Recognition of generalized network matrices

TL;DR

This thesis presents algorithms for recognizing and characterizing binet matrices, Camion bases, and certain network matrices, with polynomial time complexity improvements and new theoretical insights.

Contribution

It introduces a polynomial-time algorithm for recognizing binet matrices and characterizes Camion bases and specific network matrices.

Findings

Binet matrices can be tested in O(n^6 m) time.

A new characterization and recognition algorithm for Camion bases is provided.

Polynomial algorithms are developed for recognizing certain network matrices.

Abstract

In this PhD thesis, we deal with binet matrices, an extension of network matrices. The main result of this thesis is the following. A rational matrix A of size n times m can be tested for being binet in time O(n^6 m). If A is binet, our algorithm outputs a nonsingular matrix B and a matrix N such that [B N] is the node-edge incidence matrix of a bidirected graph (of full row rank) and A=B^{-1} N. Furthermore, we provide some results about Camion bases. For a matrix M of size n times m', we present a new characterization of Camion bases of M, whenever M is the node-edge incidence matrix of a connected digraph (with one row removed). Then, a general characterization of Camion bases as well as a recognition procedure which runs in O(n^2m') are given. An algorithm which finds a Camion basis is also presented. For totally unimodular matrices, it is proven to run in time O((nm)^2) where m=m'-n. The last result concerns specific network matrices. We give a characterization of nonnegative {r,s}-noncorelated network matrices, where r and s are two given row indexes. It also results a polynomial recognition algorithm for these matrices.