Abstract
The regular intersection emptiness problem for a decision problem P (\( int_{\mathrm {Reg}} \)(P)) is to decide whether a potentially infinite regular set of encoded P-instances contains a positive one. Since \( int_{\mathrm {Reg}} \)(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 \( int_{\mathrm {Reg}} \)-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 \( int_{\mathrm {Reg}} \)-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 \( int_{\mathrm {Reg}} \)(ILP).
Highlights
The problem Integer Linear Programming (ILP for short) asks whether a given set of inequalities with integer coefficients has an integer solution
ILP is among the first problems for which NP-hardness was shown and it is of great practical relevance in mathematical optimisation
Finding a generic characterization of NP-complete problems with a decidable intReg-problem is still an open problem. This work continues this line of research and we will focus on the NP-complete integer linear programming problem as the filter language, i. e., we investigate the problem intReg(ILP)
Summary
The problem Integer Linear Programming (ILP for short) asks whether a given set of inequalities with integer coefficients has an integer solution. If we consider regular sets of instances, this task can be formalised as checking whether a given regular language of P -instances (represented by a deterministic finite automaton) and the fixed language of positive P -instances have a non-empty intersection This was the original viewpoint of the line of research introduced by Guler et al [9,23], where this problem is called the intReg-problem of P (or intReg(P ) for short).. In contrast to these research questions, the line of work initiated in [9,23] focuses on classical (hard) computational problems as filter languages and respective decision procedures heavily take advantage of the regularity of the set of input instances. Finding a generic characterization of NP-complete problems with a decidable intReg-problem is still an open problem This work continues this line of research and we will focus on the NP-complete integer linear programming problem as the filter language, i.
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