On the decidability of finding a positive ILP-instance in a regular set of ILP-instances

Abstract

The regular intersection emptiness problem for a decision problem P ($${{\textit{int}}_{{\mathrm {Reg}}}}$$ int Reg (P)) is to decide whether a potentially infinite regular set of encoded P-instances contains a positive one. Since $${{\textit{int}}_{{\mathrm {Reg}}}}$$ int Reg (P) is decidable for some NP-complete problems and undecidable for others, its investigation provides insights in the nature of NP-complete problems. Moreover, the decidability of the $${{\textit{int}}_{{\mathrm {Reg}}}}$$ int Reg -problem is usually achieved by exploiting the regularity of the set of instances; thus, it also establishes a connection to formal language and automata theory. We consider the $${{\textit{int}}_{{\mathrm {Reg}}}}$$ int Reg -problem for the well-known NP-complete problem Integer Linear Programming (ILP). It is shown that any DFA that describes a set of ILP-instances (in a natural encoding) can be reduced to a finite core of instances that contains a positive one if and only if the original set of instances did. This result yields the decidability of $${{\textit{int}}_{{\mathrm {Reg}}}}$$ int Reg (ILP).

Properties of graphs specified by a regular language

“…A typical question could be whether there exists some planar graph in ρ(L). Solving this type of decision problems was the motivation to study regular realizability problems in [1,24,28] and, independently, calling them int Reg -problems 2 in [12,[29][30][31].…”

Automata Equipped with Auxiliary Data Structures and Regular Realizability Problems

“…RR problems have independently been studied under the name regular intersection emptiness problems [21,22]. A restricted version of RR problem (for context-free filters only) is a well-known CFL-reachability problem, which is related to problems in interprocedural program analysis [3,4,6,10,11,23].…”