18,329 research outputs found
A Parameterized Complexity Analysis of Bi-level Optimisation with Evolutionary Algorithms
Bi-level optimisation problems have gained increasing interest in the field
of combinatorial optimisation in recent years. With this paper, we start the
runtime analysis of evolutionary algorithms for bi-level optimisation problems.
We examine two NP-hard problems, the generalised minimum spanning tree problem
(GMST), and the generalised travelling salesman problem (GTSP) in the context
of parameterised complexity.
For the generalised minimum spanning tree problem, we analyse the two
approaches presented by Hu and Raidl (2012) with respect to the number of
clusters that distinguish each other by the chosen representation of possible
solutions. Our results show that a (1+1) EA working with the spanning nodes
representation is not a fixed-parameter evolutionary algorithm for the problem,
whereas the global structure representation enables to solve the problem in
fixed-parameter time. We present hard instances for each approach and show that
the two approaches are highly complementary by proving that they solve each
other's hard instances very efficiently.
For the generalised travelling salesman problem, we analyse the problem with
respect to the number of clusters in the problem instance. Our results show
that a (1+1) EA working with the global structure representation is a
fixed-parameter evolutionary algorithm for the problem
The linearization problem of a binary quadratic problem and its applications
We provide several applications of the linearization problem of a binary
quadratic problem. We propose a new lower bounding strategy, called the
linearization-based scheme, that is based on a simple certificate for a
quadratic function to be non-negative on the feasible set. Each
linearization-based bound requires a set of linearizable matrices as an input.
We prove that the Generalized Gilmore-Lawler bounding scheme for binary
quadratic problems provides linearization-based bounds. Moreover, we show that
the bound obtained from the first level reformulation linearization technique
is also a type of linearization-based bound, which enables us to provide a
comparison among mentioned bounds. However, the strongest linearization-based
bound is the one that uses the full characterization of the set of linearizable
matrices. Finally, we present a polynomial-time algorithm for the linearization
problem of the quadratic shortest path problem on directed acyclic graphs. Our
algorithm gives a complete characterization of the set of linearizable matrices
for the quadratic shortest path problem
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