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The Unconstrained Binary Quadratic Programming Problem: A Survey
Gary Kochenberger
School of Business Administration, University of Colorado at Denver, Denver, CO 80217,
gary.kochenberger@cudenver.edu, Tel:
303-921-6372
Jin-Kao Hao
LERIA, Université d'Angers, 2 Boulevard Lavoisier, 49045
Angers, France, hao@info.iniv-angers.fr
Fred Glover
OptTek Inc
Boulder, Colorado 80302, glover@opttek.com
Mark Lewis
Craig School of Business,
Missouri Western State University, St Joseph, MO, 64507, mlewis14@missouriwestern.edu
Zhipeng Lü
School of Computer Science and Technology,
Huazhong University of Science and Technology, 430074 Wuhan, China
Zhipeng.lui@gmail.com
Haibo Wang
Sanchez School of Business,
Texas A&M International University, Laredo, TX 78041, hwang@tamiu.edu
Yang Wang
LERIA, Université d'Angers, 2 Boulevard Lavoisier, 49045
Angers, France,yangw@info.iniv-angers.fr
Abstract:
In recent years the unconstrained binary quadratic program (UBQP) has grown in
importance in the field of combinatorial optimization due to its application potential and its
computational challenge. Research on UBQP has generated a wide range of solution techniques
for this basic model that encompasses a rich collection of problem types. In this paper we survey
the literature on this important model, providing an overview of the applications and solution
methods.
Keywords:
Unconstrained binary quadratic programs, combinatorial optimization, metaheuristics
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1.0 Introduction:
The Unconstrained Binary Quadratic Programming (UBQP) problem is defined by
min
. .
t
x Qx
s t x SÎ
where S represents the binary discrete set
{ }
0,1
n
or
{ }
1,1
n
-
and Q is an n-by-n square, symmetric
matrix of coefficients. This simple model is notable for embracing a remarkable range of
applications in combinatorial optimization. For example, the use of this model for representing
and solving optimization problems on graphs, facility locations problems, resources allocation
problems, clustering problems, set partitioning problems, various forms of assignment problems,
sequencing and ordering problems, and many others have been reported in the literature.
Even more remarkable is the fact that, once given a UBQP formulation, these problems
can be solved by a UBQP method which is not specialized to exploit the problem domain of any
individual class of problems, to yield solutions whose quality in many cases rivals or even
surpasses the quality of the solutions produced by the best specialized methods, while achieving
this outcome with an efficiency that likewise rivals or surpasses the efficiency of leading
specialized methods.
In this paper we survey the literature on UBQP, both its applications and solution
methods. While many important constrained nonlinear binary models have been reported in the
literature over the years, we focus our attention here on models that naturally occur in the form
of an unconstrained quadratic binary model and those that have been re-cast into the form of
UBQP. The paper is organized as follows. In section 2 we survey the range of applications that
have been reported in the literature. Section 3 then presents a survey of the solution
methodologies reported in the literature for solving UBQP. Section 4 highlights key theoretical
work and this is followed by section 5 which wraps up the paper with our summary and
conclusions.
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2.0 Applications:
Some reported applications appear naturally in the form of UBQP while others are “re-
cast” into the UBQP form by employing various transformations. In the sub-sections below we
examine these different categories of applications in turn. Within sub-sections, we present
applications in the chronological order in which they appeared in the literature to give the reader
a sense of when certain topics were addressed appeared in print, as well as progress made and
trends in solution methodologies.
2.1 Natural UBQP problems/applications
The literature on UBQP goes back to the 1960s where the topics of pseudo-boolean
functions and binary quadratic optimization were introduced by Hammer and Rudeanu (1968).
Early papers related to UBQP concern applications in finance (Laughunn (1970)), project
selection (Rhys(1970)), cluster analysis (Rao (1971)), economic analysis (Hammer and Shliffer
(1971)), traffic management (Witzgall(1975)) and computer aided design (Krarup and Pruzan
(1978)). While these applications actually take the form of constrained quadratic binary
programs, they are mentioned here due to their historical role in fostering an interest in quadratic
binary applications and also because several allow special cases that are precisely in the form of
UBQP.
More recently many interesting applications that are expressed naturally in the form of
UBQP have appeared in various papers. Barahona, Grotschell, Junger and Reinelt (1988)
formulate and solve the problem of finding ground states of spin glasses with exterior magnetic
fields, an important problem in physics, as an instance of UBQP. Computational results reveal
that the model produces high quality solutions to spin glass problems of realistic size in
reasonable amounts of computation time using 1980s technology.
Hansen and Jaumard (1990), in their work on the satisfiability problem, report their
experience using the UBQP model as an approach for representing and solving small to medium
sized Max 2-sat problems. Computational studies validated the attractiveness of this approach to
the Max 2-sat problem in terms of quickly producing high quality solutions.
Boros and Hammer (1991) discuss the use of UBQP as an approach for modeling the
Max-Cut problem. Their paper highlights the relationship between UBQP, Max-cut, Max 2-sat,
and the Weighted Signed Graph Problem. The authors also present a discussion of valid
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inequalities and facets for polyhedra that provide the basis for further computational and
theoretical work.
Alidaee, Kochenberger and Ahmadian (1994) discuss two machine scheduling problems
in the context of UBQP: (1) scheduling n jobs on a single machine to minimize total weighted
earliness and tardiness, and (2) scheduling n jobs on two parallel identical processors to
minimize weighted mean flow time. In each case, the authors show how the problems can be
modeled in a straight-forward manner as an instance of UBQP.
Pardalos and Xue (1994) indicate how the maximum clique problem can be modeled as
an instance of UBQP. The authors also discuss the relationship between the maximum clique
problem, the maximum independent set problem, and the vertex cover problem, indicating how
each can be represented by UBQP. Finally, the authors provide a survey of solution methods for
the maximum clique problem.
De Simone, Diehl, Junger, Mutzel, Reinelt, and Rinaldi (1995), as in the earlier 1988
paper by Barahona, Grotschell, Junger and Reinelt, adopt the UBQP model as a representation
for the problem of finding ground states for the spin glass problem. In this 1995 paper the
authors use the UBQP model to compute exact ground states for Ising spin glasses on 2-
dimensional grids with periodic boundary interactions, Gaussian bond distributions, and an
exterior magnetic field. Preliminary experiments with a branch and cut algorithm for optimizing
the UBQP form of the problem proved very promising, quickly producing high quality solutions
to large spin glass instances.
Bomze, Budinich, Pardalos, and Pelillo (1999) discuss the maximum clique (MC)
problem and how it can how it can be modeled in a variety of ways including a representation in
terms of UBQP. The authors provide a very broad and in depth discussion of a variety of
applications and of both exact and heuristic solution methods for the MC problem.
Computational experience with various solution approaches to the MC problem is also presented.
Iasemidis, Pardalos, Sackellares, and Shiau (2001) discuss the use of the UBQP model as
part of a process employed to predict the arrival of epileptic seizures. The entrainment between
two brain sites can be quantified from measures of electrical activity (EEG) of the brain. The
UBQP model was successful in identifying the most entrained sites leading to the optimal
location of electrode sites. In clinical trials this procedure was successful in predicting epileptic
seizures 20-40 minutes in advance of their occurrence.
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Alidaee, Glover, Kochenberger, and Rego (2005) discuss the number partitioning
problem where the objective is to assign numbers to subsets such that the sums of the numbers in
each subset are as close as possible to one another. The authors show that the n = 2 subset case
can be modeled as an instance of UBQP and that problems with n > 2 can be modeled as a
constrained version of UBQP. Extensions of the basic model along with computational
experience for the n=2 case are presented indicating the attractiveness of the approach.
Kochenberger, Glover, Alidaee, and Lewis (2005) discuss their experience with adopting
the UBQP model to represent and solve max 2-sat problems. Expanding the computational
scope reported earlier by Hansen and Jaumard (1990) on UBQP and the Max 2-sat problem, they
offer extensive computational experience on very large test problems with up to 1000 variables
and more than 10,000 clauses. Employing a basic form of tabu search to solve the UBQP
instances, best known solutions to most test problems were found in a few seconds of
computation time.
Neven, Rose and Macready (2008) discuss the use of quantum adiabatic algorithms,
which represent new approaches to NP-hard combinatorial problems, for solving the image
recognition problem. The authors indicate how the pattern recognition problem of deciding
whether two images contain the same object can be modeled as an instance of UBQP, which they
show is the general input format required by D-Wave superconducting quantum AQC
processors. Computational experience was not reported.
Pajouh, Balasundaram, and Prokepyev (2013) discuss the use of the UBQP model for
representing the maximal independent set problem. The authors present an analysis of local
maxima properties along with relations between continuous local maxima of the quadratic
formulation and the binary local maxima in the Hamming distance 1 and 2 neighborhoods.
These results are then used to construct effective local search algorithms for the maximum
independent set problem.
Kochenberger, Hao, Lu, Glover and Wang (2013) discuss the Max Cut problem and how
the UBQP model can be effectively used to model and solve large scale instances. Using a tabu
search algorithm, extensive computational testing is reported on problems with up to 10,000
variables. Comparisons with other solution methods from the literature for the max cut problem
are provided, indicating the attractiveness of the UBQP/Tabu Search approach.
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