DAG-Pathwidth: Graph Algorithmic Analyses of DAG-Type Blockchain Networks

Abstract

This paper analyzes a blockchain network forming a directed acyclic graph (DAG), called a DAG-type blockchain, from the viewpoint of graph algorithm theory. To use a DAG-type blockchain, NP-hard graph optimization problems on the DAG are required to be solved. Although various problems for undirected and directed graphs can be efficiently solved by using the notions of graph parameters, these currently known parameters are meaningless for DAGs, which implies that it is hopeless to design efficient algorithms based on the parameters for such problems. In this work, we propose a novel graph parameter for directed graphs called a DAG-pathwidth, which represents the closeness to a directed path. This is an extension of the pathwidth, a well-known graph parameter for undirected graphs. We analyze the features of the DAG-pathwidth and prove that computing the DAG-pathwidth of a DAG (directed graph in general) is NP-complete. Finally, we propose an efficient algorithm for a variant of the maximum k-independent set problem for the DAG-type blockchain when the DAG-pathwidth of the input graph is small.

Blockchain Technology and Sanitary Aspects of the Water Supply

“…Every new ledger entrance, a batch of transactions, is a block "chained" to the previous one by the previous block hash. Blockchain is a sequential linear ledger, while Directed Acyclic Graph (DAG) is a directed network ledger where no cycles or repetitions are possible, and transactions are represented as graph nodes (Kasahara et al, 2023). Distributed ledger is decentralized across a number of computing "nodes".…”

Graph Optimization Problems and Algorithms for DAG-Type Blockchains

“…We explain DAG-path-decomposition (DAG-PD) and DAG-pathwidth [ 7]. The DAG-PD is defined for a general directed graph .…”

“…The problem of finding a DAG-PD with minimum DAG-pathwidth is proved to be NP-hard [ 7]. The proof is based on a reduction from the minimum cut linear arrangement problem, which is NP-hard.…”

“…In this section, we describe an efficient algorithm for solving the maximum discord .k-independent set problem based on DAG-PD [ 7]. Assume that we are given a DAG .…”

“…Several similar metrics exist for directed graphs [ 3,6,10], but they cannot be applied to DAGs. This chapter introduces the concept of the DAG-pathwidth introduced in the paper [ 7], in which we designed an algorithm to efficiently solve some independent set problems on DAGs by performing DAG-path decomposition. Note that finding a DAG-path decomposition with a minimum DAG-pathwidth is NP-hard [ 7].…”