4 research outputs found
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 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
Geometric and LP-based heuristics for the quadratic travelling salesman problem
A generalization of the classical TSP is the so-called quadratic travelling
salesman problem (QTSP), in which a cost coefficient is associated with the
transition in every vertex, i.e. with every pair of edges traversed in
succession. In this paper we consider two geometrically motivated special cases
of the QTSP known from the literature, namely the angular-metric TSP, where
transition costs correspond to turning angles in every vertex, and the
angular-distance-metric TSP, where a linear combination of turning angles and
Euclidean distances is considered.
At first we introduce a wide range of heuristic approaches, motivated by the
typical geometric structure of optimal solutions. In particular, we exploit
lens-shaped neighborhoods of edges and a decomposition of the graph into layers
of convex hulls, which are then merged into a tour by a greedy-type procedure
or by utilizing an ILP model. Secondly, we consider an ILP model for a standard
linearization of QTSP and compute fractional solutions of a relaxation. By
rounding we obtain a collection of subtours, paths and isolated points, which
are combined into a tour by various strategies, all of them involving auxiliary
ILP models. Finally, different improvement heuristics are proposed, most
notably a matheuristic which locally reoptimizes the solution for rectangular
sectors of the given point set by an ILP approach.
Extensive computational experiments for benchmark instances from the
literature and extensions thereof illustrate the Pareto-efficient frontier of
algorithms in a (running time, objective value)-space. It turns out that our
new methods clearly dominate the previously published heuristics
Some Network Optimization Models under Diverse Uncertain Environments
Network models provide an efficient way to represent many real life problems
mathematically. In the last few decades, the field of network optimization has
witnessed an upsurge of interest among researchers and practitioners. The
network models considered in this thesis are broadly classified into four types
including transportation problem, shortest path problem, minimum spanning tree
problem and maximum flow problem. Quite often, we come across situations, when
the decision parameters of network optimization problems are not precise and
characterized by various forms of uncertainties arising from the factors, like
insufficient or incomplete data, lack of evidence, inappropriate judgements and
randomness. Considering the deterministic environment, there exist several
studies on network optimization problems. However, in the literature, not many
investigations on single and multi objective network optimization problems are
observed under diverse uncertain frameworks. This thesis proposes seven
different network models under different uncertain paradigms. Here, the
uncertain programming techniques used to formulate the uncertain network models
are (i) expected value model, (ii) chance constrained model and (iii) dependent
chance constrained model. Subsequently, the corresponding crisp equivalents of
the uncertain network models are solved using different solution methodologies.
The solution methodologies used in this thesis can be broadly categorized as
classical methods and evolutionary algorithms. The classical methods, used in
this thesis, are Dijkstra and Kruskal algorithms, modified rough Dijkstra
algorithm, global criterion method, epsilon constraint method and fuzzy
programming method. Whereas, among the evolutionary algorithms, we have
proposed the varying population genetic algorithm with indeterminate crossover
and considered two multi objective evolutionary algorithms.Comment: Thesis documen