levels. First, there is an algorithmic aspect that tells how to preprocess the problem, how to find heuristics, how to get improved bounds and how to solve the problem with all the above ingredients. Second, there is a practical aspect that uses QUBO to solve various applications from the engineering and social sciences fields including: via minimization, 2D/3D Ising models, 1D Ising chain models, image binarization, hierarchical clustering, greedy graph coloring/partitioning, MAX–2–SAT, MIN–VC, MAX–CLIQUE, MAX–CUT, graph stability and minimum k–partition.
Several families of fast heuristics for QUBO are analyzed, which include a novel probabilistic based class of methods.
It is shown that there is a unique maximal set of persistencies for the linearization model for QUBO. This set is determined in polynomial time by a maximum flow followed by the computation of the strong components of a network that has 2n+2 nodes, where n is the number of variables. The identification of the above persistencies leads to a unique decomposition of the function, such that each component can be optimized separately. To find further persistencies, two additional techniques are proposed: one is based on the second order derivatives of Hammer et al. [121]; the other technique is a probing procedure on the two possible values of the variables. These preprocessing tools work remarkably well for certain classes of problems.
We improved the Iterated Roof–Dual bound (IRD) of [51] by proposing two combinatorial methods: one was named the squeezed IRD; and the second was called the project–and–lift IRD method.
The cubic–dual bound can be found by means of linear programming by adding a set of triangle inequalities to the standard linearization, whose number is cubic in the number of variables. We show that this set can be reduced depending on the coefficients of the terms of the function. This leads to the possibility of computing the cubic–duals of larger QUBOs.

20 citations

01 Jan 2011

TL;DR: In this paper, a family of polyhedral outer-approximation approximations to the convex hull of feasible solutions for mixed-integer linear and nonlinear programs are obtained.

Abstract: This research effort focuses on the acquisition of polyhedral outer-approximations to the convex hull of feasible solutions for mixed-integer linear and mixed-integer nonlinear programs. The goal is to produce desirable formulations that have superior size and/or relaxation strength. These two qualities often have great influence on the success of underlying solution strategies, and so it is with these qualities in mind that the work of this dissertation presents three distinct contributions. The first studies a family of relatively unknown polytopes that enable the linearization of polynomial expressions involving two discrete variables. Projections of higher-dimensional convex hulls are employed to reduce the dimensionality of the requisite linearizing polyhedra. For certain lower dimensions, a complete characterization of the convex hull is obtained; for others, a family of facets is acquired. Furthermore, a novel linearization for the product of a bounded continuous variable and a general discrete variable is obtained. The second contribution investigates the use of simplicial facets in the formation of novel convex hull representations for a class of mixed-discrete problems having a subset of their variables taking on discrete, affinely independent realizations. These simplicial facets provide new theoretical machinery necessary to extend the reformulation-linearization technique (RLT) for mixed-binary and mixed-discrete programs. In doing so, new insight is provided which allows for the subsumation of previous mixed-binary and mixed-discrete RLT results. The third contribution presents a novel approach for representing functions of discrete variables and their products using logarithmic numbers of 0-1 variables in order to economize on the number of these binary variables. Here, base-2 expansions are used within linear restrictions to enforce the appropriate behavior of functions of discrete variables. Products amongst functions are handled by scaling these linear restrictions. This approach provides insight into, improves upon, and subsumes recent related linearization methods from the literature.

7 citations