CP-net Based Symbolic Model

• The CP-net based symbolic model is a qualitative framework that uses graph-based constructs and conditional preference tables to capture and aggregate preferences.
• It enforces semantic congruence through acyclic dependency graphs and consistent parent-set alignment across CP-nets.
• It offers efficient selection algorithms, including greedy and heuristic approaches, to optimize multi-agent decision-making in diverse application domains.

A CP-net based symbolic model is a qualitative preference modeling and composition framework that leverages Conditional Preference Networks (CP-nets) to capture, compare, and combine the conditional preferences of multiple agents or components. This model formalizes composability and semantic congruence in preference-based decision-making, allowing, for example, an Infrastructure-as-a-Service (IaaS) provider to select an optimal subset of consumers by analyzing and composing their qualitative requirements. The approach features rigorously defined graph-theoretic and preference-table constructs, explicit criteria for composability, and efficient (approximate and exact) algorithms to identify compatible consumer sets while minimizing computational complexity (Fattah et al., 2021).

1. Core CP-Net Constructs and Formal Definitions

The CP-net based symbolic model employs the following structures:

• Variables and Domains: Let $V = \{X_1, ..., X_n\}$ denote the set of service-attribute variables (e.g., CPU, Memory, Price). Each $X_i$ has a finite, discrete value domain $D(X_i)$ and a semantic domain $S(X_i)$ (e.g., $\{$"low", "moderate", "high"$\}$). A semantic mapping table $\mathrm{SemTable}(X_i, x)$ assigns each concrete value $x\in D(X_i)$ to a semantic level $s\in S(X_i)$.
• Dependency Graph (DDG): The CP-net is a directed acyclic graph over $V$; an edge $X_j \to X_i$ indicates $X_j$ is a parent in the conditional preference of $X_i$. The parent set of $X_i$ is denoted $\mathrm{Pa}(X_i)$.
• Conditional Preference Tables (CPTs): For each $X_i$, $\mathrm{CPT}(X_i)$ specifies, for every assignment $u$ to $\mathrm{Pa}(X_i)$, a (possibly partial) total order $\succ_u$ over $S(X_i)$. If $\mathrm{Pa}(X_i)=\varnothing$, the order is unconditional. The semantics are ceteris paribus: the specified preferences over $X_i$ hold only when $\mathrm{Pa}(X_i)=u$.

These constructs collectively form the symbolic preference model for each agent.

2. Mathematical Formulation of the Composability Model

The composability model rigorously distinguishes when two CP-nets can be merged while preserving preference semantics:

• Semantic Congruence: A composition is semantically congruent if "the relative importance order of preference attributes for each consumer is preserved without any ambiguity." This is central to ensuring meaningful aggregation of preferences.
• Composability Criteria:
  • Dependency-Graph Composability: DDGs $A$ and $B$ are composable if their union is acyclic: $$\mathrm{acyclic}(\mathrm{DDG}_A \cup \mathrm{DDG}_B)$$.
  • CPT Composability: For variable $X$, $\mathrm{CPT}_A(X)$ and $\mathrm{CPT}_B(X)$ are composable when they refer to the same variable and have identical parent sets ($\mathrm{Pa}_A(X)=\mathrm{Pa}_B(X)$), or if one CP-net lacks $X$.

The CP-nets $CP_A$ and $CP_B$ are composable (i.e., $composable(CP_A, CP_B)=\mathrm{true}$) if both the DDG union is acyclic and all CPT parent sets match where applicable:

$$composable(CP_A,CP_B) \Leftrightarrow acyclic(\mathrm{DDG}_A \cup \mathrm{DDG}_B) \wedge (\forall X)\ \mathrm{Pa}_A(X)=\mathrm{Pa}_B(X)\ \text{or one side absent}$$

3. Criteria for Semantic Congruence and Valid Composition

Semantic congruence is both necessary and sufficient for safe composition of CP-nets. It is enforced by requiring:

• Acyclic union of DDGs.
• Parent set alignment for all overlapping variables.
• Failure to meet either criterion (i.e., induced cycles in the DDG or conflicting parent sets for a variable) results in undefined or ambiguous composition.

This disciplined approach ensures that qualitative preference aggregation remains consistent, avoiding ambiguous or cyclic dependencies.

4. Selection Algorithms: Greedy and Heuristic Approaches

Two algorithms are described for selecting consumer CP-nets that maximize similarity to the provider's CP-net:

Similarity Measure Table

Measure	Formula / Property	Usage
CPT-based Similarity	$$Sim(CP_A,CP_B) = \frac{\sum_{X_i}(|CPT_A(X_i)\cap CPT_B(X_i)|\cdot\Pi_{X_j\notin Pa(X_i)}|\mathrm{SemTable}(X_j)|)}{\sum_{X_i}(|CPT_A(X_i)\cup CPT_B(X_i)|\cdot\Pi_{X_j\notin Pa(X_i)}|\mathrm{SemTable}(X_j)|)}$$	Guides selection and ranking

• Greedy-based selection: At each iteration, select the most similar and composable consumer candidate (w.r.t. current residual CP-net). Stop upon zero similarity or no further improvement. This achieves $O(N^2|V|L^2)$ time complexity (for $N$ candidates, $|V|$ variables, $L$ average CPT lines), and quickly identifies highly similar subsets when preferences are concentrated.
• Heuristic-based selection: First filters consumer CP-nets with similarity above threshold $\tau$. Performs brute-force search over all possible composable subsets from this filtered set. The smaller size ($|H|\ll N$) makes exhaustive subsearch tractable, with complexity $O(2^{|H|}\cdot\mathrm{costCompose})$ due to the combinatorial subset enumeration and compositional consistency checking.

These methods are approximate (greedy) or exact (heuristic over $H$), depending on threshold choice and candidate distribution.

5. Illustrative Examples of CP-Net Composition

The following selected example, drawn from (Fattah et al., 2021), demonstrates the composition process:

• Provider CP-net "CPU-intensive": DDG $CPU \rightarrow Memory \rightarrow Price$, with CPTs specifying top-down conditional orders (e.g., $c_3\succ c_2\succ c_1$ for CPU).
• Consumer CP-nets: Examples include "price-sensitive" $Price \rightarrow Memory \rightarrow CPU$ and "CPU-intensive" $CPU \rightarrow Memory \rightarrow Price$.
• Composability Determination: Only consumers with identical DDG structure and parent sets (e.g., provider and "CPU-intensive" consumer) are composable. Mismatched structures induce cycles and are excluded.
• Selection Outcome: Applying greedy and thresholded heuristic selection to a set of three consumer nets, only the one with matching DDG and CPTs ("CPU-intensive") is selected, aligning semantic congruence and maximizing similarity.

6. Generalization Beyond IaaS and Broader Applicability

The CP-net based symbolic model generalizes to any multi-agent, multi-attribute decision context where qualitative, conditional preferences are central. The essential requirements are:

• Shared variable (attribute) set for all agents.
• Meaningful semantic value mapping for qualitative preference alignment.
• Enforced semantic congruence—via DAG/cycle-check and CPT parent-set matching—for robust preference aggregation.

Potential application domains include:

• Group product feature configuration, matching manufacturer and consumer CP-nets.
• Team formation in staffing, balancing candidate and manager attribute preferences.
• Workflow/service orchestration, where each service has a CP-net describing quality-of-service demands.
• Collaborative filtering, with user and system CP-nets guiding recommendations under complex preference constraints.

This suggests the CP-net based symbolic model provides a generically applicable preference aggregation framework, given adherence to composability and semantic mapping requirements (Fattah et al., 2021).