Strong d-Convex Hulls in Causal Graphs & Geometry

• The paper establishes strong d-convex hulls as the minimal induced subgraph ensuring estimate collapsibility in DAGs and CPDAGs.
• It introduces algorithmic constructions—CVM, ICHA, and ISCHA—to iteratively build d-convex and strong d-convex hulls for efficient graph reduction.
• The work bridges causal inference with convex geometry, relating K-hulls and Carathéodory results to random-sample asymptotics and interdisciplinary applications.

Searching arXiv for recent and foundational papers on strong d-convex hulls and closely related notions. Strong d-convex hulls arise in at least two technically distinct settings. In causal graphical models, a strong d-convex hull is a graph-theoretic closure of a variable set $R$ inside a DAG or CPDAG, introduced to characterize the smallest retained subgraph that preserves estimate collapsibility and causal effect estimation under marginalization (Deng et al., 8 Jun 2026). In convex geometry, the closely related literature on strongly convex hulls studies $K$-hulls, that is, intersections of all translates of a fixed convex body $K$ containing a set, written as $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$ or $\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$ (Holmsen et al., 2015, Marynych et al., 2021). The two notions share the language of “strong convexity,” but they are defined on different objects, use different closure operators, and serve different theoretical purposes.

1. Graph-theoretic definition in DAGs

In the causal-inference formulation, the ambient object is a DAG $G=(V,E)$, and strong d-convexity is defined for subsets $R\subseteq V$. The construction depends on two ingredients: inducing paths and a parent-adjacency condition called linear ordering. For a path $l_{xy}$ between non-adjacent vertices $x,y\in R$, $l_{xy}$ is an inducing path of $K$0 if

$K$1

Here $K$2 denotes internal vertices and $K$3 the colliders on the path. The associated inducing structure is

$K$4

where $K$5 is the set of all directed paths from colliders on $K$6 to $K$7 (Deng et al., 8 Jun 2026).

The second ingredient is linear ordering with respect to $K$8. A vertex $K$9 is linearly ordered with respect to $K$0 if any two distinct vertices in $K$1 are adjacent, except when both belong to $K$2. A set $K$3 is linearly ordered with respect to $K$4 if every $K$5 is.

With this notation, the formal definition is: $K$6

1. there is no inducing path for $K$7 in $K$8; and
2. $K$9 is linearly ordered with respect to $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$0, where $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$1, $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$2, and $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$3.

If only condition (i) holds, the set is d-convex. The distinction is structural: d-convexity suffices for CI-collapsibility and model collapsibility, but not for estimate collapsibility. Strong d-convexity is therefore strictly stronger in the sense used by the causal paper (Deng et al., 8 Jun 2026).

2. Collapsibility, minimality, and hull existence

The causal paper distinguishes three marginalization properties: $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$4 These are, respectively, CI-collapsibility, model collapsibility, and estimate collapsibility. The paper states explicitly that estimate collapsibility implies model collapsibility, but not conversely (Deng et al., 8 Jun 2026).

The central characterization is Lemma 3.1. Under the assumption of non-triviality, for a CBN $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$5 and $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$6, the following are equivalent:

1. $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$7,
2. there exists a graph $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$8 in the Markov equivalence class of $$\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)$$9 such that $\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$0 is a terminal set in $\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$1,
3. $\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$2 is a strong d-convex set in $\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$3 (Deng et al., 8 Jun 2026).

This equivalence turns strong d-convexity into the exact graphical criterion for estimate collapsibility. The hull notion then follows from an intersection property: if $\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$4 is a collection of strong d-convex subsets containing $\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$5, then

$\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$6

is strongly d-convex. Consequently, both the d-convex hull $\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$7 and the strong d-convex hull $\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$8 exist and are unique.

The minimality theorem is the decisive closure result: $\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)$9 Here “minimal collapsible set” means that $G=(V,E)$0, while for every $G=(V,E)$1 with $G=(V,E)$2, the equality fails. Thus the strong d-convex hull is not merely a convenient reduction; it is the exact minimal induced subgraph preserving the MLE-based estimation behavior of the full model (Deng et al., 8 Jun 2026).

3. Algorithmic construction in DAGs and CPDAGs

The algorithmic construction proceeds in two layers. First, one computes the d-convex hull by absorbing vertices forced by minimal inducing structures. Second, one enforces the linear-ordering condition by adding parent vertices of problematic nodes.

A key graph-theoretic reduction states that if $G=(V,E)$3 are non-adjacent and $G=(V,E)$4 is d-convex or strongly d-convex, then the condition

$G=(V,E)$5

is equivalent to requiring that every shortest path $G=(V,E)$6 in

$G=(V,E)$7

satisfy

$G=(V,E)$8

This justifies computing shortest paths in a moralized ancestor graph rather than enumerating arbitrary inducing paths (Deng et al., 8 Jun 2026).

The three algorithms used in the paper are summarized below.

Algorithm	Role	Complexity
CVM($G=(V,E)$9)	Collect vertices on shortest inducing paths	$R\subseteq V$0 time, $R\subseteq V$1 space
ICHA($R\subseteq V$2)	Compute the d-convex hull by iterating CVM	$R\subseteq V$3 time, $R\subseteq V$4 space
ISCHA($R\subseteq V$5)	Upgrade the d-convex hull to the strong d-convex hull	$R\subseteq V$6 time, $R\subseteq V$7 space

Here $R\subseteq V$8, with $R\subseteq V$9 and $l_{xy}$0.

CVM computes shortest paths in moralized ancestor graphs for non-adjacent pairs in $l_{xy}$1 and returns the vertices on those paths. ICHA iterates CVM until closure. ISCHA alternates an ICHA step with the update

$l_{xy}$2

then enlarges $l_{xy}$3 by $l_{xy}$4 until no violation remains. The correctness theorem states that the set returned by ISCHA is a strong d-convex hull of $l_{xy}$5 (Deng et al., 8 Jun 2026).

For CPDAGs, the construction is transferred through Markov equivalence rather than by a separate partially directed algorithm. Theorem 4.4 states that if $l_{xy}$6 is a DAG Markov equivalent to a CPDAG $l_{xy}$7, then for distinct vertices $l_{xy}$8 and $l_{xy}$9, a vertex set $x,y\in R$0 is causal estimate collapsible in $x,y\in R$1 if and only if it is causal estimate collapsible in $x,y\in R$2. Operationally, one may choose any consistent DAG $x,y\in R$3, compute $x,y\in R$4, and use the resulting set for graph reduction in the CPDAG (Deng et al., 8 Jun 2026).

4. Role in causal effect estimation

The causal use of strong d-convex hulls is not to produce an adjustment set directly, but to define a minimal retained subgraph in which adjustment-based estimation remains valid. For non-adjacent $x,y\in R$5 and $x,y\in R$6, a DAG $x,y\in R$7 is causal estimate collapsible onto $x,y\in R$8 if for every non-empty valid back-door adjustment set $x,y\in R$9 in $l_{xy}$0,

$l_{xy}$1

The preserved estimand is therefore the post-intervention distribution $l_{xy}$2, and for binary $l_{xy}$3, the ACE is

$l_{xy}$4

The main validity statement is Theorem 4.1: if $l_{xy}$5 is a non-empty back-door adjustment set in $l_{xy}$6, then $l_{xy}$7 is causal estimate collapsible onto $l_{xy}$8 if $l_{xy}$9 is a strong d-convex hull of $K$00 and $K$01 (Deng et al., 8 Jun 2026). Thus the hull is a sufficient graphical certificate for reduction before causal estimation.

The paper treats the empty-adjustment case separately. If the empty set is a valid back-door adjustment set for $K$02 in $K$03, then the relevant hull must be computed in the manipulated graph $K$04, obtained by deleting all edges into $K$05. This captures omitted parents of $K$06 that only become relevant after intervention.

The CPDAG implementation is “Subgraph IDA.” The procedure is: obtain a consistent DAG via Meek’s rules; compute

$K$07

for each DAG in the equivalence class, set $K$08, let

$K$09

and, if $K$10, construct $K$11, recompute $K$12, and reset $K$13. Estimation then uses $K$14 when $K$15, and otherwise the adjustment formula

$K$16

The hull is therefore a preprocessing reduction layered on top of IDA, not a replacement for adjustment theory (Deng et al., 8 Jun 2026).

5. Worked examples, empirical behavior, and limitations

A representative example uses the DAG on $K$17 with edges

$K$18

For the target set $K$19, the paper states that there are $K$20 inducing paths between $K$21 and $K$22. A naive union of vertices on inducing structures gives $K$23, which is d-convex but not minimal. The d-convex hull is $K$24. The strong d-convex hull is larger, because $K$25 violates the linear-ordering condition with

$K$26

so $K$27 must be added, yielding

$K$28

as the final strong d-convex hull (Deng et al., 8 Jun 2026).

The same paper gives a larger reduction example on the 56-node Hailfinder Bayesian network for the target pair $K$29. ISCHA reduces the graph from 56 variables to a 16-variable subgraph. The initial AIP is

$K$30

this set is already d-convex, and further linear-ordering checks add variables such as $K$31, $K$32, and $K$33 (Deng et al., 8 Jun 2026).

The empirical results reported are consistent with the theoretical interpretation of the hull as a minimal safe retained subgraph. In benchmark probabilistic reasoning experiments, KL divergence is essentially zero ($K$34), node reduction is up to $K$35, and local inference is faster. In random-graph CPDAG experiments, recall and precision are essentially $K$36 across nearly all settings, and speedups increase with graph size, reaching around $K$37 for 100-node random graphs. On real-world Bayesian networks including Sachs, Insurance, Alarm, Hepar2, Pathfinder, and Munin1, precision remains $K$38, recall remains high ($K$39), and speedup reaches $K$40 on Munin1 (Deng et al., 8 Jun 2026).

The paper also states several scope restrictions. The framework currently applies to non-adjacent treatment-outcome pairs, does not handle latent variables, and focuses on back-door-type adjustment rather than other adjustment strategies. It also emphasizes that collapsibility may fail if the retained set is only d-convex but not strongly d-convex.

6. Relation to strongly convex hulls in geometry and to other nearby notions

A separate line of work studies strong convexity relative to a fixed convex body $K$41. In that setting, a set is $K$42-strongly convex if it is an intersection of translates of $K$43,

$K$44

and the strongly convex hull is

$K$45

This hull is defined only for sets $K$46 contained in a translate of $K$47. When $K$48 is generating, the strong-convexity Carathéodory theorem gives a $K$49 bound, and colorful as well as very colorful Carathéodory theorems hold with $K$50 color classes. The same paper shows that in $K$51, for arbitrary $K$52, the Carathéodory number can be arbitrarily large or infinite, so the generating-set assumption is not merely technical (Holmsen et al., 2015).

The random-geometry literature develops this $K$53-hull viewpoint further. For a compact set $K$54,

$K$55

and a set $K$56 is $K$57-strongly convex if $K$58. For i.i.d. uniform samples $K$59, one studies

$K$60

The paper on facial structure proves that $K$61 converges in distribution to the zero cell $K$62 of a Poisson hyperplane tessellation, while $K$63 converges to $K$64; under strict convexity and regularity, the generalized $K$65-vector of $K$66 converges in distribution, without normalization, to the ordinary $K$67-vector of $K$68, and all moments converge (Marynych et al., 2021). A later paper introduces $K$69-point peelings and recursive convex hull peelings for families of compact convex sets, together with the dual wrapping operations for intersections, and applies them to $K$70-hulls generated by random samples; under strict convexity and regularity of $K$71, the corresponding peelings and wrappings of the rescaled random objects converge in distribution to Poisson limits (Marynych et al., 29 Jun 2026).

Several nearby literatures use similar language for different objects. The planar lattice-based discrete hull

$K$72

is an ordinary Euclidean convex hull applied after filtering to lattice points, and the source explicitly states that it is not about strong $K$73-convex hulls in the usual digital-convexity sense (Har-Peled, 1 Jan 2026). The finite-element literature proves a strong discrete convex hull property for discretely $K$74-harmonic functions on acute triangulations, where an interior extreme point of $K$75 forces constancy (Diening et al., 2013). The computational-geometry literature on $K$76-convex hulls defines planar directional hulls by excluding empty translated halfplanes and wedges induced by a direction set $K$77, and the paper explicitly does not define a notion called strong $K$78-convex hull (Har-Peled, 2011).

Taken together, these sources delimit the current usage. In causal graphical models, the strong d-convex hull is the unique minimal induced subgraph guaranteeing estimate collapsibility and supporting reduced-graph causal estimation (Deng et al., 8 Jun 2026). In convex geometry, strongly convex hulls are $K$79-hulls generated by intersections of translates of a convex body, with their own Carathéodory theory, facial structure, and random-sample asymptotics (Holmsen et al., 2015, Marynych et al., 2021, Marynych et al., 29 Jun 2026). The terminological overlap is substantial, but the operators, ambient spaces, and intended applications are not the same.