2,294 research outputs found
Solution of minimum spanning forest problems with reliability constraints
We propose the reliability constrained k-rooted minimum spanning forest, a relevant optimization problem whose aim is to find a k-rooted minimum cost forest that connects given customers to a number of supply vertices, in such a way that a minimum required reliability on each path between a customer and a supply vertex is satisfied and the cost is a minimum. The reliability of an edge is the probability that no failure occurs on that edge, whereas the reliability of a path is the product of the reliabilities of the edges in such path. The problem has relevant applications in the design of networks, in fields such as telecommunications, electricity and transports. For its solution, we propose a mixed integer linear programming model, and an adaptive large neighborhood search metaheuristic which invokes several shaking and local search operators. Extensive computational tests prove that the metaheuristic can provide good quality solutions in very short computing times
The Fast Heuristic Algorithms and Post-Processing Techniques to Design Large and Low-Cost Communication Networks
It is challenging to design large and low-cost communication networks. In
this paper, we formulate this challenge as the prize-collecting Steiner Tree
Problem (PCSTP). The objective is to minimize the costs of transmission routes
and the disconnected monetary or informational profits. Initially, we note that
the PCSTP is MAX SNP-hard. Then, we propose some post-processing techniques to
improve suboptimal solutions to PCSTP. Based on these techniques, we propose
two fast heuristic algorithms: the first one is a quasilinear time heuristic
algorithm that is faster and consumes less memory than other algorithms; and
the second one is an improvement of a stateof-the-art polynomial time heuristic
algorithm that can find high-quality solutions at a speed that is only inferior
to the first one. We demonstrate the competitiveness of our heuristic
algorithms by comparing them with the state-of-the-art ones on the largest
existing benchmark instances (169 800 vertices and 338 551 edges). Moreover, we
generate new instances that are even larger (1 000 000 vertices and 10 000 000
edges) to further demonstrate their advantages in large networks. The
state-ofthe-art algorithms are too slow to find high-quality solutions for
instances of this size, whereas our new heuristic algorithms can do this in
around 6 to 45s on a personal computer. Ultimately, we apply our
post-processing techniques to update the bestknown solution for a notoriously
difficult benchmark instance to show that they can improve near-optimal
solutions to PCSTP. In conclusion, we demonstrate the usefulness of our
heuristic algorithms and post-processing techniques for designing large and
low-cost communication networks
Cost allocation in connection and conflict problems on networks: a cooperative game theoretic approach
This thesis examines settings where multiple decision makers with conflicting interests
benefit from cooperation in joint combinatorial optimisation problems. It draws on cooperative game theory, polyhedral theory and graph theory to address cost sharing in
joint single-source shortest path problems and joint weighted minimum colouring problems.
The primary focus of the thesis are problems where each agent corresponds to a
vertex of an undirected complete graph, in which a special vertex represents the common supplier. The joint combinatorial optimisation problem consists of determining the
shortest paths from the supplier to all other vertices in the graph. The optimal solution
is a shortest path tree of the graph and the aim is to allocate the cost of this shortest
path tree amongst the agents. The thesis defines shortest path tree problems, proposes
allocation rules and analyses the properties of these allocation rules. It furthermore introduces shortest path tree games and studies the properties of these games. Various core
allocations for shortest path tree games are introduced and polyhedral properties of the
core are studied. Moreover, computational results on finding the core and the nucleolus
of shortest path tree games for the application of cost allocation in Wireless Multihop
Networks are presented.
The secondary focus of the thesis are problems where each agent is interested in
having access to a number of facilities but can be in conflict with other agents. If two
agents are in conflict, then they should have access to disjoint sets of facilities. The
aim is to allocate the cost of the minimum number of facilities required by the agents
amongst them. The thesis models these cost allocation problems as a class of cooperative
games called weighted minimum colouring games, and characterises total balancedness
and submodularity of this class of games using the properties of the underlying graph
Opinion Optimization in Directed Social Networks
Shifting social opinions has far-reaching implications in various aspects,
such as public health campaigns, product marketing, and political candidates.
In this paper, we study a problem of opinion optimization based on the popular
Friedkin-Johnsen (FJ) model for opinion dynamics in an unweighted directed
social network with nodes and edges. In the FJ model, the internal
opinion of every node lies in the closed interval , with 0 and 1 being
polar opposites of opinions about a certain issue. Concretely, we focus on the
problem of selecting a small number of nodes and changing their
internal opinions to 0, in order to minimize the average opinion at
equilibrium. We then design an algorithm that returns the optimal solution to
the problem in time. To speed up the computation, we further develop a
fast algorithm by sampling spanning forests, the time complexity of which is , with being the number of samplings. Finally, we execute extensive
experiments on various real directed networks, which show that the
effectiveness of our two algorithms is similar to each other, both of which
outperform several baseline strategies of node selection. Moreover, our fast
algorithm is more efficient than the first one, which is scalable to massive
graphs with more than twenty million nodes
Casting Light on the Hidden Bilevel Combinatorial Structure of the Capacitated Vertex Separator Problem
Given an undirected graph, we study the capacitated vertex separator problem
that asks to find a subset of vertices of minimum cardinality, the removal of which induces a
graph having a bounded number of pairwise disconnected shores (subsets of vertices) of
limited cardinality. The problem is of great importance in the analysis and protection of communication or social networks against possible viral attacks and for matrix decomposition algorithms. In this article, we provide a new bilevel interpretation of the problem and model it
as a two-player Stackelberg game in which the leader interdicts the vertices (i.e., decides on
the subset of vertices to remove), and the follower solves a combinatorial optimization problem on the resulting graph. This approach allows us to develop a computational framework
based on an integer programming formulation in the natural space of the variables. Thanks
to this bilevel interpretation, we derive three different families of strengthening inequalities
and show that they can be separated in polynomial time. We also show how to extend these
results to a min-max version of the problem. Our extensive computational study conducted
on available benchmark instances from the literature reveals that our new exact method is
competitive against the state-of-the-art algorithms for the capacitated vertex separator problem and is able to improve the best-known results for several difficult classes of instances.
The ideas exploited in our framework can also be extended to other vertex/edge deletion/
insertion problems or graph partitioning problems by modeling them as two-player Stackel-
berg games and solving them through bilevel optimization
{RAMA}: {A} Rapid Multicut Algorithm on {GPU}
We propose a highly parallel primal-dual algorithm for the multicut (a.k.a. correlation clustering) problem, a classical graph clustering problem widely used in machine learning and computer vision. Our algorithm consists of three steps executed recursively: (1) Finding conflicted cycles that correspond to violated inequalities of the underlying multicut relaxation, (2) Performing message passing between the edges and cycles to optimize the Lagrange relaxation coming from the found violated cycles producing reduced costs and (3) Contracting edges with high reduced costs through matrix-matrix multiplications. Our algorithm produces primal solutions and dual lower bounds that estimate the distance to optimum. We implement our algorithm on GPUs and show resulting one to two order-of-magnitudes improvements in execution speed without sacrificing solution quality compared to traditional serial algorithms that run on CPUs. We can solve very large scale benchmark problems with up to variables in a few seconds with small primal-dual gaps. We make our code available at https://github.com/pawelswoboda/RAMA
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