On one maximum multiflow problem and related metrics

AbstractWe consider the undirected maximum multiflow (multicommodity flow) problem in the case when the commodity graph is the disjoint union of K3 and K2. We prove that if the supply graph satisfies a certain Eulerian-type condition, then the problem has an integer optimal solution. To obtain this result, we first study the corresponding dual problem on metrics and show that an optimal solution to the latter is achieved on some (2,3)-metric or some 3-cut metric

When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks

Let $G=(V,E)$ be a supply graph and $H=(V,F)$ a demand graph defined on the same set of vertices. An assignment of capacities to the edges of $G$ and demands to the edges of $H$ is said to satisfy the \emph{cut condition} if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair $(G,H)$ is called \emph{cut-sufficient} if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on $H$ within the network with capacities defined on $G$. We prove a previous conjecture, which states that when the supply graph $G$ is series-parallel, the pair $(G,H)$ is cut-sufficient if and only if $(G,H)$ does not contain an \emph{odd spindle} as a minor; that is, if it is impossible to contract edges of $G$ and delete edges of $G$ and $H$ so that $G$ becomes the complete bipartite graph $K_{2,p}$, with $p\geq 3$ odd, and $H$ is composed of a cycle connecting the $p$ vertices of degree 2, and an edge connecting the two vertices of degree $p$. We further prove that if the instance is \emph{Eulerian} --- that is, the demands and capacities are integers and the total of demands and capacities incident to each vertex is even --- then the multiflow problem has an integral solution. We provide a polynomial-time algorithm to find an integral solution in this case. In order to prove these results, we formulate properties of tight cuts (cuts for which the cut condition inequality is tight) in cut-sufficient pairs. We believe these properties might be useful in extending our results to planar graphs.Comment: An extended abstract of this paper will be published at the 44th Symposium on Theory of Computing (STOC 2012

On duality and fractionality of multicommodity flows in directed networks

In this paper we address a topological approach to multiflow (multicommodity flow) problems in directed networks. Given a terminal weight $\mu$, we define a metrized polyhedral complex, called the directed tight span $T_{\mu}$, and prove that the dual of $\mu$-weighted maximum multiflow problem reduces to a facility location problem on $T_{\mu}$. Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by $\mu$. By utilizing this duality, we establish the classifications of terminal weights admitting combinatorial min-max relation (i) for every network and (ii) for every Eulerian network. Our result includes Lomonosov-Frank theorem for directed free multiflows and Ibaraki-Karzanov-Nagamochi's directed multiflow locking theorem as special cases.Comment: 27 pages. Fixed minor mistakes and typos. To appear in Discrete Optimizatio

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

Stream Learning in Energy IoT Systems: A Case Study in Combined Cycle Power Plants

The prediction of electrical power produced in combined cycle power plants is a key challenge in the electrical power and energy systems field. This power production can vary depending on environmental variables, such as temperature, pressure, and humidity. Thus, the business problem is how to predict the power production as a function of these environmental conditions, in order to maximize the profit. The research community has solved this problem by applying Machine Learning techniques, and has managed to reduce the computational and time costs in comparison with the traditional thermodynamical analysis. Until now, this challenge has been tackled from a batch learning perspective, in which data is assumed to be at rest, and where models do not continuously integrate new information into already constructed models. We present an approach closer to the Big Data and Internet of Things paradigms, in which data are continuously arriving and where models learn incrementally, achieving significant enhancements in terms of data processing (time, memory and computational costs), and obtaining competitive performances. This work compares and examines the hourly electrical power prediction of several streaming regressors, and discusses about the best technique in terms of time processing and predictive performance to be applied on this streaming scenario.This work has been partially supported by the EU project iDev40. This project has received funding from the ECSEL Joint Undertaking (JU) under grant agreement No 783163. The JU receives support from the European Union’s Horizon 2020 research and innovation programme and Austria, Germany, Belgium, Italy, Spain, Romania. It has also been supported by the Basque Government (Spain) through the project VIRTUAL (KK-2018/00096), and by Ministerio de Economía y Competitividad of Spain (Grant Ref. TIN2017-85887-C2-2-P)