The arborescence-realization problem

AbstractA {0, 1}-matrix M is arborescence graphic if there exists an arborescence T such that the arcs of T are indexed on the rows of M and the columns of M are the incidence vectors of the arc sets of dipaths of T. If such a T exists, then T is an arborescence realization for M. This paper presents an almost-linear-time algorithm to determine whether a given {0, 1}-matrix is arborescence graphic and, if so, to construct an arborescence realization. The algorithm is then applied to recognize a subclass of the extended-Horn satisfiability problems introduced by Chandru and Hooker (1991)

A version of Tutte's polynomial for hypergraphs

Tutte's dichromate T(x,y) is a well known graph invariant. Using the original definition in terms of internal and external activities as our point of departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. In the definition, we associate activities to hypertrees, which are generalizations of the indicator function of the edge set of a spanning tree. We prove that hypertrees form a lattice polytope which is the set of bases in a polymatroid. In fact, we extend our invariants to integer polymatroids as well. We also examine hypergraphs that can be represented by planar bipartite graphs, write their hypertree polytopes in the form of a determinant, and prove a duality property that leads to an extension of Tutte's Tree Trinity Theorem.Comment: 49 page

On Adaptivity Gaps of Influence Maximization Under the Independent Cascade Model with Full-Adoption Feedback

In this paper, we study the adaptivity gap of the influence maximization problem under the independent cascade model when full-adoption feedback is available. Our main results are to derive upper bounds on several families of well-studied influence graphs, including in-arborescences, out-arborescences and bipartite graphs. Especially, we prove that the adaptivity gap for the in-arborescences is between [e/(e-1), 2e/(e-1)], and for the out-arborescences the gap is between [e/(e-1), 2]. These are the first constant upper bounds in the full-adoption feedback model. Our analysis provides several novel ideas to tackle the correlated feedback appearing in adaptive stochastic optimization, which may be of independent interest

A simple algorithm and min-max formula for the inverse arborescence problem

In 1998, Hu and Liu developed a strongly polynomial algorithm for solving the inverse arborescence problem that aims at minimally modifying a given cost-function on the edge-set of a digraph D so that an input spanning arborescence of D becomes a cheapest one. In this note, we develop a conceptually simpler algorithm along with a new min-max formula for the minimum modification of the cost-function. The approach is based on a link to a min-max theorem and a simple (two-phase greedy) algorithm by the first author from 1979 concerning the primal optimization problem of finding a cheapest subgraph of a digraph that covers an intersecting family along with the corresponding dual optimization problem, as well. (C) 2021 The Author(s). Published by Elsevier B.V

Hypergraph polynomials and the Bernardi process

Recently O. Bernardi gave a formula for the Tutte polynomial $T(x,y)$ of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial $I$ is a generalization of $T(x,1)$ to hypergraphs. We supply a Bernardi-type description of $I$ using a ribbon structure on the underlying bipartite graph $G$. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of $G$ in the same way as $I$ is. To prove this we interpret the Bernardi process as a way of dissecting the root polytope into simplices, along with a shelling order. We also show that our generalized Bernardi process gives a common extension of bijections (and their inverses) constructed by Baker and Wang between spanning trees and break divisors.Comment: 46 page