Related papers: Tropical Kirchhoff's Formula and Postoptimality in…
We consider the problem of finding a basis of a matroid with weight exactly equal to a given target. Here weights can be discrete values from $\{-\Delta,\ldots,\Delta\}$ or more generally $m$-dimensional vectors of such discrete values. We…
We consider the minimum spanning tree problem in a setting where the edge weights are stochastic from unknown distributions, and the only available information is a single sample of each edge's weight distribution. In this setting, we…
The effect of replacing a basis element on the way the basis spans other elements is studied. This leads to a new characterization of binary matroids.
Tropicalisation (with trivial coefficients) is a process that turns a polynomial equation into a combinatorial predicate on subsets of the set of variables. We show that for each minuscule representation of a simple reductive group, there…
We consider the minimum spanning tree problem with predictions, using the weight-arrival model, i.e., the graph is given, together with predictions for the weights of all edges. Then the actual weights arrive one at a time and an…
The basis exchange axiom has been a driving force in the development of matroid theory. However, the axiom gives only a local characterization of the relation of bases, which is a major stumbling block to further progress, and providing a…
A frame template over a field $\mathbb F$ describes the precise way in which a given $\mathbb F$-representable matroid is close to being a frame matroid. Our main result determines the maximum-rank projective or affine geometry that is…
Let $M$ be a matroid and let $Q$, $R$, $S$ and $T$ be subsets of the ground set such that the smallest separation that separates $Q$ from $R$ has order $k$ and the smallest separation that separates $S$ from $T$ has order $l$. We prove that…
A new electronic structure model is developed in which the ground state energy of a molecular system is given by a Hartree-Fock-like expression with parametrized one- and two-electron integrals over an extended (minimal + polarization) set…
In a stochastic probing problem we are given a universe $E$, where each element $e \in E$ is active independently with probability $p_e$, and only a probe of e can tell us whether it is active or not. On this universe we execute a process…
Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this classic problem in the fully dynamic setting, where elements…
We study set selection problems where the weights are uncertain. Instead of its exact weight, only an uncertainty interval containing its true weight is available for each element. In some cases, some solutions are universally optimal;…
In [27], we provided results on the $p$-modulus of the family of all bases of matroids and showed that it recovers various concepts in matroid theory, including strength, fractional arboricity, and principal partitions. In particular, the…
The free energy of the Random Energy Model at the transition point between ferromagnetic and spin glass phases is calculated. At this point, equivalent to the decoding error threshold in optimal codes, free energy has finite size…
The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e.\ when the goal is to decide if two common independent sets suffice or not.…
We provide an explicit algorithm to solve the idempotent analogue of the discrete Monge-Kantorovich optimal mass transportation problem with the usual real number field replaced by the tropical (max-plus) semiring, in which addition is…
We propose an exact iterative algorithm for minimization of a class of continuous cell-wise linear convex functions on a hyperplane arrangement. Our particular setup is motivated by evaluation of so-called rank estimators used in robust…
An unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition. For some particular…
In this paper, we consider the tractability of the matroid intersection problem under the minimum rank oracle. In this model, we are given an oracle that takes as its input a set of elements and returns as its output the minimum of the…
We define an independence system associated with simple graphs. We prove that the independence system is a matroid for certain families of graphs, including trees, with bases as minimal resolving sets. Consequently, the greedy algorithm on…