`Mutual Complexity' in Hyperscaling Violating Background

`Mutual Compl Additivity and subadditivity are cornerstone principles in information theory, describing how measures like entropy and channel capacities behave under combination. Additivity asserts f(A∪B)=f(A)+f(B), while subadditivity ensures f(A∪B)≤f(A)+f(B). These properties provide profound insights into data compression, error correction, and the design of communication systems, with results such as the strong subadditivity of entropy playing a vital role in both classical and quantum information theory.In this work, we explore the concept of 'mutual complexity' within hyperscaling violating backgrounds by employing the complexity equals action proposal. For two subregions, we determine the definite bulk action within each subregion, incorporating appropriate counterterms to describe holographic complexity. Our findings show that mutual complexity for two subregions exhibits subadditivity. Furthermore, we extend this framework to three subregions, introducing the notion of holographic 'tripartite complexity' and demonstrating that this new quantity exhibits superadditivity.