相关论文: Matroid complexity and non-succinct descriptions
Matroid is a generalization of many fundamental objects in combinatorial mathematics , and matroid intersection problem is a classical subject in combinatorial optimization . However , only the intersection of two matroids are well…
The control and sensing of large-scale systems results in combinatorial problems not only for sensor and actuator placement but also for scheduling or observability/controllability. Such combinatorial constraints in system design and…
We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in $\Sigma_2^p$. In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is…
Efficient deterministic algorithms to construct representations of lattice path matroids over finite fields are presented. They are built on known constructions of hierarchical secret sharing schemes, a recent characterization of…
This work demonstrates a methodology for using deep learning to discover simple, practical criteria for classifying matrices based on abstract algebraic properties. By combining a high-performance neural network with explainable AI (XAI)…
We prove that the extension complexity of the independence polytope of every regular matroid on $n$ elements is $O(n^6)$. Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a $O(n^2)$…
In mathematics and computer science, connectivity is one of the basic concepts of matroid theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related…
From the configuration of a matroid (which records the size and rank of the cyclic flats and the containments among them, but not the sets), one can compute several much-studied matroid invariants, including the Tutte polynomial and a…
Motivated by certain applications from physics, biochemistry, economics, and computer science, in which the objects under investigation are not accessible because of various limitations, we propose a trial-and-error model to examine…
We prove that for every proper minor-closed class $M$ of matroids representable over a prime field, there exists a constant-competitive matroid secretary algorithm for the matroids in $M$. This result relies on the extremely powerful…
We study the parallel (adaptive) complexity of the classic problem of finding a basis in an $n$-element matroid, given access via an \emph{independence oracle}. In this model, the algorithm may submit polynomially many independence queries…
We give a simple polynomial time approximation scheme for the weighted matroid matching problem on strongly base orderable matroids. We also show that even the unweighted version of this problem is NP-complete and not in oracle-coNP.
We present an output-sensitive algorithm for generating the whole set of flats of a finite matroid. Given a procedure, P, that decides in S_P time steps if a set is independent, the time complexity of the algorithm is O(N^2 M S_P), where N…
The main result is Theorem MAT 11 which states that every finite closure operator is the ground set of a matroid. Its base sets consist of nonredundant covers of of the closure. These are minimal subsets that determine the closure operator…
We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. As a consequence, we show that it is decidable to determine whether a constraint satisfaction…
We construct a family of independent sets for finite, atomic, and graded lattices, extending the well-known cryptomorphism between geometric lattices and matroids. This construction leads to an embedding theorem into geometric lattices that…
In the Inverse Matroid problem, we are given a matroid, a fixed basis $B$, and an initial weight function, and the goal is to minimally modify the weights -- measured by some function -- so that $B$ becomes a maximum-weight basis. The…
We introduce an algorithm for computing closure systems derived from a family of implications on a set. Semilattices presentations are explored and used in conjunction with the algorithm to compute various types of lattices freely generated…
Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. Geometric lattice has widely used in diverse fields, especially search algorithm design which plays important role…
We present a formal analysis, in Isabelle/HOL, of optimisation algorithms for matroids, which are useful generalisations of combinatorial structures that occur in optimisation, and greedoids, which are a generalisation of matroids. Although…