Exact DAG Extraction Methods

• Exact DAG extraction is the process of precisely recovering a directed acyclic graph that meets specified feasibility or optimization criteria.
• It employs various approaches such as continuous relaxations, combinatorial algorithms, and order-based parameterizations to enforce strict acyclicity and structure.
• These methods ensure exact representation or recovery in applications ranging from observational learning and graph partitioning to computational compilation.

Exact DAG extraction denotes a family of tasks in which a directed acyclic graph, or a DAG-shaped surrogate object, is required exactly rather than approximately. In the cited literature, the phrase is used in several distinct senses: exact acyclicity of an optimization variable, exact global optimization over DAGs, exact realization from explicit combinatorial specifications, exact decomposition of a digraph into a DAG-indexed structure, exact compilation of inference or arithmetic into a DAG, and exact reconstruction from a canonical sequential encoding (Yu et al., 2019, Rytíř et al., 2024, Darwiche et al., 2014, 2505.22949). As a result, the technical content of “exactness” depends on what object is being extracted and what guarantees are required.

1. Meanings of exactness

The literature does not use a single uniform notion of exact DAG extraction. In observational structure learning, exactness may refer to an acyclicity characterization that is mathematically equivalent to DAGness, or to global optimization of a score over DAGs. In realization and decomposition problems, exactness instead refers to exact feasibility with respect to prescribed combinatorial constraints. In compilation and representation problems, exactness means that the extracted DAG is a lossless representation of another object, such as an inference computation or a graph derivation (Yu et al., 2019, Berger et al., 2012, Darwiche et al., 2014, 2505.22949).

Setting	Extracted object	Exactness claim
Observational learning	DAG over variables	Exact acyclicity, exact optimization, or exact recovery under identifiability
Realization or partitioning	DAG or sink-based partition of a given graph	Exact feasibility and minimum cost
Decomposition or compilation	DAG-shaped strategy, arithmetic DAG, or query DAG	Exact representation of a game or computation
Sequence or e-graph extraction	Rule sequence or selected acyclic subgraph	Exact reconstruction or exact combinatorial optimum

A persistent distinction is between exact structural feasibility and exact recovery of the true edge set. The former concerns whether the returned object is acyclic or otherwise valid; the latter concerns whether it equals the ground-truth DAG. The two coincide only under stronger assumptions than acyclicity alone. This distinction is explicit in several papers and is central to interpreting exactness claims (Yu et al., 2019, Zhang et al., 24 Mar 2025, Rytíř et al., 2024).

2. Exactness in observational DAG learning

A major line of work studies DAG learning from observational data by replacing combinatorial search with continuous or mixed-integer optimization. In this setting, exactness first appeared as an exact acyclicity characterization. DAG-GNN learns a continuous weighted adjacency matrix $A$ under the polynomial equality constraint

$$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$

which the paper proves is equivalent to acyclicity. The method therefore supports exact acyclicity of the continuous optimization variable at feasibility, but it does not provide identifiability, consistency, or exact edge-recovery guarantees; the final discrete graph is obtained by thresholding edge weights at $0.3$ (Yu et al., 2019).

GraN-DAG extends the NOTEARS-style equality-constrained formulation to nonlinear neural conditional models. Its acyclicity condition is

$h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$

where $A_\phi$ is induced by neural-network path connectivity. This preserves the exact acyclicity-constrained formulation at the level of the optimization problem, but practical extraction still uses thresholding, Jacobian-based edge deletion, and pruning, so the final graph is post-processed rather than read off directly from the raw optimizer output (Lachapelle et al., 2019).

A broader theoretical generalization is given by analytic DAG constraints. For a nonnegative matrix $\tilde{\mathbf B}$ and an analytic function

$f(x)=c_0+\sum_{i=1}^{\infty} c_i x^i$

with $c_i>0$ for all $i>0$, the paper proves

$$\mathcal G \text{ is acyclic} \iff \operatorname{tr}[f(\tilde{\mathbf B})]=c_0 d.$$

This yields exact acyclicity constraints for the matrix exponential, the log-determinant family, inverse-series families, and derivatives, sums, and products of such functions. The exactness is therefore in the constraint itself; numerical optimization remains nonconvex and only achieves exact DAG validity when feasibility is satisfied exactly (Zhang et al., 24 Mar 2025).

A second line of work enforces exact DAGness by construction through order-based parameterizations. DAGuerreotype optimizes over topological orderings via the permutahedron and restricts admissible edges to those consistent with the ordering. Every permutation in the sparse support corresponds to an exact DAG mask, and the final graph is obtained from the mode permutation rather than from a cyclic weighted adjacency that must later be repaired (Zantedeschi et al., 2023). DP-DAG uses the factorization

$$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$0

where $$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$1 is a sampled permutation matrix and $$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$2 is upper triangular, so every sampled graph is a valid DAG at any time during training (Charpentier et al., 2022). $$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$3DAG uses a different exactness mechanism: after unconstrained stochastic steps, it projects the iterate to an order-consistent mask, and after projection and order-preserving optimization the iterate lies in the DAG-feasible set $$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$4, although the temporary pre-projection iterate may be cyclic (Ziu et al., 2024).

ExDAG moves from continuous relaxations to exact global optimization. It formulates static DAG learning as an MIQP with binary edge indicators $$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$5, weights $$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$6, big-$$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$7 support constraints, and lazy cycle-elimination inequalities

$$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$8

for every directed cycle $$h(A)=\operatorname{tr}\!\big[(I+\alpha A\circ A)^m\big]-m=0,$$9. These constraints are generated only when violated integer solutions are found, inside a branch-and-bound-and-cut procedure. In this sense ExDAG is exact with respect to the stated score-based optimization problem, and the paper reports identification of DAGs on up to 50 vertices in its experiments (Rytíř et al., 2024).

A stronger notion—exact recovery consistency rather than exact feasibility or exact optimization—is established in transfer learning for linear non-Gaussian DAGs. There the target graph is reconstructed layer by layer using precision-matrix estimation and distance covariance, and under the stated assumptions the paper proves

$0.3$0

The result applies even in settings where no auxiliary DAG is globally similar to the target DAG, provided suitable local structural informativeness holds (Ren et al., 2023).

3. Exact realization and exact optimization on prescribed graph inputs

In combinatorial realization problems, exact DAG extraction means constructing a DAG that exactly satisfies a given specification. The clearest example is the Dag realization problem: given a labeled sequence

$0.3$1

determine whether there exists a simple DAG on $0.3$2 with indegree $0.3$3 and outdegree $0.3$4 at each vertex. The paper gives a constructive exact exponential-time algorithm based on the set $0.3$5 of opposed-minimal eligible stream tuples, together with polynomial or linear-time exact algorithms for source-sink sequences, opposed sequences, and forest sequences (Berger et al., 2012).

This realization problem is exact in a strict sense. The constructed DAG must satisfy the indegree and outdegree requirements exactly, no parallel arcs are allowed, and the vertex set is labeled. The exact algorithm is recursive: after canonically sorting the sequence, it either handles a source-sink sequence greedily or branches on members of $0.3$6, decrementing source outdegrees and satisfying the indegree of the next stream tuple. If the recursion reaches the empty sequence, reversing the reductions yields an explicit DAG realization (Berger et al., 2012).

A different exact optimization problem on a given DAG is DAG Partitioning. Here the input is an arc-weighted DAG $0.3$7 and a budget $0.3$8, and the task is to delete arcs of total weight at most $0.3$9 so that each weakly connected component of the remaining graph contains exactly one sink. The paper proves that the problem can be solved in

$h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$0

time, gives linear-time executable data reduction rules, shows that Leskovec et al.’s heuristic is optimal on trees, and shows that if a width-$h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$1 tree decomposition of the underlying undirected graph is given, then DAG Partitioning is solvable in

$h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$2

time (Bevern et al., 2016).

In this setting, the extracted object is not a new DAG but an exact sink-rooted partition of an existing DAG. By Observation 2 in the paper, a partitioning set is valid if and only if each vertex in the remaining graph can reach exactly one sink. Exactness therefore means exact minimum-weight disambiguation of sink reachability, not graph discovery (Bevern et al., 2016).

4. Decompositions, closures, and hardness barriers

Another meaning of exact DAG extraction appears in decomposition theory. “Characterizing DAG-depth of Directed Graphs” does not reconstruct the original digraph as a DAG. Instead it extracts a DAG-depth decomposition $h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$3, where $h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$4 is a DAG and

$h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$5

is a surjective map to the original vertices. Validity is expressed by the Neighbor cover condition, and the central theorem states that the decomposition is valid if and only if every strategy induced by it is winning for the lift-free cops-and-robber game (Bezek, 2016).

This extracted DAG is exact only as a strategic object. The paper emphasizes that multiple copies of the same original vertex may be required, that different valid decompositions can exist, and that the closure of a decomposition is the unique maximal digraph compatible with it. Thus the decomposition and its closure are exact structural surrogates, but not exact reconstructions of the original edge set (Bezek, 2016).

The limits of exact extraction are even sharper for DAG-width. A DAG decomposition $h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$6 of a digraph $h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$7 has bags $h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$8 indexed by a DAG $h(\phi)=\operatorname{Tr} e^{A_\phi}-d=0,$9, and DAG-width is the minimum possible maximum bag size. The paper proves that there is an infinite class of graphs for which every optimal-width DAG decomposition has super-polynomial size, and that there is no polynomial-size DAG decomposition that approximates an optimal decomposition up to an additive constant. It also proves that deciding whether the DAG-width of a given graph is at most a given input parameter is PSPACE-complete (Amiri et al., 2014).

These results matter because they separate exact extraction of an optimal decomposition from exact extraction of a compact certificate. For DAG-width, exact decomposition may require super-polynomially many bags even when the optimal width itself is small enough to describe succinctly. In this literature, exactness is therefore constrained not only by computational hardness but also by the inherent size of the exact witness (Amiri et al., 2014).

5. Exact computational DAGs and e-graph extraction

In probabilistic inference, exact DAG extraction refers to compilation rather than structure learning. A Query DAG, or Q-DAG, is defined as a tuple

$A_\phi$0

where $A_\phi$1 is a directed acyclic graph whose roots are numbers or evidence-specific nodes and whose internal nodes are labeled by $A_\phi$2 or $A_\phi$3. The paper shows how standard exact inference algorithms for belief networks can be lifted into Q-DAG generation by replacing numeric arithmetic with DAG constructors $A_\phi$4 and $A_\phi$5. The compiled graph is an exact arithmetic representation of the inference computation, and online evaluation is linear in the size of the Q-DAG (Darwiche et al., 2014).

Expression DAGs in robust geometric computation play a related role. Here exact-decision number types record the computation history in an arithmetic expression DAG and reevaluate it with increasing precision until an exact decision can be made. The paper studies AM-Balancing, Move-To-Root restructuring, and parameterized MTR as post-extraction transformations that improve parallel evaluation. Exactness resides in the stored computation history and in the ability to certify signs or comparisons by reevaluation, not in graph discovery (Wilhelm, 2018).

Exact DAG extraction is also a standard phrase in e-graph optimization. In that setting the task is to choose a minimum-cost acyclic selected subgraph of an e-graph, with one selected enode per selected e-class, shared subexpressions counted once, root coverage, and acyclicity. “Answer Set Programming for Egg Extraction and More” gives a compact bottom-up ASP encoding in which acyclicity is handled through stable-model foundedness rather than an explicit reachability encoding, and reports that the naïve ASP encoding is comparable in efficiency to the well-optimised ILP-based exact DAG extraction in the extraction-gym while finding several extra optimal extractions on complex instances (Yang et al., 9 Jun 2026).

e-boost keeps the exact ILP formulation for e-graph DAG extraction but precedes it by parallel heuristic DAG-cost computation, adaptive threshold pruning, and warm-start initialization. Its final ILP is exact on the retained search space, with pruning enforced by $A_\phi$6 for pruned nodes. The paper is explicit that this is not unconditional global exactness on the original full e-graph, because pruning can remove a globally optimal enode. Exactness therefore applies to the reduced problem, while the full method is hybrid heuristic-exact (Yin et al., 18 Aug 2025).

6. Exact serialization, canonical representations, and persistent distinctions

A further sense of exact DAG extraction appears in sequence-based learning. DIGGED uses an edge-directed Neighborhood Controlled Embedding grammar

$A_\phi$7

to represent each DAG in a dataset by a unique sequence of production rules after disambiguation. The paper states that the mapping is one-to-one, onto, deterministic, and valid over the observed dataset $A_\phi$8, and that the resulting sequence is an equivalent, lossless sequential representation of the DAG. Exactness here is exact serialization and exact reconstruction, not causal discovery (2505.22949).

This dataset-relative exactness is important. The paper does not claim a universal canonical form for all DAGs, since ambiguity of general edNCE grammars is undecidable in general. Instead it proves that the output grammar is unambiguous with respect to $A_\phi$9. The exact claim is therefore bounded by the observed dataset and the learned grammar, which is sufficient for exact sequence reconstruction and generation in that regime (2505.22949).

Across these literatures, a recurrent distinction remains decisive. Exact acyclicity of a feasible point, exact global optimization of a score, exact reconstruction from a representation, and exact recovery of the true underlying graph are not interchangeable. DAG-GNN explicitly provides exact acyclicity of the optimization variable but not exact edge recovery; analytic constraints provide if-and-only-if acyclicity characterizations but do not remove nonconvex optimization; ExDAG provides exact global optimization of the stated score and connects exact recovery to identifiability assumptions (Yu et al., 2019, Zhang et al., 24 Mar 2025, Rytíř et al., 2024).

The term “Exact DAG Extraction” is therefore best understood as a family resemblance rather than a single method class. In some settings the extracted object is the DAG itself; in others it is a decomposition, a partition, an arithmetic circuit, a compiled inference graph, a grammar derivation, or an optimal acyclic subgraph of an e-graph. What is exact is always task-specific: exact feasibility, exact optimization, exact representation, or exact recovery.