Matching walks that are minimal with respect to edge inclusion

TL;DR

It is shown that enumerating the set MM(G,R), defined below, cannot be done with polynomial-delay in its input G and R, unless P=NP, and it is not the case that the preprocessing phase of any algorithm enumerating MM(G,R) must solve an NP-hard problem.

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 $Σ$, G is directed graph labeled over $Σ$, and MM(G,R) contains walks of G. First, consider the set Match(G,R) containing all walks G labeled by a word (over $Σ$) 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.

Figures (5)

• Figure 1: Start gadget
• Figure 2: Gadget for variable $x_i$, $i\in\{1,\ldots,k\}$
• Figure 3: Gadget for the clause $C_j$, $j\in\{1,\ldots,\ell\}$, decomposed as $C_j=\beta \vee \gamma \vee \delta$ with $\beta,\gamma,\delta\in\{x_1,\ldots,x_n\}\cup\{\bar{x}_1,\ldots,\bar{x}_n\}$. For instance, if $\beta=x_2$ (resp. ${\bar{x}}_5$), then $\beta^j$ refers to the vertex $x_2^j$ (resp. ${\bar{x}}_5^j$) in gadget $x_2$ (resp. in gadget $x_5$).
• Figure 4: End gadget
• Figure 5: Glu gadget

Theorems & Definitions (27)

• Definition 1
• Definition 2: Minimal-Multiset Semantics
• Lemma 3
• Lemma 4
• Theorem 5
• Remark 6
• Remark 7
• Remark 8
• Remark 9
• Lemma 10