On the Containment Problem for Linear Sets

TL;DR

This paper investigates the computational complexity of the containment problem for linear sets, showing it is $ ext{log}$-hard in $ ext{Pi}_2^p$ for 1-dimensional cases with binary encoding, but solvable in polynomial time when restricted to unary encoding.

Contribution

It establishes the complexity classification of the containment problem for 1-dimensional linear sets, highlighting the impact of encoding and dimension restrictions.

Findings

Containment problem for 1-dimensional linear sets is $ ext{log}$-hard in $ ext{Pi}_2^p$ with binary encoding.

The problem becomes polynomial-time solvable with unary encoding and dimension 1.

Complexity varies significantly with encoding and dimension restrictions.

Abstract

It is well known that the containment problem (as well as the equivalence problem) for semilinear sets is $\log$-complete in $\Pi_2^p$. It had been shown quite recently that already the containment problem for multi-dimensional linear sets is $\log$-complete in $\Pi_2^p$ (where hardness even holds for a unary encoding of the numerical input parameters). In this paper, we show that already the containment problem for $1$-dimensional linear sets (with binary encoding of the numerical input parameters) is $\log$-hard (and therefore also $\log$-complete) in $\Pi_2^p$. However, combining both restrictions (dimension $1$ and unary encoding), the problem becomes solvable in polynomial time.