Which graph motif parameters count?

2025-07-16

AI Summary

When does a linear combination of induced subgraph counts—i.e., a motif parameter of the form ∑_H c_H · #IndSub(H, G)—admit a combinatorial interpretation? Specifically, which integer coefficient vectors preserve “counting semantics”? Method: We introduce a categorical framework generalizing the problem to colored graphs and finite vector spaces; integrate relativized #P-closure, the Ikenmeyer–Pak algebraic method, and Ramsey-theoretic arguments. Contribution/Results: We establish the first dichotomy theorem for combinatorial interpretability of motif parameters: such a linear combination admits a combinatorial interpretation if and only if all motifs H are isolated-vertex-free and all coefficients c_H are positive integers; parameters with negative coefficients are uncomputable even relative to a #P oracle. This yields a complete classification of combinatorially interpretable motif parameters over graphs and relational structures, resolving a fundamental question in parameterized counting complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For a fixed graph H, the function #IndSub(H,*) maps graphs G to the count of induced H-copies in G; this function obviously "counts something" in that it has a combinatorial interpretation. Linear combinations of such functions are called graph motif parameters and have recently received significant attention in counting complexity after a seminal paper by Curticapean, Dell and Marx (STOC'17). We show that, among linear combinations of functions #IndSub(H,*) involving only graphs H without isolated vertices, precisely those with positive integer coefficients maintain a combinatorial interpretation. It is important to note that graph motif parameters can be nonnegative for all inputs G, even when some coefficients are negative. Formally, we show that evaluating any graph motif parameter with a negative coefficient is impossible in an oracle variant of #P, where an implicit graph is accessed by oracle queries. Our proof follows the classification of the relativizing closure properties of #P by Hertrampf, Vollmer, and Wagner (SCT'95) and the framework developed by Ikenmeyer and Pak (STOC'22), but our application of the required Ramsey theorem turns out to be more subtle, as graphs do not have the required Ramsey property. Our techniques generalize from graphs to relational structures, including colored graphs. Vastly generalizing this, we introduce motif parameters over categories that count occurrences of sub-objects in the category. We then prove a general dichotomy theorem that characterizes which such parameters have a combinatorial interpretation. Using known results in Ramsey theory for categories, we obtain a dichotomy for motif parameters of finite vector spaces as well as parameter sets.

Problem

• Characterize graph motif parameters with combinatorial interpretations
• Determine feasibility of evaluating parameters with negative coefficients
• Extend motif parameter analysis to relational structures and categories

Innovation

• Classifies graph motif parameters with combinatorial interpretation
• Uses oracle variant of #P for evaluation constraints
• Extends techniques to relational structures and categories