## 1 Introduction
The concept of the Sixth-Generation (6G) communication system and the technologies surrounding it arose as a counter to the weaknesses of its predecessor [21]. The deployment and operation of massive sensors and actuators requires the synthesis of specifications from both mMTC and uRLLC into the massive Ultra-reliable Low-Latency Communication (mULC) paradigm. The 6G innovative architecture aims to successfully operate IoE services by simultaneously delivering high reliability, low latency, and high data rates to connected heterogeneous devices. Its design aims to facilitate the E2E communication, control, and computing functionality requirements of emerging IoE applications. Thus, it intends to achieve user-centric networking alongside service-centric provisioning by incorporating the end-user's specifications and demands. This may be achievable by combining service-oriented technologies like NS and user-centric approaches like digital twining [22]. However, the concepts and techniques that will comprehensively define the provisioning of 6G-based IoE applications are still under development and scientific debate, to reach a firm consensus [23; 24]. The ability to successfully specify and handle user-defined performance metrics among other extreme requirements is of paramount importance [22; 24]. The proposed framework aspires to fit in as an optimal resource allocation enabler for IoE user-centric applications with heterogeneous requirements by achieving multidimensional performance-oriented path evaluations and rankings. By integrating primary metrics that individually indicate a distinctive critical aspect of performance for candidate paths, the proposal of this study incorporates the required innovation for 6G-based orchestration deployment. Thus, it inherently supports the successful facilitation of both _ultra-reliable and low-latency_ and _high-speed and delay-tolerant_ applications, which is an expected feature of future telecommunications networks [25]. It therefore exhibits high levels of flexibility in supporting a wide range of emerging use case needs. Moreover, its architecture-independent nature due to the utilization of robust algebraic principles renders this solution scalable by design. By following the PA guidelines, both single-constraint and multi-constraint resource allocations can be realized successfully. PA allows metrics to be formulated analytically in order to be applicable to any optimization objective regardless of its orientation. This diligent descriptive capability, enriched with PA properties that ensure valid and optimal candidate solutions, unlocks the potential of a complete network element classification scheme. Notwithstanding this, the proposed solution is further developed by incorporating a novel weighted mechanism that allows it to adapt to each specific variant of user-centric applications and handle user-defined resource allocation requirements. This integration acts like the key to become a successful enabler for user-configurable and user-controllable VNR management, providing guaranteed granularity of embedding options. The output of this mechanism can either be deterministic or derive from Artificial Intelligence (AI) or Machine Learning (ML) managing systems. While AI/ML techniques are envisioned as foundational pillars of the 6G initiative, their de facto integration faces its own challenges. Important issues such as high variance, lack of explainability, and complex online training must be primarily addressed before fully adopting AI and ML as the cornerstones for orchestrating NGN resource allocations [26]. Therefore, the review of existing exact and heuristic performance optimization frameworks must precede the development of the forthcoming intelligent but still immature ones. The contribution of this work can be summarized in the following points:  
* The core primary metrics that can define different performance-oriented aspects of a candidate path are identified and, by the means of a Path Algebra (PA), transformed into proper algebraic structures.