Integrated information has been introduced as a metric to quantify the amount of information generated by a system beyond the information generated by its individual elements. While the metrics associated with the Greek letter ϕ require the calculation of the interaction of an exponential number of sub-divisions of the system, most of these numerical approaches related to the metric are based on the basics of classical information theory and perturbation analysis. Here we introduce and sketch alternative approaches to connect algorithmic complexity and integrated information based on the concept of algorithmic perturbation rooted in algorithmic information dynamics and its concept of programmability. We hypothesize that if an object is algorithmic random or algorithmic simple, algorithmic random perturbations will have little to no effect to the internal capabilities of a system to produce integrated information but when an object is more integrated the object will also display elements able to perturb the object and increase or decrease its algorithmic randomness. We sketch some of these ideas related to an object integrated information value and its algorithmic information content. We propose that such an algorithmic perturbation test quantifying compression sensitivity may provide a system with a means to extract explanations–causal accounts–of its own behavior hence making IIT and associated measure ϕ more explainable and interpretable. Our technique may reduce the number of calculations to arrive at some estimations with algorithmic perturbation guiding a more efficient search. Our work sets the stage for a systematic exploration and further investigation of the connections between algorithmic complexity and integrated information at the level of both theory and practice.