55 research outputs found
Average value of solutions of the bipartite quadratic assignment problem and linkages to domination analysis
In this paper we study the complexity and domination analysis in the context
of the \emph{bipartite quadratic assignment problem}. Two variants of the
problem, denoted by BQAP1 and BQAP2, are investigated. A formula for
calculating the average objective function value of all solutions
is presented whereas computing the median objective function value is shown to
be NP-hard. We show that any heuristic algorithm that produces a solution with
objective function value at most has the domination ratio at
least . Analogous results for the standard \emph{quadratic
assignment problem} is an open question. We show that computing a solution
whose objective function value is no worse than that of
solutions of BQAP1 or
solutions of BQAP2, is NP-hard for any fixed natural numbers and such
that . However, a solution with the domination number
for BQAP1 and
for BQAP2, can be found in
time
A characterization of linearizable instances of the quadratic minimum spanning tree problem
We investigate special cases of the quadratic minimum spanning tree problem
(QMSTP) on a graph that can be solved as a linear minimum spanning
tree problem. Characterization of such problems on graphs with special
properties are given. This include complete graphs, complete bipartite graphs,
cactuses among others. Our characterization can be verified in time.
In the case of complete graphs and when the cost matrix is given in factored
form, we show that our characterization can be verified in time.
Related open problems are also indicated
The generalized vertex cover problem and some variations
In this paper we study the generalized vertex cover problem (GVC), which is a
generalization of various well studied combinatorial optimization problems. GVC
is shown to be equivalent to the unconstrained binary quadratic programming
problem and also equivalent to some other variations of the general GVC. Some
solvable cases are identified and approximation algorithms are suggested for
special cases. We also study GVC on bipartite graphs and identify some
polynomially solvable cases. We show that GVC on bipartite graphs is equivalent
to the bipartite unconstrained 0-1 quadratic programming problem. Integer
programming formulations of GVC and related problems are presented and
establish half-integrality property on some variables for the corresponding
linear programming relaxations. We also discuss special cases of GVC where all
feasible solutions are independent sets or vertex covers. These problems are
observed to be equivalent to the maximum weight independent set problem or
minimum weight vertex cover problem along with some algorithmic results.Comment: 24 page
Bottleneck flows in networks
The bottleneck network flow problem (BNFP) is a generalization of several
well-studied bottleneck problems such as the bottleneck transportation problem
(BTP), bottleneck assignment problem (BAP), bottleneck path problem (BPP), and
so on. In this paper we provide a review of important results on this topic and
its various special cases. We observe that the BNFP can be solved as a sequence
of maximum flow problems. However, special augmenting path based
algorithms for the maximum flow problem can be modified to obtain algorithms
for the BNFP with the property that these variations and the corresponding
maximum flow algorithms have identical worst case time complexity. On unit
capacity network we show that BNFP can be solved in . This improves the best available
algorithm by a factor of . On unit capacity simple graphs, we
show that BNFP can be solved in time. As a consequence
we have an algorithm for the BTP with unit arc
capacities
Representations of quadratic combinatorial optimization problems: A case study using the quadratic set covering problem
The objective function of a quadratic combinatorial optimization problem
(QCOP) can be represented by two data points, a quadratic cost matrix Q and a
linear cost vector c. Different, but equivalent, representations of the pair
(Q, c) for the same QCOP are well known in literature. Research papers often
state that without loss of generality we assume Q is symmetric, or
upper-triangular or positive semidefinite, etc. These representations however
have inherently different properties. Popular general purpose 0-1 QCOP solvers
such as GUROBI and CPLEX do not suggest a preferred representation of Q and c.
Our experimental analysis discloses that GUROBI prefers the upper triangular
representation of the matrix Q while CPLEX prefers the symmetric representation
in a statistically significant manner. Equivalent representations, although
preserve optimality, they could alter the corresponding lower bound values
obtained by various lower bounding schemes. For the natural lower bound of a
QCOP, symmetric representation produced tighter bounds, in general. Effect of
equivalent representations when CPLEX and GUROBI run in a heuristic mode are
also explored. Further, we review various equivalent representations of a QCOP
from the literature that have theoretical basis to be viewed as strong and
provide new theoretical insights for generating such equivalent representations
making use of constant value property and diagonalization (linearization) of
QCOP instances.Comment: 36 page
Heuristic algorithms for the bipartite unconstrained 0-1 quadratic programming problem
We study the Bipartite Unconstrained 0-1 Quadratic Programming Problem (BQP)
which is a relaxation of the Unconstrained 0-1 Quadratic Programming Problem
(QP). Applications of the BQP include mining discrete patterns from binary
data, approximating matrices by rank-one binary matrices, computing cut-norm of
a matrix, and solving optimization problems such as maximum weight biclique,
bipartite maximum weight cut, maximum weight induced subgraph of a bipartite
graph, etc. We propose several classes of heuristic approaches to solve the BQP
and discuss a number of construction algorithms, local search algorithms and
their combinations. Results of extensive computational experiments are reported
to establish the practical performance of our algorithms. For this purpose, we
propose several sets of test instances based on various applications of the
BQP. Our algorithms are compared with state-of-the-art heuristics for QP which
can also be used to solve BQP with reformulation. We also study theoretical
properties of the neighborhoods and algorithms. In particular, we establish
complexity of all neighborhood search algorithms and establish tight worst-case
performance ratio for the greedy algorithm.Comment: 17 page
Combinatorial Optimization Problems with Interaction Costs: Complexity and Solvable Cases
We introduce and study the combinatorial optimization problem with
interaction costs (COPIC). COPIC is the problem of finding two combinatorial
structures, one from each of two given families, such that the sum of their
independent linear costs and the interaction costs between elements of the two
selected structures is minimized. COPIC generalizes the quadratic assignment
problem and many other well studied combinatorial optimization problems, and
hence covers many real world applications. We show how various topics from
different areas in the literature can be formulated as special cases of COPIC.
The main contributions of this paper are results on the computational
complexity and approximability of COPIC for different families of combinatorial
structures (e.g. spanning trees, paths, matroids), and special structures of
the interaction costs. More specifically, we analyze the complexity if the
interaction cost matrix is parameterized by its rank and if it is a diagonal
matrix. Also, we determine the structure of the intersection cost matrix, such
that COPIC is equivalent to independently solving linear optimization problems
for the two given families of combinatorial structures
The Quadratic Minimum Spanning Tree Problem and its Variations
The quadratic minimum spanning tree problem and its variations such as the
quadratic bottleneck spanning tree problem, the minimum spanning tree problem
with conflict pair constraints, and the bottleneck spanning tree problem with
conflict pair constraints are useful in modeling various real life
applications. All these problems are known to be NP-hard. In this paper, we
investigate these problems to obtain additional insights into the structure of
the problems and to identify possible demarcation between easy and hard special
cases. New polynomially solvable cases have been identified, as well as NP-hard
instances on very simple graphs. As a byproduct, we have a recursive formula
for counting the number of spanning trees on a -accordion and a
characterization of matroids in the context of a quadratic objective function
A characterization of Linearizable instances of the Quadratic Traveling Salesman Problem
We consider the linearization problem associated with the quadratic traveling
salesman problem (QTSP). Necessary and sufficient conditions are given for a
cost matrix of QTSP to be linearizable. It is shown that these conditions
can be verified in time. Some simpler sufficient conditions for
linearization are also given along with related open problems
Bilinear Assignment Problem: Large Neighborhoods and Experimental Analysis of Algorithms
The bilinear assignment problem (BAP) is a generalization of the well-known
quadratic assignment problem (QAP). In this paper, we study the problem from
the computational analysis point of view. Several classes of neigborhood
structures are introduced for the problem along with some theoretical analysis.
These neighborhoods are then explored within a local search and a variable
neighborhood search frameworks with multistart to generate robust heuristic
algorithms. Results of systematic experimental analysis have been presented
which divulge the effectiveness of our algorithms. In addition, we present
several very fast construction heuristics. Our experimental results disclosed
some interesting properties of the BAP model, different from those of
comparable models. This is the first thorough experimental analysis of
algorithms on BAP. We have also introduced benchmark test instances that can be
used for future experiments on exact and heuristic algorithms for the problem.Comment: Corrected typos. Figures are now vector graphics (instead of raster
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