Topological Sorting under Regular Constraints

TL;DR

This paper investigates the complexity of constrained topological sorting problems with regular language constraints, identifying tractable cases and establishing NP-hardness for others, with implications for scheduling and program verification.

Contribution

It characterizes the complexity of CTS[K] and CSh[K] problems for various language classes, introducing new techniques and conjecturing a dichotomy based on language properties.

Findings

CTS[K] is tractable for unions of monomials.

CTS[K] is NP-hard for K = (ab)^*.

CSh[K] is tractable for group languages.

Abstract

We introduce the constrained topological sorting problem (CTS): given a regular language K and a directed acyclic graph G with labeled vertices, determine if G has a topological sort that forms a word in K. This natural problem applies to several settings, e.g., scheduling with costs or verifying concurrent programs. We consider the problem CTS[K] where the target language K is fixed, and study its complexity depending on K. We show that CTS[K] is tractable when K falls in several language families, e.g., unions of monomials, which can be used for pattern matching. However, we show that CTS[K] is NP-hard for K = (ab)^* and introduce a shuffle reduction technique to show hardness for more languages. We also study the special case of the constrained shuffle problem (CSh), where the input graph is a disjoint union of strings, and show that CSh[K] is additionally tractable when K is a group language or a union of district group monomials. We conjecture that a dichotomy should hold on the complexity of CTS[K] or CSh[K] depending on K, and substantiate this by proving a coarser dichotomy under a different problem phrasing which ensures that tractable languages are closed under common operators.