On the matching arrangement of a graph, improper weight function problem and its application

TL;DR

This paper explores the improper weight function problem in graph matching arrangements, demonstrating its NP-completeness and proposing a cryptographic application based on this problem.

Contribution

It introduces the improper weight function problem, proves its NP-completeness, and constructs a knapsack-like cryptosystem utilizing this problem.

Findings

Improper weight function problem is NP-complete.

Finite field method can compute characteristic polynomials of matching arrangements.

A cryptosystem based on the improper weight function problem is proposed.

Abstract

This article presents examples of an application of the finite field method for the computation of the characteristic polynomial of the matching arrangement of a graph. Weight functions on edges of a graph with weights from a finite field are divided into proper and improper functions in connection with proper colorings of vertices of the matching polytope of a graph. An improper weight function problem is introduced, a proof of its NP-completeness is presented, and a knapsack-like public key cryptosystem is constructed based on the improper weight function problem.