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

Approximating the generalized terminal backup problem via half-integral multiflow relaxation

We consider a network design problem called the generalized terminal backup problem. Whereas earlier work investigated the edge-connectivity constraints only, we consider both edge- and node-connectivity constraints for this problem. A major contribution of this paper is the development of a strongly polynomial-time 4/3-approximation algorithm for the problem. Specifically, we show that a linear programming relaxation of the problem is half-integral, and that the half-integral optimal solution can be rounded to a 4/3-approximate solution. We also prove that the linear programming relaxation of the problem with the edge-connectivity constraints is equivalent to minimizing the cost of half-integral multiflows that satisfy flow demands given from terminals. This observation presents a strongly polynomial-time algorithm for computing a minimum cost half-integral multiflow under flow demand constraints

Scikit-Multiflow: A Multi-output Streaming Framework

Scikit-multiflow is a multi-output/multi-label and stream data mining framework for the Python programming language. Conceived to serve as a platform to encourage democratization of stream learning research, it provides multiple state of the art methods for stream learning, stream generators and evaluators. scikit-multiflow builds upon popular open source frameworks including scikit-learn, MOA and MEKA. Development follows the FOSS principles and quality is enforced by complying with PEP8 guidelines and using continuous integration and automatic testing. The source code is publicly available at https://github.com/scikit-multiflow/scikit-multiflow.Comment: 5 pages, Open Source Softwar