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Orbit-finite linear programming

Arka Ghosh, Piotr Hofman, Sławomir Lasota

TL;DR

This work provides a decision procedure for checking if an orbit-finite system of linear inequalities has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set.

Abstract

An infinite set is orbit-finite if, up to permutations of the underlying structure of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of linear inequalities. As our principal contribution we provide a decision procedure for checking if such a system has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set. We also show undecidability of these problems in case when only integer solutions are considered. Therefore orbit-finite linear programming is decidable, while orbit-finite integer linear programming is not.

Orbit-finite linear programming

TL;DR

This work provides a decision procedure for checking if an orbit-finite system of linear inequalities has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set.

Abstract

An infinite set is orbit-finite if, up to permutations of the underlying structure of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of linear inequalities. As our principal contribution we provide a decision procedure for checking if such a system has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set. We also show undecidability of these problems in case when only integer solutions are considered. Therefore orbit-finite linear programming is decidable, while orbit-finite integer linear programming is not.
Paper Structure (21 sections, 33 theorems, 119 equations, 1 figure)

This paper contains 21 sections, 33 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on orbit-finite sets
  3. Orbit-finite (integer) linear programming
  4. Results
  5. Polynomially-parametrised inequalities
  6. Monotonic polynomially-parametrised inequalities
  7. Finitely setwise-supported sets
  8. Decidability of real solvability
  9. Preliminaries
  10. Idea of the reduction
  11. Reduction of Fin-Ineq-Solv$(\mathbb{R})$ to Almost-all-Poly-Ineq-Solv
  12. Optimisation problems
  13. Polynomially-parametrised maximisation problem
  14. Reduction of Fin-Ineq-Max$(\mathbb{R})$ to Almost-all-Poly-Ineq-Max
  15. Undecidability of integer solvability
  16. ...and 6 more sections

Key Result

Lemma 8

Let $T\subseteq_\text{fin} \mathbb{A}$. $T$-orbits $U\subseteq \mathbb{A}^{(n)}$ are exactly sets of the form where $I \subseteq \left\{ 1, \ldots, n \right\}$, $|I| = \ell$, and $u\in T^{(\ell)}$. The projection $\Pi_{n,I} : \mathbb{A}^{(n)} \to \mathbb{A}^{(\ell)}$ is defined in the expected way.

Figures (1)

  • Figure 1: Diagram of reductions between solvability problems.

Theorems & Definitions (57)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 47 more