Computational Complexity of Edge Coverage Problem for Constrained Control Flow Graphs

TL;DR

This paper analyzes the complexity of achieving edge coverage in control flow graphs with various constraints, revealing polynomial-time solutions for positive constraints and NP-completeness for others, with fixed-parameter tractability for negative constraints.

Contribution

It classifies the computational complexity of edge coverage problems with five types of constraints and provides an FPT algorithm for negative constraints.

Findings

Positive constraints are decidable in polynomial time.

Negative, MAX ONCE, ONCE, and ALWAYS constraints are NP-complete.

An FPT algorithm is proposed for negative constraints based on the number of constraints.

Abstract

The article studies edge coverage for control flow graphs extended with explicit constraints. Achieving a given level of white-box coverage for a given code is a classic problem in software testing. We focus on designing test sets that achieve edge coverage \textit{while respecting additional constraints} between vertices. The paper analyzes how such constraints affect both the feasibility and computational complexity of edge coverage. The paper discusses five types of constraints. POSITIVE constraints require at least one test path where a given vertex precedes another. NEGATIVE constraints forbid any such test path. ONCE constraints require exactly one test path with a single occurrence of one vertex before another. MAX ONCE constraints allow such precedence in at most one test path. ALWAYS constraints require every test path containing a given vertex to also contain another vertex later on the same path. Each type models a different test requirement, such as mandatory flows, semantic exclusions, or execution cost limits. We investigate the computational complexity of finding a test set that achieves edge coverage and respects a given set of constraints. For POSITIVE constraints, the existence of an edge covering test set is decidable in polynomial time by extending standard edge coverage constructions with additional paths for each constraint. For NEGATIVE, MAX ONCE, ONCE, and ALWAYS constraints, the decision problem is NP-complete. The proofs rely on polynomial reductions from variants of SAT. The NP-completeness results hold even for restricted graph classes, including acyclic graphs, for all these four constraints. Finally, we study the fixed-parameter tractability of the NEGATIVE constraint. Although the general problem is NP-complete, the paper presents an FPT algorithm with respect to the number of constraints.