Quantifier Elimination Meets Treewidth

TL;DR

This paper introduces a treewidth-based dynamic programming framework that significantly reduces the complexity of quantifier elimination methods like FME and CAD for formulas with sparse variable dependency graphs, especially when the treewidth is small.

Contribution

It presents a novel approach leveraging treewidth and dynamic programming to improve quantifier elimination efficiency, extending to general procedures and demonstrating exponential complexity reduction.

Findings

Framework achieves exponential complexity improvement for fixed treewidth.

Preliminary experiments show better performance on low-treewidth instances.

Method outperforms heuristic approaches on sparse arithmetic benchmarks.

Abstract

In this paper, we address the complexity barrier inherent in Fourier-Motzkin elimination (FME) and cylindrical algebraic decomposition (CAD) when eliminating a block of (existential) quantifiers. To mitigate this, we propose exploiting structural sparsity in the variable dependency graph of quantified formulas. Utilizing tools from parameterized algorithms, we investigate the role of treewidth, a parameter that measures the graph's tree-likeness, in the process of quantifier elimination. A novel dynamic programming framework, structured over a tree decomposition of the dependency graph, is developed for applying FME and CAD, and is also extensible to general quantifier elimination procedures. Crucially, we prove that when the treewidth is a constant, the framework achieves a significant exponential complexity improvement for both FME and CAD, reducing the worst-case complexity bound from doubly exponential to single exponential. Preliminary experiments on sparse linear real arithmetic (LRA) and nonlinear real arithmetic (NRA) benchmarks confirm that our algorithm outperforms the existing popular heuristic-based approaches on instances exhibiting low treewidth.