Relational Circuit Complexity

• Relational circuit complexity is a framework that uses {∪, ×}-circuits with decomposability and smoothness to succinctly represent relational query outputs.
• It leverages structural parameters like treewidth and submodular width to establish tight upper and lower bounds on circuit size for database queries.
• The field integrates techniques from communication complexity, graph theory, and algebraic methods to address efficiency and tractability in query processing.

Relational circuit complexity studies the size and structure of Boolean or set-valued circuit representations for relational problems, particularly those arising from database theory, constraint satisfaction, and knowledge compilation. This field investigates how properties of relational queries or input structures—captured by notions such as treewidth and submodular width—determine the efficiency and succinctness of circuit-based representations for solution sets, such as query results or sets of homomorphisms. Distinct from classical (function) circuit complexity, relational circuit complexity hinges on decomposing solution sets via union and join, typically formalized as $\{\cup, \times\}$-circuits with decomposability and smoothness properties. Core results expose tight upper and lower bounds on the circuit size required for representing the output of relational queries, and reveal complexity-theoretic dichotomies anchored in structural width parameters.

1. Fundamental Definitions and Circuit Models

Relational circuit complexity formalizes the succinct representation of query results using $\{\cup, \times\}$-circuits, also known as d-representations or decomposable negation normal forms (DNNFs). A $\{\cup, \times\}$-circuit $C$ over $(X, D)$ is a directed acyclic graph, where each source node is labeled by a literal $x \mapsto d$ ($x \in X, d \in D$), each internal node is labeled by $\cup$ or $\times$ with arity two, and the following constraints hold:

• Decomposability: For any $\times$-gate $g$ with children $g_1, g_2$, $$\mathrm{Var}(g_1) \cap \mathrm{Var}(g_2) = \emptyset$$.
• Smoothness: For any $\cup$-gate $g$ with children $g_1, g_2$, $\mathrm{Var}(g_1) = \mathrm{Var}(g_2)$.

Such a circuit computes, at each gate, a relation (set of partial assignments). The sink gate computes the set of homomorphisms or, equivalently, the result of a join query. The size of a circuit is its number of gates. These models enable a compositional construction of sets via syntactic union and Cartesian product (join), mirroring the semantics of relational algebra (Berkholz et al., 26 Mar 2025).

2. Structural Width Parameters: Treewidth and Submodular Width

The efficiency of relational circuit representations is fundamentally determined by structural parameters of the query's hypergraph:

• Treewidth ($\mathrm{tw}(H)$): Defined through tree decompositions, it measures, for a hypergraph $H = (V, E)$, how closely $H$ approximates a tree: $$\mathrm{tw}(H) = \min_{(T, \beta)} \max_{t \in T} |\beta(t)| - 1$$, where $(T, \beta)$ is a tree decomposition.
• Submodular Width ($\mathrm{subw}(H)$): Generalizes treewidth by optimizing tree decompositions with respect to monotone submodular functions: $$\mathrm{subw}(H) = \sup_{f \in \mathcal{S}(H)} \min_{(T, \beta)} \max_{t \in T} f(\beta(t))$$, where $\mathcal{S}(H)$ is the set of monotone submodular functions $f: 2^V \to [0,1]$ with $f(\emptyset) = 0$ and $f(e) \le 1$ for all $e \in E$.

The connections among width parameters yield, for bounded-arity queries, $$\mathrm{tw}(H) \leq \mathrm{fhtw}(H) \leq \mathrm{subw}(H)$$, and $\mathrm{fhtw}(H) = \Theta(\mathrm{tw}(H))$ (Berkholz et al., 26 Mar 2025).

3. Upper and Lower Bounds on Representational Complexity

Upper Bounds

For any join query $Q$ with hypergraph $H$ and database size $N$, one can construct a $\{\cup, \times\}$-circuit of size $O(N^{(1+\delta)\cdot \mathrm{subw}(H)})$ (for arbitrarily small $\delta > 0$); with bounded relation arity, the bound tightens to $N^{O(\mathrm{tw}(H))}$ (Berkholz et al., 26 Mar 2025). These bounds are achieved via algorithmic constructions based on fractional hypertree or submodular-width-guided decompositions, e.g., the deterministic methods of Olteanu–Závodný and the flow-based decomposition due to Marx.

Unconditional Lower Bounds

Two major unconditional lower bounds, proven via the balanced rectangle-cover method in communication complexity, are established:

Join Arity	Lower Bound on Circuit Size	Structural Parameter
Bounded	$N^{\Omega(\mathrm{tw}(Q))}$	Treewidth
Unbounded	$N^{\Omega(\mathrm{subw}(Q)^{1/4})}$	Submodular Width

These bounds hold for infinitely many database instances and are tight (up to constant factors in the exponent) in the bounded-arity regime. The proofs integrate structural properties from graph theory (e.g., excluded-grid, separator theorems) and information-theoretic rectangle size arguments (Berkholz et al., 26 Mar 2025).

4. Rectangle-Cover and Communication Complexity Techniques

The lower bounds rely on the correspondence between circuit size and covers of the query result by balanced combinatorial rectangles. Any $\{\cup, \times\}$-circuit computing $S \subseteq D^X$ of size $s$ yields a cover of $S$ by $s$ rectangles. Lower bounding the size of each rectangle—by showing that any rectangle must be small under appropriate weighting—implies a lower bound on the size of the covering family and, hence, on the minimum required circuit size. For bounded-arity joins, the matching and separator-based arguments demonstrate that every balanced rectangle projects only a polylogarithmic set of values in certain coordinates, necessitating exponentially many rectangles (Berkholz et al., 26 Mar 2025).

5. Dichotomy Theorems and Sharp Complexity Transitions

The main dichotomy asserts:

• If the submodular width of a query $Q$ is bounded, there exist polynomial-size $\{\cup, \times\}$-circuits for $Q$ across all database instances (more precisely, the circuit size is fixed-parameter tractable in $|Q|$ and polynomial in $N$).
• If the submodular width is unbounded, then for some infinite family of database instances, every circuit must have superpolynomial size—at least $N^{\Omega(\mathrm{subw}(Q)^{1/4})}$ (Berkholz et al., 26 Mar 2025).

In the bounded-arity setting with subw$(Q) \approx$ tw$(Q)$, this is a tw-dichotomy: bounded treewidth entails efficient circuits, while unbounded treewidth forces superpolynomial size. Submodular width strictly generalizes treewidth to higher-arity settings and, therefore, governs the boundary of tractability for succinct relational circuit representations.

6. Extensions, Methodological Connections, and Algebraic Perspectives

Relational circuit complexity is deeply linked to advances in Boolean function complexity, algebraic CSP theory, and knowledge compilation. The algebraic approach to circuit complexity based on polymorphisms and partial polymorphisms (Nordh, 2016) characterizes the deterministic and nondeterministic circuit size for representable relations, providing a correspondence between algebraic closure properties and minimal circuit covers. Communication complexity arguments, such as the Karchmer–Wigderson paradigm, underpin many lower bound frameworks and can be captured within the rectangle-cover technique fundamental to relational circuit lower bounds (Berkholz et al., 26 Mar 2025).

Theoretical analyses of message-passing GNNs and symmetric EGNN circuits build on similar representations—constant-depth threshold circuits over relational inputs—revealing that the expressive barrier for such models is sharply determined by the depth and width constraints, with explicit separation from $\mathsf{NC}^1$ decision problems such as graph connectivity unless $\mathsf{TC}^0 = \mathsf{NC}^1$ (Li et al., 11 Jan 2025, Cao et al., 7 Oct 2025).

7. Key Formulas, Parameters, and Open Problems

Essential quantitative parameters and expressions include:

• Circuit Size Upper Bound: $|C| = O(N^{(1+\delta)\cdot \mathrm{subw}(Q)})$; $O(N^{O(\mathrm{tw}(Q))})$ if arity bounded.
• Circuit Size Lower Bound: $|C| \geq N^{\Omega(\mathrm{tw}(Q))}$ (bounded arity), $|C| \geq N^{\Omega(\mathrm{subw}(Q)^{1/4})}$ (arbitrary arity).
• Rectangle Size Bound: $|R| \leq \prod_{x \in X} |\mathrm{Proj}_x(R)|$ for $R \subseteq D^X$.
• Dichotomy Cutoff: polynomial-size iff $\mathrm{subw}(Q)$ bounded.

A major unresolved problem is the removal of a logarithmic factor gap in decision-complexity lower bounds for bounded-arity joins (i.e., improving from $N^{\Omega(\mathrm{tw}/\log \mathrm{tw})}$ to $N^{\Omega(\mathrm{tw})}$), which remains open despite tight representation-theoretic bounds (Berkholz et al., 26 Mar 2025).

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References:

• "Factorised Representations of Join Queries: Tight Bounds and a New Dichotomy" (Berkholz et al., 26 Mar 2025)
• "Polymorphisms and Circuit Complexity" (Nordh, 2016)
• "On the Computational Capability of Graph Neural Networks: A Circuit Complexity Bound Perspective" (Li et al., 11 Jan 2025)
• "Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspective" (Cao et al., 7 Oct 2025)
• "Formulas vs. Circuits for Small Distance Connectivity" (Rossman, 2013)