The Theory and Practice of Computing the Bus-Factor

TL;DR

This paper introduces a unified, domain-agnostic framework for measuring the bus-factor in projects, modeling it as a combinatorial optimization problem and proposing a new robustness-inspired measure with efficient approximation algorithms.

Contribution

It formalizes bus-factor as a family of NP-hard problems, introduces a novel robustness-based measure, and provides linear-time approximation algorithms for practical computation.

Findings

The robustness-based measure aligns well with project management expectations.

The new measure captures project fragmentation and loss of coverage.

Approximation algorithms enable scalable computation of bus-factor estimates.

Abstract

The bus-factor is a measure of project risk with respect to personnel availability, informally defined as the number of people whose sudden unavailability would cause a project to stall or experience severe delays. Despite its intuitive appeal, existing bus-factor measures rely on heterogeneous modeling assumptions, ambiguous definitions of failure, and domain-specific artifacts, limiting their generality, comparability, and ability to capture project fragmentation. In this paper, we develop a unified, domain-agnostic framework for bus-factor estimation by modeling projects as bipartite graphs of people and tasks and casting the computation of the bus-factor as a family of combinatorial optimization problems. Within this framework, we formalize and reconcile two complementary interpretations of the bus-factor, redundancy and criticality, corresponding to the Maximum Redundant Set and the Minimum Critical Set, respectively, and prove that both formulations are NP-hard. Building on this theoretical foundation, we introduce a novel bus-factor measure inspired by network robustness. Unlike prior approaches, the proposed measure captures both loss of coverage and increasing project fragmentation by tracking the largest connected set of tasks under progressive contributor removal. The resulting measure is normalized, threshold-free, and applicable across domains; we show that its exact computation is NP-hard as well. We further propose efficient linear-time approximation algorithms for all considered measures. Finally, we evaluate their behavior through a sensitivity analysis based on controlled perturbations of project structures, guided by expectations derived from project management theory. Our results show that the robustness-based measure behaves consistently with these expectations and provides a more informative and stable assessment of project risk than existing alternatives.