The Fine-Grained Complexity of CFL Reachability

TL;DR

This paper investigates the exact polynomial time complexity of CFL reachability problems, providing lower bounds for various classes, especially Dyck-$k$ languages, under widely believed conjectures, advancing the understanding of fine-grained complexity in static analysis.

Contribution

It establishes the first fine-grained lower bounds for CFL reachability, especially for Dyck-$k$ languages, under popular complexity conjectures, filling a gap in the complexity landscape.

Findings

Cubic lower bounds for Dyck-$k$ languages on sparse graphs

New lower bounds for Andersen's Pointer Analysis

Clarification of the complexity landscape for CFL reachability

Abstract

Many problems in static program analysis can be modeled as the context-free language (CFL) reachability problem on directed labeled graphs. The CFL reachability problem can be generally solved in time $O(n^3)$, where $n$ is the number of vertices in the graph, with some specific cases that can be solved faster. In this work, we ask the following question: given a specific CFL, what is the exact exponent in the monomial of the running time? In other words, for which cases do we have linear, quadratic or cubic algorithms, and are there problems with intermediate runtimes? This question is inspired by recent efforts to classify classic problems in terms of their exact polynomial complexity, known as {\em fine-grained complexity}. Although recent efforts have shown some conditional lower bounds (mostly for the class of combinatorial algorithms), a general picture of the fine-grained complexity landscape for CFL reachability is missing. Our main contribution is lower bound results that pinpoint the exact running time of several classes of CFLs or specific CFLs under widely believed lower bound conjectures (Boolean Matrix Multiplication and $k$-Clique). We particularly focus on the family of Dyck-$k$ languages (which are strings with well-matched parentheses), a fundamental class of CFL reachability problems. We present new lower bounds for the case of sparse input graphs where the number of edges $m$ is the input parameter, a common setting in the database literature. For this setting, we show a cubic lower bound for Andersen's Pointer Analysis which significantly strengthens prior known results.