Algorithm for B-partitions, parameterized complexity of the matrix determinant and permanent

10/10/2018
by   Ranveer Singh, et al.
IIT Rajasthan
Placement cell at Indian Statistical Institute
0

Every square matrix A=(a_uv)∈C^n× n can be represented as a digraph having n vertices. In the digraph, a block (or 2-connected component) is a maximally connected subdigraph that has no cut-vertex. The determinant and the permanent of a matrix can be calculated in terms of the determinant and the permanent of some specific induced subdigraphs of the blocks in the digraph. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Such partitions of the digraph are called the B-partitions. In this paper, first, we develop an algorithm to find the B-partitions. Next, we analyze the parameterized complexity of matrix determinant and permanent, where, the parameters are the sizes of blocks and the number of cut-vertices of the digraph. We give a class of combinations of cut-vertices and block sizes for which the parametrized complexities beat the state of art complexities of the determinant and the permanent.

READ FULL TEXT

page 1

page 2

page 3

page 4

03/16/2020

On the parameterized complexity of 2-partitions

We give an FPT algorithm for deciding whether the vertex set a digraph D...
12/30/2019

Computing 2-twinless blocks

Let G=(V,E)) be a directed graph. A 2-twinless block in G is a maximal v...
07/21/2018

How to sample connected K-partitions of a graph

A connected undirected graph G=(V,E) is given. This paper presents an al...
05/22/2023

Testing Isomorphism of Graphs in Polynomial Time

Given a graph G, the graph [G] obtained by adding, for each pair of vert...
07/19/2022

Efficient Constructions for the Győri-Lovász Theorem on Almost Chordal Graphs

In the 1970s, Győri and Lovász showed that for a k-connected n-vertex gr...
01/31/2019

On (2n/3-1)-resilient (n,2)-functions

A {00,01,10,11}-valued function on the vertices of the n-cube is called ...
08/03/2022

Network homophily via multi-dimensional extensions of Cantelli's inequality

Homophily is the principle whereby "similarity breeds connections". We g...

Please sign up or login with your details

Forgot password? Click here to reset