Matching walks that are minimal with respect to edge inclusion

TL;DR

This paper proves that enumerating minimal walks conforming to a regular expression in a labeled directed graph cannot be done with polynomial delay unless P=NP, highlighting computational complexity limitations.

Contribution

It establishes the NP-hardness of polynomial-delay enumeration of minimal walks matching a regular expression in a graph, despite some walks being computable efficiently.

Findings

Enumeration cannot be polynomial-delay unless P=NP.

Some minimal walks can be computed in polynomial time.

The problem's complexity is linked to well-known NP-hardness results.

Abstract

In this paper we show that enumerating the set MM(G,R), defined below, cannot be done with polynomial-delay in its input G and R, unless P=NP. R is a regular expression over an alphabet $\Sigma$, G is directed graph labeled over $\Sigma$, and MM(G,R) contains walks of G. First, consider the set Match(G,R) containing all walks G labeled by a word (over $\Sigma$) that conforms to $R$. In general, M(G,R) is infinite, and MM(G,R) is the finite subset of Match(G,R) of the walks that are minimal according to a well-quasi-order <. It holds w<w' if the multiset of edges appearing in w is strictly included in the multiset of edges appearing in w'. Remarkably, the set MM(G,R) contains some walks that may be computed in polynomial time. Hence, it is not the case that the preprocessing phase of any algorithm enumerating MM(G,R) must solve an NP-hard problem.