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

A Combinatorial Formula for Principal Minors of a Matrix with Tree-metric Exponents and Its Applications

Let $T$ be a tree with a vertex set $\{ 1,2,\dots, N \}$. Denote by $d_{ij}$ the distance between vertices $i$ and $j$. In this paper, we present an explicit combinatorial formula of principal minors of the matrix $(t^{d_{ij}})$, and its applications to tropical geometry, study of multivariate stable polynomials, and representation of valuated matroids. We also give an analogous formula for a skew-symmetric matrix associated with $T$.Comment: 16 page