Graph of Convex Sets

• Graph of convex sets (GCS) is a framework that associates graph vertices with convex sets to integrate combinatorial properties and convex optimization.
• It encodes structural constraints through convex graph invariants and invariant sets, enabling efficient relaxations such as spectral and semidefinite programming methods.
• The framework unifies problems like graph deconvolution, hypothesis testing, and graph generation by leveraging robust optimization and permutation symmetry.

A graph of convex sets (GCS) is a mathematical framework in which vertices of a graph are associated with convex sets—typically subsets of a vector space or matrix space—and structural or combinatorial properties are encoded or enforced through convexity constraints. This paradigm allows the paper, description, and solution of a variety of graph-theoretic and optimization problems by leveraging the tractability of convex sets and functions. It unites discrete graph invariants and combinatorial structures with convex analysis and optimization, enabling sophisticated formulations and strong computational procedures for previously intractable problems.

1. Foundations: Convex Graph Invariants and Invariant Convex Sets

Convex graph invariants form the theoretical core of GCS. A convex graph invariant is a real-valued function $f$ defined on the adjacency matrix $A$ of a graph (considered as an $n\times n$ symmetric matrix in the space $S$), that is both:

• convex with respect to $A$, and
• invariant under relabelings of the graph (that is, $f(A) = f(\Pi A \Pi^\top)$ for all $A \in S$ and all $n \times n$ permutation matrices $\Pi$).

Examples of convex graph invariants include:

• the sum of edge weights (with $P = \frac{1}{2}(J-I)$, $J$ the all-ones matrix);
• maximum node degree (using matrices $P$ with appropriately chosen structure);
• MAXCUT value (expressed as $$\max_{y \in \{\pm1\}^n} \frac{1}{4} \sum_{ij} A_{ij}(1 - y_i y_j)$$, or, in practice, via its semidefinite relaxation);
• spectral invariants, such as the sum of the $k$ largest eigenvalues.

Each invariant can be enforced or bounded using convex constraints, giving rise to invariant convex sets $C \subseteq S$ such that $A \in C \implies \Pi A \Pi^\top \in C$ for all $\Pi$. These sets form the "nodes" in the graph of convex sets, and can be combined or intersected to encode complex structural requirements (e.g., connectivity through constraints on Laplacian eigenvalues, absence of specific subgraphs via bounded $\Theta_P(A)$, or prescribed degree sequences).

2. Representation and Computation via Elementary Invariants

A key representation theorem establishes that every convex graph invariant $f$ can be written as a supremum over elementary convex graph invariants: $$f(A) = \sup_{P \in \mathcal{P}} \{\Theta_P(A) - \alpha_P\}$$ for some collection of symmetric matrices $\mathcal{P}$ and constants $\alpha_P$. Here, the elementary invariant is defined as: $$\Theta_P(A) = \max_{\Pi \in \mathrm{Sym}(n)} \langle P, \Pi A \Pi^\top \rangle$$ This formulation is analogous to expressing a convex function as the supremum over affine functionals.

In general, direct computation of $\Theta_P(A)$ is NP-hard, equivalent to solving a Quadratic Assignment Problem (QAP). The paper describes tractable relaxations—most notably spectral relaxations, where the finite group of permutations is replaced by the orthogonal group, yielding: $$\Lambda_P(A) = \max_{V \in O(n)} \langle P, V A V^\top \rangle = \lambda(P)^\top \lambda(A)$$ with $\lambda(\cdot)$ denoting the vector of ordered eigenvalues. Further, semidefinite programming (SDP) relaxations can provide tighter approximations for specific invariants, crucial for efficient computation and optimization.

3. Advantages and Tradeoffs: Convex Versus Nonconvex Invariants

While many graph invariants of structural interest are nonconvex (e.g., counts of specific subgraphs, exact eigenvalue positions), convex invariants afford significant computational and theoretical advantages:

• Convexity ensures that optimization problems imposed over these sets are tractable and robust—replacing combinatorial explosion with efficient numerical methods.
• While convex invariants typically encode aggregate rather than pointwise properties (e.g., sum of top $k$ eigenvalues versus individual ones; bounding numbers of triangles via surrogate constraints), the loss of granularity is offset by substantial algorithmic benefit.

The framework is "complete" in the sense that for any two non-isomorphic graphs, there is some elementary convex invariant (potentially constructed from the symmetries) that distinguishes them, although any fixed family of convex invariants may blur or conflate certain detailed features.

4. Unified Convex Formulations for Classic Graph Problems

By encoding structural properties via convex graph invariants and assembling these as convex sets, GCS enables convex formulations for problems traditionally cast as highly nonconvex or combinatorial:

• Graph deconvolution: Given $A = A_1 + A_2$ where $A_1 \in \mathcal{C}(\mathcal{G}_1)$, $A_2 \in \mathcal{C}(\mathcal{G}_2)$ (convex hulls of graphs with unknown labelings), one poses the deconvolution as a convex optimization over the convex hulls subject to observed summation constraints.
• Hypothesis testing: Given two families of graphs represented by different invariant convex sets, the problem of deciding which family better explains a sample reduces to optimizing a linear functional over the convex sets and selecting the maximizing one.
• Graph generation: Constructing graphs with prescribed global properties is rephrased as constructing a matrix in the intersection of invariant convex sets corresponding to the desired invariants, often via random convex maximization to select an extreme point.

This convex-analytic perspective enables the formulation of practical, tractable relaxations for these and related problems, fundamentally transforming their computational profile.

5. Relation to Robust Optimization and Permutation Symmetry

A central insight is the analogy between invariance under node relabeling and uncertainty in robust optimization. The convex hull of a graph under node permutation actions, denoted

$$\mathcal{C}(\mathcal{G}) = \operatorname{conv} \{\Pi A \Pi^\top : \Pi \in \mathrm{Sym}(n)\}$$

serves as the minimal invariant convex set containing all possible adjacency representations of a graph. This is almost exactly the role of uncertainty sets (permutohulls) in robust optimization frameworks.

Consequently, optimizing over all invariants simultaneously is equivalent to optimizing over this convex hull, which aligns with robust worst-case guarantees and naturally accounts for label uncertainty. SDP relaxations and other convex optimization tools ubiquitous in robust optimization are thus immediately applicable, reinforcing the methodological connection.

6. Implications and Expressiveness of the GCS Framework

The GCS approach—encoding graph-theoretic and structural constraints as invariant convex sets—provides a unifying language and analytic toolset for a wide spectrum of graph problems. Key implications include:

• Strong expressiveness: Many structural features (connectivity, degree constraints, spectral properties, forbidden subgraphs) can be enforced via convex invariants and hence incorporated into GCS formulations.
• Computational tractability: Even though some elementary invariants and sets are intractable to compute directly (NP-hard), convex relaxations, spectral methods, and SDPs provide efficient and scalable approximations with strong theoretical bounds.
• Unified methodology: Problems as different as deconvolution, generation, hypothesis testing, and robust analysis are addressed within the same conceptual and algorithmic framework.

Moreover, by translating challenging combinatorial graph questions into convex optimization over structured, permutation-invariant convex sets, GCS bridges the discrete/continuous divide and allows the application of powerful tools in convex analysis and optimization theory to combinatorial domains.

7. Connections, Limitations, and Future Research

The GCS paradigm establishes deep links between graph invariants, convex analysis, permutation symmetry, and robust optimization, but also reveals some inherent limitations:

• Information loss: Specifically, convex relaxations may "blur" or aggregate features compared to nonconvex invariants, and tractable computation of invariants for large graphs still poses theoretical and practical challenges.
• Tightness of relaxations: While many SDPs and spectral relaxations are empirically effective, understanding the precise approximation guarantees and developing tighter (or problem-specific) relaxations remains an active area.
• Extension to finer structure: Further research can focus on extending GCS to encode more granular combinatorial substructures, or to develop new classes of tractable relaxations for hard invariants.

The connection to robust optimization suggests broader applicability to problems with inherent symmetry or uncertainty; the design of convex sets to capture novel invariants, and the expansion of GCS-style formulations to other combinatorial settings, represent promising directions.

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By recasting graph-theoretic properties in terms of convex graph invariants and their associated invariant convex sets, the Graph of Convex Sets framework enables expressive, tractable, and robust formulations for a wide array of classical and modern graph problems, grounded in the mathematical principles of convex analysis and equipped for scalable computation (Chandrasekaran et al., 2010).